Integrability of the ${\rm Ad}{{{\rm S}}_{5}}\times {{{\rm S}}^{5}}$ superstring and its deformations

The ${\rm SU}(N)$ chiral Gross–Neveu model is a model of N interacting Dirac fermions with Lagrangian²⁸

$$\begin{eqnarray}&&\qquad {{\mathcal{L}}_{cGN}}={{\bar{\psi }}_{a}}i{{\psi }^{a}}+\frac{1}{2}g_{s}^{2}\left( {{({{{\bar{\psi }}}_{a}}{{\psi }^{a}})}^{2}}-{{({{{\bar{\psi }}}_{a}}{{\gamma }_{5}}{{\psi }^{a}})}^{2}} \right)-\frac{1}{2}g_{v}^{2}{{({{\bar{\psi }}_{a}}{{\gamma }_{\mu }}{{\psi }^{a}})}^{2}}\;,\end{eqnarray} \tag{ 2.38 }$$

where a = 1,...,N labels the N Dirac spinors. This Lagrangian has ${\rm U}(N)\times {\rm U}{{(1)}_{c}}$ symmetry, where viewed as an N-component vector the spinors transform in the fundamental representation of ${\rm U}(N)$, and ${\rm U}{{(1)}_{c}}$ denotes the chiral symmetry $\psi \to {{{\rm e}}^{{\rm i}\theta {{\gamma }_{5}}}}\psi$. Typically the chiral Gross–Neveu Lagrangian is written with g_v = 0.²⁹

This model was initially studied as it shares some features with QCD, namely it is asymptotically free and has dynamical mass generation. The subtle point however is that chiral symmetry naively prevents the fermions from gaining mass, as there is no spontaneous symmetry breaking in two dimensions [67]. Yet this was precisely what is observed in the large N limit [68], with even a massless particle present in the spectrum. The resolution of this apparent mismatch is nicely discussed in [69], the upshot being that the physical fermions in this model carry no chirality, and that while there is a chiral massless particle in the spectrum it is not a Goldstone boson. The full spectrum of this theory contains $N-1$ ${\rm SU}(N)$ multiplets of interacting massive fermions, and massless excitations with ${\rm U}(1)$ charge that decouple completely [69–71]. Interestingly, this model is related to the sine–Gordon model we briefly mentioned in section 2.1 at $\beta =8\pi$ by bosonization [72]. This model possesses an infinite number of conserved charges [73], and is integrable in the sense we just discussed.

Before moving on we should introduce a bit of notation. The conventional variables in relativistic scattering theory are the Mandelstam variables s, t and u, but only one of these is independent in two dimensions. Hence we can write the S-matrix as a function of $s={{({{p}_{1}}+{{p}_{2}})}^{2}}$. Without going into details, continuing s into the complex plane the S-matrix has branch cuts from ${{({{m}_{1}}+{{m}_{2}})}^{2}}$ to positive infinity, and ${{({{m}_{1}}-{{m}_{2}})}^{2}}$ to negative infinity (see e.g. [62]). By definition, the physical scattering values lie just above the first of these branch cuts, thereby singling out this copy of the s-plane as the physical one, known as the physical sheet. It turns out to be possible to resolve these branch cuts by introducing a (relative) rapidity variable u as³⁰

$$\begin{eqnarray}&&s=m_{1}^{2}+m_{2}^{2}+2{{m}_{1}}{{m}_{2}}{\rm cosh} \frac{\pi u}{2}\;,\end{eqnarray} \tag{ 2.40 }$$

in terms of which the physical copy of the s-plane is mapped to the physical strip ${\rm Im}(u)=(0,2i)$. This variable $u={{u}_{1}}-{{u}_{2}}$ is nothing but the difference of the physical rapidity variables of the particles, related to energy and momentum as

$$\begin{eqnarray}&&{{E}_{i}}={{m}_{i}}{\rm cosh} \frac{\pi {{u}_{i}}}{2}\;,\;\;\;{{p}_{i}}={{m}_{i}}{\rm sinh} \frac{\pi {{u}_{i}}}{2}\;,\end{eqnarray} \tag{ 2.41 }$$

where m is the mass of the fermions. Branch cut issues aside, another reason to introduce these variables is that Lorentz boosts act additively on the rapidity, and therefore by Lorentz invariance the two-body S-matrix is a function of the difference of the particles' rapidities only. Furthermore, note that this parametrization uniformizes the relativistic dispersion relation

$$\begin{eqnarray}&&{{E}^{2}}-{{p}^{2}}={{m}^{2}}\;,\end{eqnarray} \tag{ 2.42 }$$

meaning that it is completely parametrized by u. Note that in these variables two-particle bound states have $u\in (0,2i)$, corresponding to $s\in ({{({{m}_{1}}-{{m}_{2}})}^{2}},{{({{m}_{1}}+{{m}_{2}})}^{2}})$ on the physical sheet. We should also mention that crossing should take us from physical values of s to physical values of t which by the usual $i\epsilon$ prescription corresponds to $u\to 2i-u$.³¹

Coming back to our model, for simplicity we will focus on the case N = 2 where we have only one ${\rm SU}(2)$ multiplet of interacting fermions of mass m. Unitarity, ${\rm SU}(2)$ symmetry, the crossing relations, and requiring a minimal number of poles and zeros of the S-matrix³² , together fix the two body S-matrix of this model to be [16]³³

$$\begin{eqnarray}&&{{S}_{ab}}(u)={{S}^{ff}}(u)R_{ab}^{-1}(u),\end{eqnarray} \tag{ 2.43 }$$

where

$$\begin{eqnarray}&&{{S}^{ff}}(u)=-\frac{\Gamma (1-\frac{u}{4i})\Gamma (\frac{1}{2}+\frac{u}{4i})}{\Gamma (1+\frac{u}{4i})\Gamma (\frac{1}{2}-\frac{u}{4i})}\;,\;\;\;{\rm and}\;\;\;R(u)=\frac{1}{u+2i}\left( u{{1}_{ab}}+2i{{P}_{ab}} \right),\end{eqnarray} \tag{ 2.44 }$$

where P is the permutation operator³⁴ . As we saw above, to get the Bethe–Yang equations from this S-matrix we will first need to diagonalize the associated transfer matrix

$$\begin{eqnarray}&&{{T}^{g}}(u|\{{{u}_{i}}\})={\rm T}{{{\rm r}}_{a}}\mathcal{T}_{a}^{g}(u|\{{{u}_{i}}\})={\rm T}{{{\rm r}}_{a}}\;{{g}_{a}}\mathop{\prod }\limits_{i=1}^{{{N}_{f}}}R_{ai}^{-1}(u-{{u}_{i}})={\rm T}{{{\rm r}}_{a}}\;{{g}_{a}}\mathop{\prod }\limits_{i=1}^{{{N}_{f}}}{{R}_{ai}}({{u}_{i}}-u)\;,\end{eqnarray} \tag{ 2.45 }$$

where we take out the scalar factor S^ff for soon-to-be-obvious reasons, we note that ${{R}_{ab}}(u)={{R}_{ba}}(u)$ for this particular model, and we have allowed ourselves quasi-periodic boundary conditions for generality. ${{N}_{f}}$ denotes the number of physical fermions in a given state. As our S-matrix is ${\rm SU}(2)$ symmetric any quasi-periodic boundary condition associated to a $g\in {\rm SU}(2)$ is compatible with integrability. Without loss of generality we can restrict to elements of the Cartan subgroup since the eigenvalues of the transfer matrix depend only on these (see section 4.2.1 for details). As such we take g to be of the form

$$\begin{eqnarray}g=\left( \begin{array}{ccccccccccccccc} {{{\rm e}}^{{\rm i}\alpha }} & 0 \\ 0 & {{{\rm e}}^{-{\rm i}\alpha }} \\ \end{array} \right).\end{eqnarray} \tag{ 2.46 }$$

Note that these boundary conditions break the ${\rm SU}(2)$ symmetry down to ${\rm U}(1)$. We already mentioned that the transfer matrix for an integrable quantum field theory has an associated discrete integrable model. In this case (2.45) is the transfer matrix for the spin-$\frac{1}{2}$ Heisenberg XXX spin chain. As we will encounter more complicated integrable spin chains later, let us discuss this simple example in some detail.

The Heisenberg XXX spin chain is described by the Hamiltonian

$$\begin{eqnarray}&&H=-\frac{J}{4}\mathop{\sum }\limits_{i=1}^{{{N}_{f}}}\left( {{{\vec{\sigma }}}_{i}}\cdot {{{\vec{\sigma }}}_{i+1}}-1 \right)\;,\end{eqnarray} \tag{ 2.47 }$$

where $\vec{\sigma }$ is the vector of Pauli matrices, and ${{\sigma }_{{{N}_{f}}+1}}={{g}^{-1}}{{\sigma }_{1}}g$. This Hamiltonian acts on a Hilbert space given by ${{N}_{f}}$ copies of ${{\mathbb{C}}^{2}}$, one for each site i. Identifying $(1,0)$ as $|\uparrow \rangle$ and $(0,1)$ as $|\downarrow \rangle ,$ states in this Hilbert space can be viewed as chains of spins, in this case quasi-periodic. This Hamiltonian (2.47) can be obtained from the transfer matrix (2.45) as³⁵

$$\begin{eqnarray}&&H=iJ\frac{{\rm d}}{{\rm d}u}{\rm log} {{T}^{g}}(u|\{0\}){{|}_{u=0}}\;.\end{eqnarray} \tag{ 2.48 }$$

From the definition of the R-matrix and the transfer matrix it is not hard to see that the transfer matrix is a polynomial in u. Combining this with the commutativity of the transfer matrix, we can expand the transfer matrix in powers of u to get a set of manifestly mutually commuting operators, all furthermore commuting with the Hamiltonian (2.48). This makes contact with the notion of integrability we discussed for continuous systems in sections 2.1 and 2.2. The transfer matrix and hence the Heisenberg Hamiltonian can be diagonalized by the Bethe ansatz in either coordinate or algebraic form. Here we will briefly but explicitly illustrate the algebraic Bethe ansatz, as we will encounter it again. Note that we will be keeping non-zero u_is in our transfer matrix since we are really considering the chiral Gross–Neveu model; this corresponds to an inhomogeneous spin chain.

The algebraic Bethe ansatz is based on the fundamental commutation relations (2.33). To write them out explicitly, we will first write the monodromy matrix ${{\mathcal{T}}_{g}}$ of equation (2.45) explicitly as a matrix in auxiliary space

$$\begin{eqnarray}{{\mathcal{T}}^{g}}(u|\{{{u}_{i}}\})=\left( \begin{array}{ccccccccccccccc} A(u) & B(u) \\ C(u) & D(u) \\ \end{array} \right),\end{eqnarray} \tag{ 2.49 }$$

where we suppressed the dependence on u_i on the right-hand side. Explicitly writing out the fundamental commutation relations directly gives (recall that we are working with ${{R}^{-1}}$)

$$\begin{eqnarray}\begin{array}{rcl} [B(v),B(w)] & = & 0\;, \\ A(v)B(w) & = & f(v-w)B(w)A(v)+g(v-w)B(v)A(w), \\ D(v)B(w) & = & f(w-v)B(w)D(v)-g(v-w)B(v)D(w), \\ \end{array}\end{eqnarray} \tag{ 2.50 }$$

where

$$\begin{eqnarray}&&f(u)=\frac{u+2i}{u}\;,\;\;{\rm and}\;\;\;g(u)=-\frac{2i}{u}\;.\end{eqnarray} \tag{ 2.51 }$$

Note that these commutation relations do not contain u_is. The crucial point is now to view these commutation relations as an analogue of the harmonic oscillator algebra $N{{a}^{\dagger }}={{a}^{\dagger }}(N+1)$, $Na=a(N-1)$, with the number operator $N={{a}^{\dagger }}a$. Concretely, we will make the ansatz that B creates eigenstates of the transfer matrix $A+D$ by acting on a vacuum $\mid 0\rangle$ which is annihilated by C. To show that such a vacuum exists, we note that the R-matrix acts upper triagonally in auxiliary space on the vector $|\uparrow \rangle$ in space i since

and ${{\sigma }_{+}}|\uparrow \rangle =0$.³⁶ This means that if we take $\mid 0\rangle =|\uparrow \ldots \uparrow \rangle$ the monodromy matrix $\mathcal{T}$ acts as

$$\begin{eqnarray}\mathcal{T}|0\rangle =\left( \begin{array}{ccccccccccccccc} A & B \\ C & D \\ \end{array} \right)|0\rangle =\left( \begin{array}{ccccccccccccccc} {{{\rm e}}^{{\rm i}\alpha }} & \bullet \\ 0 & {{{\rm e}}^{-{\rm i}\alpha }}\mathop{\prod }\limits_{i}{{(f({{u}_{{\rm i}}}-u))}^{-1}} \\ \end{array} \right)|0\rangle \;,\end{eqnarray} \tag{ 2.52 }$$

from which we see that $C|0\rangle =0$, $A|0\rangle ={{{\rm e}}^{{\rm i}\alpha }}|0\rangle$, and $D|0\rangle ={{{\rm e}}^{-{\rm i}\alpha }}\mathop{\prod }\limits_{i}{{(f({{u}_{i}}-u))}^{-1}}|0\rangle$; the bullet point • denotes terms irrelevant for our considerations. In terms of field theory the existence of this vacuum simply means that there is a one-particle-type sector which is closed under scattering, which is of course obvious from the form of the S-matrix (2.43). We will act with creation operators on this vacuum to construct the other eigenstates of the transfer matrix. Let us consider an arbitrary state obtained by acting with ${{N}_{a}}$ creation operators

$$\begin{eqnarray}&&|\{{{v}_{1}},\ldots ,{{v}_{{{N}_{a}}}}\}\rangle \equiv B({{v}_{1}})\ldots B({{v}_{{{N}_{a}}}})|0\rangle \;,\end{eqnarray} \tag{ 2.53 }$$

and insist that this is in fact an eigenstate of the transfer matrix $A(u)+D(u)$. Using the commutation relations (2.50) we can commute A(u) and D(u) through to the vacuum where we understand their action. When we do so however, there will also be many contributions from the terms in the commutation relations which exchange rapidities. This means that unless these terms cancel between A and D, we are not creating an eigenstate. Explicitly we have

$$\begin{eqnarray}\begin{array}{rcl} A(u)|\left\{ {{v}_{1}},\ldots ,{{v}_{{{N}_{a}}}} \right\}\rangle & = & {{{\rm e}}^{{\rm i}\alpha }}\mathop{\prod }\limits_{i=1}^{{{N}_{a}}}f(u-{{v}_{i}})B({{v}_{1}})\ldots B({{v}_{{{N}_{a}}}})|0\rangle \\ {} & {} & +{{{\rm e}}^{{\rm i}\alpha }}\mathop{\sum }\limits_{j}{{M}_{j}}B({{v}_{1}})\ldots \hat{B}({{v}_{j}})\ldots B({{v}_{{{N}_{a}}}})B(u)|0\rangle , \\ D(u)|\{{{v}_{1}},\ldots ,{{v}_{{{N}_{a}}}}\}\rangle & = & {{{\rm e}}^{-{\rm i}\alpha }}\mathop{\prod }\limits_{i=1}^{{{N}_{a}}}f({{v}_{i}}-u)\mathop{\prod }\limits_{j=1}^{{{N}_{f}}}{{\left( f({{u}_{j}}-u) \right)}^{-1}}B({{v}_{1}})\ldots B({{v}_{{{N}_{a}}}})|0\rangle \\ {} & {} & +{{{\rm e}}^{-{\rm i}\alpha }}\mathop{\sum }\limits_{j}{{\tilde{M}}_{j}}B({{v}_{1}})\ldots \hat{B}({{v}_{j}})\ldots B({{v}_{{{N}_{a}}}})B(u)|0\rangle , \\ \end{array}\end{eqnarray} \tag{ 2.54 }$$

where M_j and ${{\tilde{M}}_{j}}$ are coefficients of the terms where A respectively D was commuted with the jth B using the term in the commutation relations that exchanges the rapidities; the hat on B denotes that the operator is removed from the product. Noting that M₁ and ${{\tilde{M}}_{1}}$ can be easily computed, and that by commutativity of the Bs we can simply replace v₁ by v_j in the result to generalize it, we get

$$\begin{eqnarray}\begin{array}{rcl} {{M}_{j}} & = & g(u-{{v}_{j}})\mathop{\prod }\limits_{i\ne j}^{{{N}_{a}}}f({{v}_{j}}-{{v}_{i}}), \\ {{\tilde{M}}_{j}} & = & -g(u-{{v}_{j}})\mathop{\prod }\limits_{i\ne j}^{{{N}_{a}}}f({{v}_{i}}-{{v}_{j}})\mathop{\prod }\limits_{m=1}^{{{N}_{f}}}{{\left( f({{u}_{m}}-{{v}_{j}}) \right)}^{-1}}. \\ \end{array}\end{eqnarray} \tag{ 2.55 }$$

Now we can cancel the unwanted terms against each other by insisting that the v_j satisfy a set of so-called Bethe equations

$$\begin{eqnarray}&&\mathop{\prod }\limits_{i\ne qj}^{{{N}_{a}}}f({{v}_{j}}-{{v}_{i}})={{{\rm e}}^{-2i\alpha }}\mathop{\prod }\limits_{i\ne j}^{{{N}_{a}}}f({{v}_{i}}-{{v}_{j}})\mathop{\prod }\limits_{m=1}^{{{N}_{f}}}{{\left( f({{u}_{m}}-{{v}_{j}}) \right)}^{-1}}\;.\end{eqnarray} \tag{ 2.56 }$$

Shifting ${{\tilde{v}}_{l}}={{v}_{l}}+i$ and dropping the tilde this can be rewritten as

$$\begin{eqnarray}&&\mathop{\prod }\limits_{m=1}^{{{N}_{f}}}\frac{{{v}_{j}}-{{u}_{m}}-i}{{{v}_{j}}-{{u}_{m}}+i}={{{\rm e}}^{-2i\alpha }}\mathop{\prod }\limits_{i\ne j}^{{{N}_{a}}}\frac{{{v}_{j}}-{{v}_{i}}-2i}{{{v}_{j}}-{{v}_{i}}+2i}\;.\end{eqnarray} \tag{ 2.57 }$$

The corresponding eigenvalue of the transfer matrix is given by

$$\begin{eqnarray}&&\lambda (u|\{{{v}_{j}}\},\{{{u}_{m}}\})={{{\rm e}}^{{\rm i}\alpha }}\mathop{\prod }\limits_{i=1}^{{{N}_{a}}}\frac{u-{{v}_{i}}+i}{u-{{v}_{i}}-i}+{{{\rm e}}^{-{\rm i}\alpha }}\mathop{\prod }\limits_{m=1}^{{{N}_{f}}}\frac{u-{{u}_{m}}}{u-{{u}_{m}}-2i}\mathop{\prod }\limits_{i=1}^{{{N}_{a}}}\frac{u-{{v}_{i}}-3i}{u-{{v}_{i}}-i}\;,\end{eqnarray} \tag{ 2.58 }$$

where again the v_l have been shifted. Note that the Bethe equations also follow by insisting that the residues of the superficial poles in λ at $u={{v}_{j}}+i$ vanish, which must be the case since by construction T has no poles there³⁷ . Solutions to the Bethe equations with repeating rapidities u_i = u_j would require further constraints to be consistent and should in fact be discarded. In the framework of the coordinate Bethe ansatz this would be obvious as the ansatz for the wave-function manifestly vanishes in such situations.

We see that we can create eigenstates of the transfer matrix by acting with a number of creation operators on the vacuum, provided the associated rapidities satisfy equation (2.57). To give a physical interpretation to these eigenstates let us consider their quantum numbers. Using equation (2.48) we can easily find the spin chain energy of a general eigenstate to be

$$\begin{eqnarray}&&E(\{{{v}_{l}}\})=-iJ\frac{{\rm d}}{{\rm d}u}{\rm log} \lambda (u\mid \{{{v}_{j}}\},\{0\}){{\mid }_{u=0}}=2J\mathop{\sum }\limits_{i}\frac{1}{v_{i}^{2}+1}\;,\end{eqnarray} \tag{ 2.59 }$$

where we note that the vacuum has zero energy. We see that the energy is a sum of different contributions dependent on v_i only. Similarly, we can consider the spin chain momentum identified through³⁸

$$\begin{eqnarray}&&p=i{\rm log} \lambda (0|\{0\}),\end{eqnarray} \tag{ 2.60 }$$

which gives

$$\begin{eqnarray}&&p(\{{{v}_{j}}\})=i\mathop{\sum }\limits_{j}{\rm log} \frac{{{v}_{j}}-i}{{{v}_{j}}+i}\;;\end{eqnarray} \tag{ 2.61 }$$

again a sum of individual contributions³⁹ . Finally we can consider the spin of these eigenstates. In the periodic limit $\alpha \to 0$ it immediately follows from the ${\rm SU}(2)$ symmetry of the S-matrix that the monodromy matrix commutes with the total action of $\mathfrak{s}\mathfrak{u}(2)$ ⁴⁰

$$\begin{eqnarray}&&[\mathcal{T}_{a}^{g}(v|\{{{v}_{1}},\ldots ,{{v}_{{{N}_{a}}}}\}),{{\sigma }_{i;a}}+\mathop{\sum }\limits_{j=1}^{{{N}_{a}}}{{\sigma }_{i;j}}]=0\;,\end{eqnarray} \tag{ 2.62 }$$

where ${{\sigma }_{i;j}}$ denotes the ith pauli matrices acting in space j. For our general quasi-periodic boundary conditions we can still sensibly talk about the ${\rm U}(1)$ charge or total z spin, and defining ${{S}_{3}}=\frac{1}{2}\sum _{h=1}^{{{N}_{a}}}{{\sigma }_{3;j}}$ and writing things out we get

$$\begin{eqnarray}&&[{{S}_{3}},B]=-B\;,\end{eqnarray} \tag{ 2.63 }$$

showing that B lowers the spin by one unit. Since the vacuum has spin ${{N}_{f}}/2$, this means that an eigenstate with ${{N}_{a}}$ v_js has spin ${{N}_{f}}/2-{{N}_{a}}$. This shows that by all accounts these eigenstates can be interpreted as states of ${{N}_{a}}$ particles, the energy, momentum and charge being a sum of the individual particles' contributions, where B now really creates these particles out of the vacuum. These particles are called magnons, and their dispersion relation is

$$\begin{eqnarray}&&E=J(1-{\rm cos} (p)),\end{eqnarray} \tag{ 2.64 }$$

where we have introduced p for the spin chain momentum of a single magnon cf equation (2.61). From the point of view of our field theory these excitations do not carry energy or momentum. They do carry ${\rm SU}(2)$ spin however, and correspond to 'exciting' a spin down fermion in a sea of spin up fermions. Since they only excite the auxiliary quantum numbers we will call them auxiliary excitations with associated auxiliary rapidities.

Let us note that in the periodic limit $\alpha \to 0$ the states generated in this fashion are highest weight states with respect to $\mathfrak{s}\mathfrak{u}(2)$. For the vacuum this follows immediately while for excited states this follows by the (fundamental) commutation relations (2.62) and the Bethe equations (2.57), see for example [74]. Note that this means we cannot act with more than $M/2$ creation operators to generate an eigenstate since highest weight states have positive spin. We do not need to either since flipping all spins is a symmetry of the spin chain Hamiltonian. Similarly, at the level of the chiral Gross–Neveu model it is clear that the labeling of the two fermionic species is inconsequential. Including the descendants of our highest weight states given by the algebraic Bethe ansatz we obtain a complete set of eigenstates for the Heisenberg Hamiltonian and our transfer matrix. These descendants can also be seen in the Bethe ansatz by allowing ourselves to consider irregular rapidities; adding an extra particle with infinite rapidity immediately satisfies its own equation, and does not change any of the others. In the quasi-periodic case there are no descendants, and correspondingly these infinite rapidity 'solutions' have turned into extra regular solutions with finite rapidities.

Now we are ready to write the Bethe–Yang equation (2.34) for our specific model. By the algebraic Bethe ansatz we just found a description of the eigenvalues of our transfer matrix, and setting $u={{u}_{j}}$ in (2.58) we obtain the Bethe–Yang equations

$$\begin{eqnarray}&&{{{\rm e}}^{{\rm i}{{p}_{j}}L}}\mathop{\prod }\limits_{m=1}^{{{N}_{f}}}{{S}^{ff}}({{u}_{j}}-{{u}_{m}})\mathop{\prod }\limits_{i=1}^{{{N}_{a}}}\frac{{{u}_{j}}-{{v}_{i}}+i}{{{u}_{j}}-{{v}_{i}}-i}=-1\;,\end{eqnarray} \tag{ 2.65 }$$

where we have re-instated the scalar factor S^ff,⁴¹ and the v_i satisfy the auxiliary Bethe equations

$$\begin{eqnarray}&&\mathop{\prod }\limits_{m=1}^{{{N}_{f}}}f\frac{{{v}_{i}}-{{u}_{m}}-i}{{{v}_{i}}-{{u}_{m}}+i}={{{\rm e}}^{-2{\rm i}\alpha }}\mathop{\prod }\limits_{j\ne i}^{{{N}_{a}}}\frac{{{v}_{i}}-{{v}_{j}}-2i}{{{v}_{i}}-{{v}_{j}}+2i}\;.\end{eqnarray} \tag{ 2.66 }$$

For future reference we would like to introduce the notation

$$\begin{eqnarray}&&{{S}^{11}}(w)\equiv \frac{w-2i}{w+2i}\;,\;\;\;{{S}^{f1}}(w)\equiv \frac{w+i}{w-i}\;,\;\;\;{{S}^{1f}}(w)\equiv \frac{w+i}{w-i}\;,\end{eqnarray} \tag{ 2.67 }$$

so that these Bethe–Yang equations read

$$\begin{eqnarray}&&{{{\rm e}}^{{\rm i}{{p}_{j}}L}}\mathop{\prod }\limits_{m=1}^{{{N}_{f}}}{{S}^{ff}}({{u}_{j}}-{{u}_{m}})\mathop{\prod }\limits_{i=1}^{{{N}_{a}}}{{S}^{f1}}({{u}_{j}}-{{v}_{i}})=-1\;,\end{eqnarray} \tag{ 2.68 }$$

$$\begin{eqnarray}&&{{{\rm e}}^{-2{\rm i}\alpha }}\mathop{\prod }\limits_{m=1}^{{{N}_{f}}}{{S}^{1f}}({{v}_{i}}-{{u}_{m}})\mathop{\prod }\limits_{j=1}^{{{N}_{a}}}{{S}^{11}}({{v}_{i}}-{{v}_{j}})=-1\;.\end{eqnarray} \tag{ 2.69 }$$

The relevance of this notation will become apparent in section 2.5.2.

The solutions of these auxiliary equations simply parametrize the possible eigenvalues of the transfer matrix; once we fix the ${\rm SU}(2)$ spin there is a fixed number of possible eigenvalues for the transfer matrix depending on the physical rapidities only, concretely obtained by substituting the solution of equation (2.66) in equation (2.65). Choosing one of these amounts to choosing one of the possible states with given spin. In this way we can think of auxiliary excitations as exciting the spin of the fermion. Note that as discussed above we should only consider states with ${{N}_{a}}\leqslant {{N}_{f}}/2$. Solving these equations for a given state gives its energy as $E={{\sum }_{j}}m{\rm cosh} \frac{\pi {{u}_{j}}}{2}$. Solving these equations for some two-particle states of spin one and zero gives the energy values shown in figure 7.

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The first step towards computing the finite size spectrum of our model will be to compute the ground state energy. This can be done exactly thanks to a clever idea by Zamolodchikov [29]. To describe this idea we begin by recalling that the ground state energy is the leading low temperature contribution to the Euclidean partition function

$$\begin{eqnarray}&&Z(\beta ,L)=\mathop{\sum }\limits_{n}{{{\rm e}}^{-\beta {{E}_{n}}}}\sim {{{\rm e}}^{-\beta {{E}_{0}}}}\;,\;\quad {\rm as}\;\;\beta \equiv \frac{1}{T}\to \infty \;.\end{eqnarray} \tag{ 2.70 }$$

This partition function can be computed from our original quantum field theory by Wick rotating $\tau \to \tilde{\sigma }=i\tau$ and considering a path integral over fields periodic in $\tilde{\sigma }$ with period β. In geometrical terms we are putting the theory on a torus which in the zero temperature limit degenerates to the cylinder we started with. Analytically continuing $\tilde{\sigma }$ back to τ takes us back to our original Lorentzian theory, however we could also analytically continue $\sigma \to \tilde{\tau }=-i\sigma$. This gives us a Lorentzian theory where the role of space and time have been interchanged with respect to the original theory; it gives us its mirror model ⁴² . Putting it geometrically, we could consider Hamiltonian evolution along either of the two cycles of the torus. Note that at the level of the Hamiltonian and the momentum the mirror transformation corresponds to

$$\begin{eqnarray}&&H\to i\tilde{p},\quad p\to i\tilde{H},\end{eqnarray} \tag{ 2.71 }$$

where mirror quantities are denoted with a tilde⁴³ . To emphasize its role as the mirror volume, let us from now on denote the inverse string temperature by $R(=\beta )$. In principle we can compute the Euclidean partition function both through our original model at size L and temperature $1/R$ and through the mirror model at size R and temperature $1/L$. These ideas are illustrated in figure 8.

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To compute the ground state energy of our model then, we can equivalently compute the infinite volume partition function of our mirror model at finite temperature or just its (generalized) (Helmholtz) free energy ($\tilde{F}$) since

$$\begin{eqnarray}&&Z={{{\rm e}}^{-L\tilde{F}}}\;.\end{eqnarray} \tag{ 2.72 }$$

More precisely, cf equation (2.70) the ground state energy is the (generalized) free energy density of the mirror model

$$\begin{eqnarray}&&{{E}_{0}}=\frac{\tilde{F}}{R}\;.\end{eqnarray} \tag{ 2.73 }$$

The key point is that we are considering the mirror model in the infinite volume limit where we can use factorized scattering and the asymptotic Bethe ansatz of the previous section, since any exponential corrections to them can be safely neglected⁴⁴ . The price we have to pay is dealing with a finite temperature. Fortunately the TBA will allow us to do precisely this.

Before moving on to a computation of the free energy we should be a little careful about the boundary conditions in our model. Firstly, where fermions are concerned we should note that the Euclidean partition function is only the proper statistical mechanical partition function used above provided the fermions are anti-periodic in imaginary time. Turning things around, if the fermions are periodic on the circle then from the mirror point of view they will be periodic in imaginary time, so that our goal in the mirror theory is not to compute the standard statistical mechanical partition function but rather what is known as Wittenʼs index

$$\begin{eqnarray}&&{{Z}_{W}}={\rm Tr}\left( {{(-1)}^{F}}{{{\rm e}}^{-L\tilde{H}}} \right),\end{eqnarray} \tag{ 2.74 }$$

where F is the fermion number operator. Continuing along these lines, if we consider quasi-periodic boundary conditions instead of (anti-)periodic boundary conditions we can expect a more general operator to enter in the trace. In fact we can view quasi-periodic boundary conditions as a line defect running along the space-time cylinder of our theory, and upon a double Wick rotation this induces a so-called defect operator in the partition function of the mirror theory [29, 75] as illustrated in figure 9. This defect operator is of course just the inverse⁴⁵ of the operator g we introduced in equation (2.35), meaning we want to compute

$$\begin{eqnarray}&&Z={\rm Tr}\left( {{g}^{-1}}\;{{(-1)}^{F}}{{{\rm e}}^{-L\tilde{H}}} \right).\end{eqnarray} \tag{ 2.75 }$$

Since the boundary conditions we will consider are compatible with integrability, this defect operator commutes with the mirror S-matrix and Hamiltonian. As such we can arrange the excitations of the Bethe ansatz to be eigenstates of this operator, so that the defect operator will effectively add (imaginary) chemical potential terms to the mirror Hamiltonian. Concretely, denoting the eigenvalue of g on a particle of type χ as ${{{\rm e}}^{{\rm i}{{\alpha }_{\chi }}}}$, we need to add ${{\sum }_{\chi }}i{{\alpha }_{\chi }}{{N}_{\chi }}/L$ to the free energy, where ${{N}_{\chi }}$ is the number operator for particle type χ.⁴⁶

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We just saw that we would like to compute the free energy of our mirror model in the infinite size limit. In this limit, the ideas of factorized scattering and the asymptotic Bethe ansatz apply, so in principle we know the possible momentum distributions of the particles. As we also know their dispersion, we ought to be able to compute the free energy $F=E-TS$. The procedure to actually do this computation goes under the name of the TBA [30].

Let us denote the system size of our (mirror) model by $R(=\beta )$ and the number of particles by K. Then in the limit $R\to \infty$ the partition function is dominated by states with a finite density $K/R$. Since in a nested system the type of particle is determined by the auxiliary excitations, we should also keep their densities finite to allow all particle types to contribute to the partition function. Hence in order to proceed we need to understand what happens to the Bethe–Yang equations when the system size and excitation numbers become large. In turns out that in this limit the solutions to the Bethe–Yang equations arrange themselves in certain patterns in the complex plane which have an interpretation in terms of (auxiliary) bound states.

To understand this intuitively, let us consider a hypothetical theory we know to have physical bound states. At large but finite lengths there is no reason for the momenta solving the Bethe–Yang equations to be real and in general there are in fact solutions with complex momenta. At the same time our equations should reproduce the continuum of scattering states in the infinite volume limit. This means that unless all momenta become real in this limit, the resulting configuration has to have a different particle content than we naively thought—some particles have formed one or more bound states. As we will see the auxiliary excitations can form similar configurations, which by analogy we interpret as auxiliary bound states. Another motivation for this interpretation comes from the spin chain interpretation of the auxiliary system, where these configurations really have the interpretation of physical bound states.

Determining the precise spectrum of excitations in the thermodynamic limit is the nontrivial step in the derivation of the TBA equations, differing from model to model. Rather than attempting a semi-general discussion we will discuss these ideas in detail for the simple example of the chiral Gross–Neveu model. By considering the XXX spin chain as both the corresponding auxiliary problem as well as a physical model on its own, we can kill two birds with one stone by discussing the bound states of magnons in the XXX spin chain⁴⁷ . After this we hope that the general story will be quite readily accepted. Note that the mirror Bethe–Yang equations for the chiral Gross–Neveu model are given by (2.65) and (2.66) with $L\to R$ and $\alpha \to 0$.

We will begin our discussion with magnon bound states in the XXX spin chain. In the u_m = 0 limit the Bethe equations (2.69) can be written as

$$\begin{eqnarray}&&{{{\rm e}}^{{\rm i}{{p}_{i}}{{N}_{f}}}}\mathop{\prod }\limits_{j=1}^{{{N}_{a}}}{{S}^{11}}({{v}_{i}}-{{v}_{j}})=-1\;,\end{eqnarray} \tag{ 2.76 }$$

and we have identified the spin chain momentum p via equation (2.61); ${{S}^{1f}}({{v}_{i}})={{{\rm e}}^{i{{p}_{i}}}}$. We would like to understand the type of solutions these equations can have, specifically as we take the system size ${{N}_{f}}$ to infinity. For real momenta nothing particular happens in these equations, and we simply get many more possible solutions as ${{N}_{f}}$ grows. If we consider a solution with complex momenta however⁴⁸ , say a state with ${\rm Im}({{p}_{1}})\gt 0$, we have an immediate problem:

$$\begin{eqnarray}&&{{{\rm e}}^{{\rm i}{{p}_{q}}{{N}_{f}}}}\to 0\;,\;\;\;\;\;\;{\rm as}\;\;\;{{N}_{f}}\to \infty \;.\end{eqnarray} \tag{ 2.77 }$$

We see that the only way a solution containing p₁ can exist in this limit is if this zero is compensated by a pole in one of the S¹¹ (equation (2.67)), which can be achieved by setting

$$\begin{eqnarray}&&{{v}_{2}}={{v}_{1}}+2i\;.\end{eqnarray} \tag{ 2.78 }$$

At this point we have doctored up the equation for p₁, but we have introduced potential problems in the equation for p₂. Whether there is a problem can be determined by multiplying the equations for p₁ and p₂ so that the singular contributions of their relative S-matrix cancel out

$$\begin{eqnarray*}&&{{{\rm e}}^{{\rm i}({{p}_{1}}+{{p}_{2}}){{N}_{f}}}}\mathop{\prod }\limits_{i\ne 1}^{{{N}_{a}}}{{S}^{11}}({{v}_{1}}-{{v}_{i}})\mathop{\prod }\limits_{i\ne 2}^{{{N}_{a}}}{{S}^{11}}({{v}_{2}}-{{v}_{i}})={{{\rm e}}^{{\rm i}({{p}_{1}}+{{p}_{2}}){{N}_{f}}}}\mathop{\prod }\limits_{i\ne 1,2}^{{{N}_{a}}}{{S}^{11}}({{v}_{1}}-{{v}_{i}}){{S}^{11}}({{v}_{2}}-{{v}_{i}})=1\;.\end{eqnarray*}$$

If the sum of their momenta is real this equation is fine, and this set of momenta can be part of a solution to the Bethe equations. In terms of the rapidities this solution looks like

$$\begin{eqnarray}&&{{v}_{1}}=v-i\;,\;\;\;{{v}_{2}}=v+i\;,\;\;\;v\in \mathbb{R}\;.\end{eqnarray} \tag{ 2.79 }$$

On the other hand, if the sum of our momenta has positive imaginary part we are still in trouble⁴⁹ . In this case, since we should avoid coincident rapidities, the only way to fix the equation is to have a third particle in the solution, with rapidity

$$\begin{eqnarray}&&{{v}_{3}}={{v}_{2}}+2i\;.\end{eqnarray} \tag{ 2.80 }$$

As before, if now the total momentum is real the equations are consistent and these three rapidities can be part of a solution. If not, we continue this process and create a bigger configuration or run off to infinity. These configurations in the complex rapidity plane are known as Bethe strings, illustrated in figure 10. Since in our spin chain p has positive imaginary part in the lower half of the complex rapidity plane and vice versa, strings of any size can be generated in this fashion by starting appropriately far below the real line⁵⁰ . Concretely, a Bethe string with Q constituents and rapidity v is given by the configuration

$$\begin{eqnarray}&&\{{{v}_{Q}}\}\equiv \{v-(Q+1-2j)i|j=1,\ldots ,Q\}\;,\end{eqnarray} \tag{ 2.81 }$$

where $v\in \mathbb{R}$ is called the center of the string. Full solutions of the Bethe equation in the limit $M\to \infty$ can be built out of these string configurations.

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We would like to give these (Bethe) strings the interpretation of bound states as they have less energy than sets of individual real magnons⁵¹ . For example, the energy of the two-string (2.79) is given by

$$\begin{eqnarray}&&{{E}_{2}}(v)=E({{v}_{1}})+E({{v}_{2}})=-2J\left( \frac{1}{{{\left( v-i \right)}^{2}}+1}+\frac{1}{{{\left( v+i \right)}^{2}}+1} \right)=-2J\frac{2}{{{v}^{2}}+{{2}^{2}}}\;,\end{eqnarray} \tag{ 2.82 }$$

which is less than that of any two-particle state made out of real particles;

$$\begin{eqnarray}&&{{E}_{2}}(v)\lt E({{\tilde{v}}_{1}})+E({{\tilde{v}}_{2}})\;\;\;{\rm for}\;v,{{\tilde{v}}_{1,2}}\in \mathbb{R}\;\;\;({\rm real}\ {\rm momenta}).\end{eqnarray} \tag{ 2.83 }$$

Similarly, the energy of a Q-string is lower than that of Q separate particles and is given by

$$\begin{eqnarray}&&{{E}_{Q}}(v)=\mathop{\sum }\limits_{{{v}_{j}}\in \{{{v}_{Q}}\}}E({{v}_{j}})=-2J\frac{Q}{{{v}^{2}}+{{Q}^{2}}}\;.\end{eqnarray} \tag{ 2.84 }$$

This is most easily shown by noting that

$$\begin{eqnarray}&&E(v)=J\frac{{\rm d}p(v)}{{\rm d}v}\;,\end{eqnarray} \tag{ 2.85 }$$

and the fact that the expression for the momentum of a Q-string is particularly simple

$$\begin{eqnarray}&&{{p}_{Q}}(v)=i{\rm log} \frac{v-Qi}{v+Qi}\;,\end{eqnarray} \tag{ 2.86 }$$

as follows by cancelling numerators and denominators in the product as indicated.

We have just determined that the possible solutions of the Bethe equations in the limit ${{N}_{f}}\to \infty$ are built out of elementary objects called Bethe strings (a one-string being a normal magnon). Interpreting them as bound states, we see that the spectrum thus obtained is reflected by an appropriate pole in the two-particle S-matrix. This example is not a field theory, but in the next section we will see that the bound state spectrum of the (mirror) light-cone superstring has the exact same pattern in appropriate variables. So far so good, but ultimately we are interested in thermodynamic limits, meaning we should take ${{N}_{f}}\to \infty$ with ${{N}_{a}}/{{N}_{f}}\leqslant 1/2$ fixed; the number of magnons goes to infinity as well. In this limit the analysis above is clearly no longer even remotely rigorous since an ever growing product of magnon S-matrices with complex momenta can perfectly mimic the role of the pole in our story for example. Still, since such solutions seem rather atypical and at least low magnon density solutions should essentially conform to the string picture, we can hypothesize that the string complexes should contain 'most' of the possible solutions, in the sense that they are the ones that give measurable contributions to the free energy. Indeed in the XXX spin chain there are examples of solutions that do not approach string complexes in the thermodynamic limit [76–78], but nonetheless the free energy is captured correctly by taking only string configurations into account [79]. The assumption that all thermodynamically relevant solutions to the Bethe equations are built up out of such string configurations, and which form these configurations take, goes under the name of the string hypothesis. More details and references on the string hypothesis can for example be found in section four of [80].

Now we would like to come to the chiral Gross–Neveu model where the XXX spin chain is an auxiliary system. Because the chiral Gross–Neveu model has no bound states the only solutions to the Bethe equations correspond to fundamental excitations with real momenta with correspondingly real rapidities, also in the thermodynamic limit⁵² . However, in the thermodynamic limit $R\to \infty$ keeping ${{N}_{f}}/R$ and ${{N}_{a}}/{{N}_{f}}$ fixed, we are also effectively taking the infinite length limit of the spin chain, in fact its thermodynamic limit. In this case the auxiliary equations become

$$\begin{eqnarray}&&{{\left( {{S}^{1f}}({{v}_{i}}) \right)}^{{{N}_{f}}}}\mathop{\prod }\limits_{j\ne i}^{{{N}_{a}}}{{S}^{11}}({{v}_{i}}-{{v}_{j}})=1\to \mathop{\prod }\limits_{m=1}^{{{N}_{f}}}{{S}^{1f}}({{v}_{i}}-{{u}_{m}})\mathop{\prod }\limits_{j\ne i}^{{{N}_{a}}}{{S}^{11}}({{v}_{i}}-{{v}_{j}})=1\;,\end{eqnarray} \tag{ 2.87 }$$

but since all u_m are real $|{{S}^{1f}}({{v}_{i}}-{{u}_{m}})|\gt 1$ when $|{{S}^{1f}}({{v}_{i}})|\gt 1$ and vice versa, the above analysis for the spin chain is not affected. Therefore will make the string hypothesis that the solutions of the Bethe–Yang equations of the chiral Gross–Neveu model in the thermodynamic limit consist of

• fermions with real momenta
• strings of auxiliary magnons of any length with real center.

The constituent rapidities of a Q string with center v are given by equation (2.81). While they carry no energy in the chiral Gross–Neveu model and hence cannot have a 'binding energy', we will still refer to and think of these patterns of auxiliary excitations as bound states of magnons.

In general the string hypothesis for a given model may contain bound states of physical particles, bound states of auxiliary particles like in the chiral Gross–Neveu model, and even bound states of physical and auxiliary particles as happens in the Hubbard model [81]. Whatever the type of bound state, it appears to always be possible to find some appropriate rapidity type variable in terms of which the bound states look like Bethe strings. The allowed lengths for typical string complexes may be constrained in quite complicated ways however [82]. We will encounter examples of all this in the later sections.

We would like to group terms in the (auxiliary) Bethe equations (2.65) and (2.66) in accordance with the string hypothesis; having a thermodynamic limit in mind, the N magnons of a given solution of the Bethe equations should arrange themselves into combinations of string complexes. In other words, denoting the number of bound states of length Q occurring in a given configuration by N_Q we have

$$\begin{eqnarray}&&\mathop{\prod }\limits_{j=1}^{{{N}_{a}}}\to \mathop{\prod }\limits_{Q=1}^{\infty }\mathop{\prod }\limits_{l=1}^{{{N}_{Q}}}\mathop{\prod }\limits_{j\in \{{{v}_{Q,l}}\}},\end{eqnarray} \tag{ 2.88 }$$

under the constraint

$$\begin{eqnarray}&&\mathop{\sum }\limits_{Q=1}^{\infty }Q{{N}_{Q}}={{N}_{a}}\;.\end{eqnarray} \tag{ 2.89 }$$

Taking this into account we can appropriately represent the Bethe equations (2.68) and (2.69) as

$$\begin{eqnarray}&&{{{\rm e}}^{{\rm i}{{p}_{j}}L}}\mathop{\prod }\limits_{m=1}^{{{N}_{f}}}{{S}^{ff}}({{u}_{j}}-{{u}_{m}})\mathop{\prod }\limits_{Q=1}^{\infty }\mathop{\prod }\limits_{l=1}^{{{N}_{Q}}}{{S}^{fQ}}({{u}_{j}}-{{v}_{Q,l}})=-1,\end{eqnarray} \tag{ 2.90 }$$

$$\begin{eqnarray}&&\mathop{\prod }\limits_{m=1}^{{{N}_{f}}}{{S}^{1f}}({{v}_{i}}-{{u}_{m}})\mathop{\prod }\limits_{Q=1}^{\infty }\mathop{\prod }\limits_{l=1}^{{{N}_{Q}}}{{S}^{1Q}}({{v}_{i}}-{{v}_{Q,l}})=-1,\end{eqnarray} \tag{ 2.91 }$$

where

$$\begin{eqnarray}&&{{S}^{\chi Q}}(v-{{w}_{Q}})\equiv \mathop{\prod }\limits_{{{w}_{j}}\in \{{{w}_{Q}}\}}{{S}^{\chi 1}}(v-{{w}_{j}})\;,\quad \chi =f,1.\end{eqnarray} \tag{ 2.92 }$$

At this point not all ${{N}_{a}}$ auxiliary Bethe equations are independent anymore, as some magnons are bound in strings. We already saw that we can get the Bethe equation for the center of a bound state by taking a product over the Bethe equations of its constituents, so that our complete set of Bethe equations becomes

$$\begin{eqnarray}&&{{{\rm e}}^{{\rm i}{{p}_{j}}L}}\mathop{\prod }\limits_{m\ne j}^{{{N}_{f}}}{{S}^{ff}}({{u}_{j}}-{{u}_{m}})\mathop{\prod }\limits_{Q=1}^{\infty }\mathop{\prod }\limits_{l=1}^{{{N}_{Q}}}{{S}^{fQ}}({{u}_{j}}-{{v}_{Q,l}})=-1,\end{eqnarray} \tag{ 2.93 }$$

$$\begin{eqnarray}&&\mathop{\prod }\limits_{m=1}^{{{N}_{f}}}{{S}^{Pf}}({{v}_{P,r}}-{{u}_{m}})\mathop{\prod }\limits_{Q=1}^{\infty }\mathop{\prod }\limits_{l=1}^{{{N}_{Q}}}{{S}^{PQ}}({{v}_{P,r}}-{{v}_{Q,l}})=-1,\end{eqnarray} \tag{ 2.94 }$$

where

$$\begin{eqnarray}&&{{S}^{P\chi }}({{v}_{P}}-w)\equiv \mathop{\prod }\limits_{{{v}_{i}}\in \{{{v}_{P}}\}}{{S}^{1\chi }}({{v}_{i}}-w)\;,\quad \chi =f,Q.\end{eqnarray} \tag{ 2.95 }$$

The physical interpretation of these expressions is of course that they are the scattering amplitudes between the particles indicated by superscripts. These products of constituent S-matrices typically simplify, but their concrete expression is not important for our considerations (yet); what is important is that they exist and only depend on the centers of the strings, or in other words, the bound state momenta. The process of obtaining the bound state S-matrices from fundamental ones in this way of course corresponds to the discussion of fusion in section 2.2, just applied at the diagonalized level.

To proceed, we will work with the Bethe–Yang equations in logarithm form. Through a fixed choice of branch for the logarithm this introduces an integer I in each equation which labels the possible solutions of the equation

$$\begin{eqnarray}&&2\pi I_{j}^{f}=R{{p}_{j}}-i\mathop{\sum }\limits_{m=1}^{{{N}_{f}}}{\rm log} {{S}^{ff}}({{u}_{j}}-{{u}_{m}})-i\mathop{\sum }\limits_{Q=1}^{\infty }\mathop{\sum }\limits_{l=1}^{{{N}_{Q}}}{\rm log} {{S}^{fQ}}({{u}_{j}}-{{v}_{Q,l}})\;,\end{eqnarray} \tag{ 2.96 }$$

$$\begin{eqnarray}&&-2\pi I_{r}^{P}=-i\mathop{\sum }\limits_{m=1}^{{{N}_{f}}}{\rm log} {{S}^{Pf}}({{v}_{P,r}}-{{u}_{m}})-i\mathop{\prod }\limits_{Q=1}^{\infty }\mathop{\prod }\limits_{l=1}^{{{N}_{Q}}}{\rm log} {{S}^{PQ}}({{v}_{P,r}}-{{v}_{Q,l}})\;.\end{eqnarray} \tag{ 2.97 }$$

We choose to define the integer in the second equation with a minus sign for reasons we will explain shortly. In the thermodynamic limit the solutions to these equations become dense

$$\begin{eqnarray}&&{{u}_{i}}-{{u}_{j}}\sim \mathcal{O}(1/R),\;\;\;\;\;{{u}_{i}}-{{v}_{j}}\sim \mathcal{O}(1/R)\;\;\;\;\;{\rm and}\;\;\;\;\;{{v}_{i}}-{{v}_{j}}\sim \mathcal{O}(1/R).\end{eqnarray} \tag{ 2.98 }$$

With this in mind it may be sensible to generalize the integers I to functions of the relevant rapidity (momentum), known as counting functions, interpolating between the integers corresponding to different solutions of the Bethe-equations. Concretely

$$\begin{eqnarray}&&R{{c}^{f}}(u)=\frac{R}{2\pi }p(u)+\frac{1}{2\pi i}\mathop{\sum }\limits_{m=1}^{{{N}_{f}}}{\rm log} {{S}^{ff}}(u-{{u}_{m}})+\frac{1}{2\pi i}\mathop{\sum }\limits_{Q=1}^{\infty }\mathop{\sum }\limits_{l=1}^{{{N}_{Q}}}{\rm log} {{S}^{fQ}}(u-{{v}_{Q,l}})\;,\end{eqnarray} \tag{ 2.99 }$$

$$\begin{eqnarray}&&R{{c}^{P}}(u)=-\frac{1}{2\pi i}\mathop{\sum }\limits_{m=1}^{{{N}_{f}}}{\rm log} {{S}^{Pf}}(u-{{u}_{m}})-\frac{1}{2\pi i}\mathop{\sum }\limits_{Q=1}^{\infty }\mathop{\sum }\limits_{l=1}^{{{N}_{Q}}}{\rm log} {{S}^{PQ}}(u-{{v}_{Q,l}})\;,\end{eqnarray} \tag{ 2.100 }$$

so that

$$\begin{eqnarray}&&R{{c}^{f}}({{u}_{k}})=I_{k}^{f}\;,\;\;\;\;\;{\rm and}\;\;\;\;\;R{{c}^{P}}({{v}_{l}})=I_{l}^{P}\;.\end{eqnarray} \tag{ 2.101 }$$

Importantly we assume that these counting functions are monotonically increasing functions of u provided their leading terms are⁵³ , and here indeed we have

$$\begin{eqnarray}&&\frac{{\rm d}p}{{\rm d}u}\gt 0\;,\quad {\rm and}\;\quad \frac{1}{2\pi i}\frac{{\rm dlog}\;{{S}^{Pf}}}{{\rm d}u}\lt 0\;.\end{eqnarray} \tag{ 2.102 }$$

This is the reason for our sign choice above. Clearly in general we have

$$\begin{eqnarray}&&c({{w}_{i}})-c({{w}_{j}})=\frac{{{I}_{i}}-{{I}_{j}}}{R}\;.\end{eqnarray} \tag{ 2.103 }$$

For a given solution of the Bethe–Yang equations some of the integers are said to be 'occupied' and some to be 'vacant' if there is, respectively there is not, a corresponding rapidity in the solution. Since the solutions become dense we will introduce an associated particle density ρ for the rapidities that are taken in a solution, and a hole density $\bar{\rho }$ for the ones that are not. The density of states in 'integer space' is one (in units of ${{R}^{-1}}$), which in rapidity space means

$$\begin{eqnarray}&&{{\rho }^{f}}(u)+{{\bar{\rho }}^{f}}(u)=\frac{{\rm d}{{c}^{f}}(u)}{{\rm d}u}\;,\end{eqnarray} \tag{ 2.104 }$$

$$\begin{eqnarray}&&{{\rho }^{P}}(v)+{{\bar{\rho }}^{P}}(v)=\frac{{\rm d}{{c}^{P}}(v)}{{\rm d}v}\;.\end{eqnarray} \tag{ 2.105 }$$

Explicitly taking the derivative of the counting functions gives us the thermodynamic analogue of the Bethe–Yang equations

$$\begin{eqnarray}&&{{\rho }^{f}}(u)+{{\bar{\rho }}^{f}}(u)=\frac{1}{2\pi }\frac{{\rm d}p(u)}{{\rm d}u}+{{K}^{ff}}\star {{\rho }^{f}}(u)-{{K}^{fQ}}\;\star \;{{\rho }_{Q}}(u)\;,\end{eqnarray} \tag{ 2.106 }$$

$$\begin{eqnarray}&&{{\rho }^{P}}(v)+{{\bar{\rho }}^{P}}(v)={{K}^{Pf}}\;\star \;{{\rho }^{f}}(u)-{{K}^{PQ}}\;\star \;{{\rho }^{Q}}(u),\end{eqnarray} \tag{ 2.107 }$$

where we implicitly sum over repeated indices and we have replaced the sums over roots by integrals over densities, resulting in convolutions defined as

$$\begin{eqnarray}&&K\star f(u)\equiv \int _{-\infty }^{\infty }{\rm d}v\;K(u,v)f(v).\end{eqnarray} \tag{ 2.108 }$$

The kernels K are defined as the logarithmic derivatives of the scattering amplitudes

$$\begin{eqnarray}&&{{K}^{\chi }}(u,v)\equiv \pm \frac{1}{2\pi i}\frac{{\rm d}}{{\rm d}u}{\rm log} {{S}^{\chi }}(u,v),\end{eqnarray} \tag{ 2.109 }$$

where χ denotes an arbitrary set of particle labels. The sign is chosen such that the kernels are positive, in this case requiring minus signs for K^fP and K^Mf54 . From this point on we will distinguish the two arguments even though the S-matrices and kernels we have seen so far are all of difference form.

The remainder of the derivation simply amounts in finding an expression for the generalized free energy given these constraints on the density of states. The energy density (per unit length) is just

$$\begin{eqnarray}&&e=\int _{-\infty }^{\infty }{\rm d}u\;\mathcal{E}(u){{\rho }^{f}}(u),\end{eqnarray} \tag{ 2.110 }$$

where in this case

$$\begin{eqnarray}&&\mathcal{E}(u)=m{\rm cosh} \frac{\pi u}{2}\;,\end{eqnarray} \tag{ 2.111 }$$

while the entropy in an interval $\Delta u$ for each particle type is of course

$$\begin{eqnarray}&&\Delta S(\rho )={\rm log} \frac{(R(\rho (u)+\bar{\rho }(u))\Delta u)!}{(R\rho (u)\Delta u)!(R\bar{\rho }(u)\Delta u)!}\end{eqnarray} \tag{ 2.112 }$$

which by use of Stirlingʼs formula ($R\rho (\bar{\rho })\Delta u\gg 1$) gives us the entropy density as

$$\begin{eqnarray}&&s=\mathop{\sum }\limits_{i\in \{f,P\}}\int _{-\infty }^{\infty }{\rm d}u\;{{\rho }^{i}}{\rm log} \left( 1+\frac{{{{\bar{\rho }}}^{i}}}{{{\rho }^{i}}} \right)+{{\bar{\rho }}^{i}}{\rm log} \left( 1+\frac{{{\rho }^{i}}}{{{{\bar{\rho }}}^{i}}} \right)\;.\end{eqnarray} \tag{ 2.113 }$$

Next, since we are considering twisted boundary conditions we should add chemical potentials corresponding to the defect operator g, in this case given by equation (2.46). Also, we should not forget about the ${{(-1)}^{F}}={{{\rm e}}^{{\rm i}\pi {{N}_{f}}}}$ in our trace. The eigenvalue of g on the fundamental fermions is just ${{\rm e}^{{\rm i}\alpha }}$, so taking into account also their fermionic nature we should add $i(\alpha +\pi ){{N}_{f}}T$ to the free energy. As auxiliary particles represent flipping the spin of a fermion they carry double the opposite ${\rm SU}(2)$ charge, meaning we should add $-2i\alpha {{N}_{1}}T$ for them. Bound states of auxiliary particles of course carry the charge of their constituents, so that for these we add $-2i\alpha Q{{N}_{Q}}T$. Of course the particle densities are just given by

$$\begin{eqnarray}&&{{n}^{i}}=\int _{-\infty }^{\infty }{\rm d}u\;{{\rho }^{i}}\;,\end{eqnarray} \tag{ 2.114 }$$

Putting this together we get an expression for the generalized free energy

$$\begin{eqnarray}&&F=R(e-Ts-{{\mu }_{f}}{{n}^{f}}-{{\mu }_{Q}}{{n}^{Q}}),\end{eqnarray} \tag{ 2.115 }$$

where ${{\mu }_{f}}=-i(\alpha +\pi )T$ and ${{\mu }_{Q}}=2iQ\alpha T$. Being in thermodynamic equilibrium however means

$$\begin{eqnarray}&&\delta F=0\;.\end{eqnarray} \tag{ 2.116 }$$

Here the Bethe–Yang equations come in by giving us the hole densities as functions of the particle densities. Varying equations (2.106) and (2.107) we get

$$\begin{eqnarray}&&\delta {{\rho }^{f}}+\delta {{\bar{\rho }}^{f}}={{K}^{ff}}\;\star \;\delta {{\rho }_{f}}-{{K}^{fQ}}\;\star \;\delta {{\rho }_{Q}}\;,\end{eqnarray} \tag{ 2.117 }$$

$$\begin{eqnarray}&&\delta {{\rho }^{P}}+\delta {{\bar{\rho }}^{P}}={{K}^{Pf}}\;\star \;\delta {{\rho }_{f}}-{{K}^{PQ}}{\mkern 1mu} \star {\mkern 1mu} \delta {{\rho }_{Q}}\;,\end{eqnarray} \tag{ 2.118 }$$

Writing this schematically as

$$\begin{eqnarray}&&\delta {{\rho }^{i}}+\delta {{\bar{\rho }}^{i}}={{K}^{ij}}{\mkern 1mu} \star {\mkern 1mu} \delta {{\rho }^{j}}\;,\end{eqnarray} \tag{ 2.119 }$$

after a little algebra we get the variation of the entropy

$$\begin{eqnarray}&&\frac{\delta s}{\delta {{\rho }^{j}}(u)}={\rm log} \frac{{{{\bar{\rho }}}^{j}}}{{{\rho }^{j}}}(u)+{\rm log} \left( 1+\frac{{{\rho }^{i}}}{{{{\bar{\rho }}}^{i}}} \right)\tilde{\star }\;{{K}^{ij}}(u),\end{eqnarray} \tag{ 2.120 }$$

where $\tilde{\star }$ denotes 'convolution' from the right

$$\begin{eqnarray}&&f\;\tilde{\star }\;K(u)\equiv \int _{-\infty }^{\infty }{\rm d}vf(v)K(v-u).\end{eqnarray} \tag{ 2.121 }$$

The variation of the other terms is immediate, and $\delta F=0$ results in the TBA equations

$$\begin{eqnarray}&&{\rm log} \frac{{{{\bar{\rho }}}^{j}}}{{{\rho }^{j}}}=\frac{{{\mathcal{E}}_{j}}-{{\mu }_{j}}}{T}-{\rm log} \left( 1+\frac{{{\rho }^{i}}}{{{{\bar{\rho }}}^{i}}} \right)\star \;{{K}^{ij}}\;,\end{eqnarray} \tag{ 2.122 }$$

where by conventional abuse of notation we dropped the tilde on the 'convolution'. We will henceforth denote the combination $\frac{{{{\bar{\rho }}}^{j}}}{{{\rho }^{j}}}$ by the Y-functions Y_j, meaning the TBA equations read

$$\begin{eqnarray}&&{\rm log} {{Y}_{j}}=\frac{{{\mathcal{E}}_{j}}-{{\mu }_{j}}}{T}-{\rm log} \left( 1+\frac{1}{{{Y}_{i}}} \right)\star {{K}^{ij}}\;.\end{eqnarray} \tag{ 2.123 }$$

Taking into account the generalized form of equations (2.106) and (2.107), on a solution of the TBA equations the free energy of the mirror model is given by

$$\begin{eqnarray}&&F=-R\int _{-\infty }^{\infty }{\rm d}u\frac{1}{2\pi }\frac{{\rm d}{{p}_{j}}}{{\rm d}u}{\rm log} \left( 1+\frac{1}{{{Y}_{j}}} \right),\end{eqnarray} \tag{ 2.124 }$$

giving the ground state energy of our original model as

$$\begin{eqnarray}&&{{E}_{0}}=-\int _{-\infty }^{\infty }{\rm d}u\frac{1}{2\pi }\frac{{\rm d}{{p}_{j}}}{{\rm d}u}{\rm log} \left( 1+\frac{1}{{{Y}_{j}}} \right).\end{eqnarray} \tag{ 2.125 }$$

Specifying our schematic notation to the chiral Gross–Neveu model by comparing equations (2.117), (2.118) and (2.119) gives

$$\begin{eqnarray}&&{\rm log} {{Y}_{f}}=L\mathcal{E}+i(\alpha +\pi )-{\rm log} \left( 1+\frac{1}{{{Y}_{f}}} \right)\star {{K}^{ff}}-{\rm log} \left( 1+\frac{1}{{{Y}_{Q}}} \right)\star {{K}^{Qf}}\;,\end{eqnarray} \tag{ 2.126 }$$

$$\begin{eqnarray}&&{\rm log} {{Y}_{P}}=-2i\alpha P+{\rm log} \left( 1+\frac{1}{{{Y}_{Q}}} \right)\star {{K}^{QP}}+{\rm log} \left( 1+\frac{1}{{{Y}_{f}}} \right)\star {{K}^{fP}}\;,\end{eqnarray} \tag{ 2.127 }$$

where we note that in the setting of the mirror trick the temperature $T=1/L$. The ground state energy is given by

$$\begin{eqnarray}&&{{E}_{0}}=-\int _{-\infty }^{\infty }{\rm d}u\frac{1}{2\pi }\frac{{\rm d}p}{{\rm d}u}{\rm log} \left( 1+\frac{1}{{{Y}_{f}}} \right).\end{eqnarray} \tag{ 2.128 }$$

At this point the generalization to an arbitrary model is hopefully almost obvious, with the exception of the string hypothesis which depends on careful analysis of the Bethe–Yang equations for a particular model. If we have this however, we can readily determine the complete set of Bethe–Yang equations analogous to the procedure to arrive at equations (2.93) and (2.94). From there we immediately get the analogue of equations (2.117) and (2.118) by a logarithmic derivative⁵⁵ . This is all we need to specify the general TBA equations (2.123) to a given model.

In problems where there are bound states the TBA equations can typically be rewritten in a simpler fashion. The intuitive picture why this should be possible is illustrated in figure 11. Since we obtain all S-matrices by fusing over the constituents, provided S has no branch cuts the figure tells us

$$\begin{eqnarray}&&\frac{{{S}^{\chi Q+1}}(v,u){{S}^{\chi Q-1}}(v,u)}{{{S}^{\chi Q}}(v,u+i){{S}^{\chi Q}}(v,u-i)}=1\;,\end{eqnarray} \tag{ 2.129 }$$

where $\chi$ is any particle type and we have reinstated a dependence on two arguments for future reference. In other words, (the logs of) our S-matrices satisfy a discrete Laplace equation. The associated kernels should then satisfy

$$\begin{eqnarray}&&{{K}^{\chi Q}}(v,u+i)+{{K}^{\chi Q}}(v,u-i)-({{K}^{\chi Q+1}}(v,u)+{{K}^{\chi Q-1}}(v,u))=0\;.\end{eqnarray} \tag{ 2.130 }$$

However, when we shift u by $\pm i$ we may generate a pole in $K(v,u+i)$ for some real value of v. This can lead to a discontinuity in integrals involving K such as those in the TBA equations. Therefore we need to understand what exactly we mean by this equation. The way this is done is to introduce the kernel s

$$\begin{eqnarray}&&s(u)=\frac{1}{4{\rm cosh} \frac{\pi u}{2}}\;,\end{eqnarray} \tag{ 2.131 }$$

and the operator ${{s}^{-1}}$ that in hindsight will properly implement our shifts

$$\begin{eqnarray}&&f\star {{s}^{-1}}(u)=\mathop{{\rm lim} }\limits_{\epsilon \to 0}\left( f(u+i-i\epsilon )+f(u-i+i\epsilon ) \right),\end{eqnarray} \tag{ 2.132 }$$

which satisfy

$$\begin{eqnarray}&&s\star {{s}^{-1}}(u)=\delta (u).\end{eqnarray} \tag{ 2.133 }$$

Note that ${{s}^{-1}}$ has a large null space, so that $f\star {{s}^{-1}}\star s\ne f$ in general (we will see examples of this soon). This kernel can now be used to define

$$\begin{eqnarray}&&\left( K+1 \right)_{PQ}^{-1}={{\delta }_{P,Q}}-{{I}_{PQ}}s\;,\end{eqnarray} \tag{ 2.134 }$$

where ${{I}_{PQ}}={{\delta }_{P,Q+1}}+{{\delta }_{P,Q-1}}$. In other words, the kernel K^PQ introduced above satisfies

$$\begin{eqnarray}&&{{K}^{PQ}}-({{K}^{PQ+1}}+{{K}^{PQ-1}})\star s=s\;{{I}_{PQ}}\;,\end{eqnarray} \tag{ 2.135 }$$

which we can easily prove by Fourier transformation (see appendix A.3.2). Similarly we have

$$\begin{eqnarray}&&{{K}^{fQ}}-({{K}^{fQ+1}}+{{K}^{fQ-1}})\star s=s\;{{\delta }_{Q1}}\;.\end{eqnarray} \tag{ 2.136 }$$

If the TBA equations contain other types of kernels these typically also reduce to something nice under the action of ${{(K+1)}^{-1}}$; we will see examples of this in later sections.

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With these identities we can rewrite the auxiliary TBA equations (2.127) for the chiral Gross–Neveu model as

$$\begin{eqnarray}&&{\rm log} {{Y}_{Q}}={\rm log} (1+{{Y}_{Q+1}})(1+{{Y}_{Q-1}})\star s+{{\delta }_{Q,1}}{\rm log} \left( 1+\frac{1}{{{Y}_{f}}} \right)\star s\;,\end{eqnarray} \tag{ 2.137 }$$

so that the infinite sums have disappeared. The chemical potentials (twists) have disappeared completely from these equations as well, since $1\star s=1/2$ and hence $c\;Q$ is in the null-space of $(K+1)_{PQ}^{-1}$ for any constant c. This shows us that the canonical TBA equations carry more information than the simplified TBA equations. As we will discuss in more detail in section 4 the twists modify the large u asymptotics of the Y -functions, and once these are specified the simplified and canonical TBA equations are equivalent. The infinite sum in the main TBA equation can now be removed by noting that similarly to K^fQ, K^Qf satisfies

$$\begin{eqnarray}&&{{K}^{Qf}}-{{I}_{QP}}s\star {{K}^{Pf}}=s{{\delta }_{Q1}}\;.\end{eqnarray} \tag{ 2.138 }$$

Then by rewriting the above simplified equations as

$$\begin{eqnarray}&&{\rm log} {{Y}_{Q}}-{{I}_{QP}}{\rm log} {{Y}_{P}}\star s={{I}_{QP}}{\rm log} \left( 1+\frac{1}{{{Y}_{P}}} \right)\star s+{{\delta }_{Q,1}}{\rm log} \left( 1+\frac{1}{{{Y}_{f}}} \right)\star s\;,\end{eqnarray} \tag{ 2.139 }$$

integrating with K^Qf and using equation (2.138) we get

$$\begin{eqnarray}&&{\rm log} {{Y}_{1}}\star s+i\alpha ={\rm log} \left( 1+\frac{1}{{{Y}_{Q}}} \right)\star {{K}^{Qf}}-{\rm log} \left( 1+\frac{1}{{{Y}_{1}}} \right)\star s+{\rm log} \left( 1+\frac{1}{{{Y}_{f}}} \right)\star s\star {{K}^{1f}}\;,\end{eqnarray} \tag{ 2.140 }$$

or in other words

$$\begin{eqnarray}&&{\rm log} \left( 1+\frac{1}{{{Y}_{Q}}} \right)\star {{K}^{Qf}}=i\alpha +{\rm log} \left( 1+{{Y}_{1}} \right)\star s-{\rm log} \left( 1+\frac{1}{{{Y}_{f}}} \right)\star s\star {{K}^{1f}}\;.\end{eqnarray} \tag{ 2.141 }$$

We will for the moment skip over the technical details leading to the $i\alpha$ contribution in this identity; they will be discussed in section 4 and appendix A.4.1. The main TBA equation (2.126) then becomes

$$\begin{eqnarray}&&{\rm log} {{Y}_{f}}=L\mathcal{E}+i\pi -{\rm log} \left( 1+{{Y}_{1}} \right)\star s\;,\end{eqnarray} \tag{ 2.142 }$$

upon noting that magically enough the Y_f contribution drops out completely thanks to ${{K}^{ff}}=s\star {{K}^{1f}}$.⁵⁶ For uniformity we can define ${{Y}_{0}}\equiv Y_{f}^{-1}$ and get

$$\begin{eqnarray}&&{\rm log} {{Y}_{M}}={\rm log} (1+{{Y}_{M+1}})(1+{{Y}_{M-1}})\star s-{{\delta }_{M,0}}(L\mathcal{E}+i\pi )\end{eqnarray} \tag{ 2.143 }$$

with ${{Y}_{M}}\equiv 0$ for $M\lt 0$. This form of the TBA equations goes under the name of simplified TBA equations. To finish what we started, we can now apply ${{s}^{-1}}$ to these equations to get

$$\begin{eqnarray}&&Y_{M}^{+}\;Y_{M}^{-}=(1+{{Y}_{M+1}})(1+{{Y}_{M-1}}),\end{eqnarray} \tag{ 2.144 }$$

where the ± denote shifts in the argument by $\pm i$; ${{f}^{\pm }}(u)\equiv f(u\pm i)$ (note that the energy and $i\pi$ are in the null space of ${{s}^{-1}}$, at least mod $2\pi i$). These equations are known as the Y-system [83].

In general, the structure of simplified TBA equations and Y-systems can be represented diagrammatically by graphs. For example, in this case equations (2.143) and (2.144) can be represented by figure 12.

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It is important to note that in this process we lose information at each step along the way; both ${{(K+1)}^{-1}}$ and ${{s}^{-1}}$ have null-spaces. Therefore the simplified TBA equations are only equivalent to the canonical TBA equations provided we specify additional information on the Y-functions such as their large u asymptotics. An alternative but equivalent specification often encountered in the literature is to give the large Q asymptotics of the Y_Q functions. We will discuss both of these in more detail in section 4. The Y-system requires even further specifications to really correspond to a particular model. For example the Y-system for the XXX spin chain is given by dropping Y₀ altogether, but this is nothing but the chiral Gross-Neveu Y-system again, just shifting the label M by one unit.

At this point we have actually done something quite impressive; we have found a system of equations we can solve (admittedly numerically) to find the exact finite volume ground state energy of a two dimensional field theory. It would be great if we could extend this approach to the entire spectrum. If we look back at our arguments however, we are immediately faced with a big conceptual problem; the mirror trick and infinite volume limit work nicely precisely for the ground state and the ground state only! Still it is hard to believe that a set of complicated TBA equations knows about the ground state only, especially since they are derived from the mirror Bethe–Yang equations which are just an analytic continuation away from describing the complete large volume spectrum. In this section we will take an approach often taken in physics; we will analytically continue from one part of a problem to another, in this case from the ground state energy to excited state energies. The idea that excited states can be obtained by analytic continuation is an old one, see for example [85] in the case of the quantum anharmonic oscillator.

Before moving on, we would like to motivate and illustrate these ideas on the simple quantum mechanical problem⁵⁷

$$\begin{eqnarray}H\psi =E\psi \;,\;\;\;\;\;{\rm with}\;\;\;\;\;H=\left( \begin{array}{ccccccccccccccc} 1 & 0 \\ 0 & -1 \\ \end{array} \right)+\lambda \left( \begin{array}{ccccccccccccccc} 0 & 1 \\ 1 & 0 \\ \end{array} \right).\end{eqnarray} \tag{ 2.145 }$$

After considerable effort we realize that the spectrum in this model is given by

$$\begin{eqnarray}&&E(\lambda )=\pm \sqrt{1+{{\lambda }^{2}}}\;,\end{eqnarray} \tag{ 2.146 }$$

and hence the ground state energy is $-\sqrt{1+{{\lambda }^{2}}}$. Allowing ourselves to analytically continue in the coupling constant we realize that the equation for the ground state energy has branch points at $\lambda =\pm i$. This has the interesting consequence that by analytically continuing around either of these branch points and coming back to the real line we do not quite get back the ground state energy, but rather the energy of the excited state. This is illustrated in figure 13(a). The message we can take away from this, as stated in [86], is that by analytically continuing a parameter around a 'closed contour'⁵⁸ we end up back at the same problem although our eigenvalue may have changed. Because it is still the same problem, if the continued eigenvalue is changed it must be one of the other eigenvalues of the problem. Note that this does not imply all eigenvalues can be found this way; the spectrum may well split into distinct sectors closed under analytic continuation.

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Let us now forget about the description of this simple example in terms of linear algebra, and suppose for the sake of the argument that in solving our spectral problem we had obtained

$$\begin{eqnarray}&&E(\lambda )=-\int _{-1}^{1}{\rm d}z\;\frac{1}{2\pi i}f(z)g(z)-1\;.\end{eqnarray} \tag{ 2.147 }$$

where

$$\begin{eqnarray}&&f(z)=\frac{1}{z-i/\lambda }\;,\;\;\;\;{\rm and}\;\;\;\;g(z)=2\lambda \sqrt{1-{{z}^{2}}}\;.\end{eqnarray} \tag{ 2.148 }$$

We can determine that this integral has branch points at $\lambda =\pm i$ without knowing anything about f(z) other than that it is meromorphic with a single pole at $i/\lambda$ (with finite residue). Conceptually we consider g(z) to be some nice known function, while f is not explicitly known. Analytically continuing the integral in λ we get a function that is well defined everywhere except for the half-lines $i\lambda \gt 1$ and $i\lambda \lt -1$ where this pole moves into the domain of integration. Continuing around the point $\lambda =i$ as in figure 13(a), nothing happens when we first cross the line $\operatorname{Re}(\lambda )=0$ but when we cross the second time, the pole moves through our integration contour on the real line, dragging the contour along as illustrated in figure 13(b). We can rewrite the resulting contour integral in terms of our original one by picking up the residue, resulting in

$$\begin{eqnarray}&&{{E}^{c}}(\lambda )=-\int _{-1}^{1}{\rm d}z\;\frac{1}{2\pi i}f(z)g(z)+g(i/\lambda )-1\;.\end{eqnarray} \tag{ 2.149 }$$

Since ${{E}^{c}}(\lambda )-E(\lambda )=g(i/\lambda )\ne 0$ there must be a branch point inside this contour. In this integral picture we do not need to know the precise analytic expression of E or equivalently f to determine the excited state expression for the energy. All we need to know is the pole structure of f relative to the integration contour.

Inspired by this simple example, we can try to analytically continue our expression for the ground state energy, equation (2.125), in some appropriate variable and see whether we encounter any changes in the description. We could try continuing in the mass variable of the chiral Gross–Neveu model for example. This method of obtaining excited state TBA equations was proposed and successfully applied to the scaling Lee–Yang model in [87]. They observed that in the process of analytic continuation the Y-functions solving the TBA equations undergo non-trivial monodromies. They also noted that changes in the form of the TBA equations are possible if singular points of $1+1/Y$ move in the complex plane during this analytic continuation. These changes are analogous to the changes in the energy formula of our simple example. In this case the integral is a typical term on the right-hand side of the TBA equations

$$\begin{eqnarray}&&y(u)\equiv {\rm log} \left( 1+\frac{1}{Y} \right)\star K(u).\end{eqnarray} \tag{ 2.150 }$$

where we recall that ⋆ denotes (right) convolution on the real line. If there is a singular point

$$\begin{eqnarray}&&Y({{u}^{*}})=-1\;,\end{eqnarray} \tag{ 2.151 }$$

and its location ${{u}^{*}}$ crosses the real line during the analytic continuation, we can pick up the residue just as in our simple example to get

$$\begin{eqnarray}&&{{y}^{c}}(u)={\rm log} \left( 1+\frac{1}{Y} \right)\star K(u)\pm {\rm log} S({{u}^{*}},u),\end{eqnarray} \tag{ 2.152 }$$

where we recall that $K(v,u)=\frac{1}{2\pi i}\frac{d}{dv}{\rm log} S(v,u)$ and the sign is positive for singular points that cross hit the contour from below and negative for those that hit it from above. If Y vanishes at a particular point, this leads to the same considerations, just resulting in an opposite sign. If we wanted to do this at the level of the simplified equations, all we need is the S-matrix associated to s, which is given by

$$\begin{eqnarray}&&S(u)=-{\rm tanh} \frac{\pi }{4}(u-i).\end{eqnarray} \tag{ 2.153 }$$

Note that the energy itself is also determined by an integral equation in the TBA approach, meaning it can change explicitly as well as implicitly through the solution of a changed set of TBA equations.

The upshot of this is that the excited state TBA equations obtained this way differ from those of the ground state by the addition of ${\rm log} S$ terms, which we will call driving terms ⁵⁹ . It should not matter whether we consider this procedure at the level of the canonical equations or at the level of the simplified equations, and indeed the results agree because of the S-matrix analogue of identities like equation (2.135). The case of the Y-system is a bit more peculiar however, since there the distinguishing features of an excited state completely disappear. This follows directly from the fact that the S-matrix (2.153) naturally vanishes under application of ${{s}^{-1}}$. From this we see that whatever excited state TBA equations we obtain this way, the Y-system equations are the same as those of the ground state; the Y-system is universal ⁶⁰ .

This last point is closely related to other approaches of obtaining equations that describe excited state energies. In some cases it is possible to construct a functional analogue of the Y-system directly. If we can then get satisfactory insight into the analytic structure of the corresponding objects, we can 'integrate' these functional relations to obtain integral equations describing the energy of excited states [89–92]. Depending on how these functional equations are integrated we can obtain equations of TBA form but also various other forms that can be more efficient in actual calculations. The latter equations generically go under the name of 'nonlinear integral equations' [89], but depending on the context are also called 'Klümper–Pearce' [89, 93–95] or 'Destri–de Vega' [96, 97] equations. While not obvious from their form, these different types of equations should be equivalent [98, 99].

We will employ an amalgamation of these ideas in the next sections to find excited state TBA equations, going under the name of the contour deformation trick. We will come back to this again, but the basic idea goes as follows. It turns out we will have a candidate solution of the Y-system for an excited state with some limited regime of applicability. We then assume that the form of the TBA equations for an excited state is uniform and does not change outside of the regime of applicability of our candidate solution. Next, drawing lessons from the analytic continuation story above we expect that the only changes in the equation should be the addition of possible driving terms. Furthermore, although our limited solution only gives us a static picture, we expect that we can qualitatively view these terms as if coming from singular points that crossed the integration contour. Since in this picture such singular points would have dragged the contour along with them, we expect that an excited state TBA equation should be of the same form as the ground state, except with modified integration contours. Analyzing the analytic structure of the candidate solution will allow us to consistently define these contours in such a way that the TBA equations are satisfied, and by taking the integration contours back to the real line we can explicitly pick up the corresponding driving terms. Coming back to our simple example, it would be as if internal consistency of the problem (perhaps in the form of some other equation) told us that the natural integration contour for the excited state was not $(-1,1)$, but a contour that starts at one and finishes at minus one while enclosing $i/\lambda$ between itself and the real line. Such a contour is of course equivalent to the red contour in figure 13(b) obtained by direct analytic continuation.

In the next section we will discuss how the program of this section applies to the ${\rm Ad}{{{\rm S}}_{5}}\times {{{\rm S}}^{5}}$ superstring.

At this point we have introduced the necessary notation to discuss crossing in a little more detail. When we consider crossing, we are considering analytically continuing some of the particles involved in our scattering process to their corresponding anti-particles. At the level of the torus, similar to the relativistic case this corresponds to the shift $z\to z+{{\omega }_{2}}$ if we cross the first particle and $z\to z-{{\omega }_{2}}$ if we cross the second [27]. Indeed, the energy and momentum change sign under this transformation⁶⁷ . The crossing equation (2.21) (using unitarity) turns into a constraint on the overall phase of our S-matrix, i.e. ${{S}_{\mathfrak{s}\mathfrak{l}(2)}}$ or ${{S}_{\mathfrak{s}\mathfrak{u}(2)}}$ depending on our choice of vacuum. In terms of the dressing phase σ it reads

$$\begin{eqnarray}&&\sigma ({{z}_{k}},{{z}_{l}})\sigma ({{z}_{k}},{{z}_{l}}-{{\omega }_{2}})=\frac{x_{k}^{+}}{x_{k}^{-}}h({{z}_{k}},{{z}_{l}}),\end{eqnarray} \tag{ 3.26 }$$

or by unitarity

$$\begin{eqnarray}&&\sigma ({{z}_{k}}+{{\omega }_{2}},{{z}_{l}})\sigma ({{z}_{k}},{{z}_{l}})=\frac{x_{l}^{-}}{x_{l}^{+}}h({{z}_{k}},{{z}_{l}}),\end{eqnarray} \tag{ 3.27 }$$

where

$$\begin{eqnarray}&&h({{z}_{k}},{{z}_{l}})=\frac{x_{k}^{-}-x_{l}^{+}}{x_{k}^{+}-x_{l}^{+}}\frac{1-\frac{1}{x_{k}^{-}x_{l}^{-}}}{1-\frac{1}{x_{k}^{+}x_{l}^{-}}}\;.\end{eqnarray} \tag{ 3.28 }$$

From this equation we manifestly see that the dressing phase is not periodic, since

$$\begin{eqnarray}&&\sigma ({{z}_{k}},{{z}_{l}}-2{{\omega }_{2}})=\frac{h({{z}_{k}},{{z}_{l}}-{{\omega }_{2}})}{h({{z}_{k}},{{z}_{l}})}\sigma ({{z}_{k}},{{z}_{l}}),\end{eqnarray} \tag{ 3.29 }$$

and

$$\begin{eqnarray}&&\sigma ({{z}_{k}}+2{{\omega }_{2}},{{z}_{l}})=\frac{h({{z}_{k}}+{{\omega }_{2}},{{z}_{l}})}{h({{z}_{k}},{{z}_{l}})}\sigma ({{z}_{k}},{{z}_{l}}).\end{eqnarray} \tag{ 3.30 }$$

We should note that the dressing 'phase' σ is not unitary in the mirror theory, while of course the total scalar factor is. The nice object to split off from the mirror scalar factor (particularly with regard to bound states discussed below) is the so-called improved dressing phase

$$\begin{eqnarray}&&\Sigma ({{z}_{k}},{{z}_{l}})\equiv \frac{1-\frac{1}{x_{k}^{+}x_{l}^{-}}}{1-\frac{1}{x_{k}^{-}x_{l}^{+}}}\sigma ({{z}_{k}},{{z}_{l}}).\end{eqnarray} \tag{ 3.31 }$$

Like the string theory, the mirror theory has bound state excitations of any length [31]. Contrary to the string theory bound states however, they are bound states of fermions and transform in totally anti-symmetric short representations⁶⁸ . They are supported by poles in the mirror S-matrix which occur when⁶⁹

$$\begin{eqnarray}&&{{x}^{-}}({{\tilde{p}}_{i}})={{x}^{+}}({{\tilde{p}}_{i+1}}).\end{eqnarray} \tag{ 3.32 }$$

Similarly to the string S-matrix, other poles in the mirror S-matrix do not correspond to BPS configurations and should not correspond to bound states.

We saw above that the torus plays a role analogous to (twice) the physical strip in relativistic models as far as the dispersion is considered. However, as the physical strip is defined in terms of a relative rapidity it does not immediately have an analogue in our non-relativistic model. To define what we will call the physical region on the torus we should consider bound states [31]. For relativistic models all bound states lie on the imaginary axis of the rapidity plane within in the physical strip, once in the s and once in the t-channel. In other words, the constituent rapidities of a two-particle s-channel bound state lie between the lines ${\rm Im}({{u}_{j}})=\pm 1$, a region we can call the physical strip for the individual rapidities of a two-particle bound state. In this spirit we will try to define the physical region of the string and mirror theory as the region on the torus which contains the constituents of all bound states of the string, respectively mirror theory.

If we were to consider the locations of string and mirror bound states on the torus we would quickly realize that it is natural to divide the torus in regions separated by the contours $|{{x}^{\pm }}|=1$ and ${\rm Im}({{x}^{\pm }})=0$. Concretely then, we can illustrate the torus with these divisions as in figure 14. In fact, the upshot of a careful analysis of the bound state equations for both the string and mirror theory [31] shows that the constituents of a two-particle mirror bound state move on the torus along the curves $|{{x}^{\pm }}|=1$ inside the region ${\rm Im}({{x}^{\pm }})\lt 0$ as the bound state momentum $\tilde{p}$ grows. The point where the bound state constituents hit the lines ${\rm Im}({{x}^{\pm }})=0$ corresponds to a critical value of the momentum, after which the bound state constituents move along the lines ${\rm Im}({{x}^{\pm }})=0,$ but in an asymmetric fashion. We have illustrated this by the orange curves in figure 15. In fact, beyond the critical value of momentum there are two possible bound state solutions on the torus; the other solution corresponds to vertically flipping the light-green region. If we insist there should be only one bound state solution in the physical region of the mirror theory, it becomes very natural to consider the lines ${\rm Im}({{x}^{\pm }})=0$ as the boundary of this region. Analyzing multi-particle bound states we would moreover find that there are ${{2}^{Q-1}}$ bound state solutions for a Q-particle bound state on the torus (on top of multiple choices beyond critical values), but that restricting ourselves to ${\rm Im}({{x}^{\pm }})\lt 0$ we find precisely one [31]. Similar comments apply to bound states of the string theory if we interchange the role of $|{{x}^{\pm }}|$ and ${\rm Im}({{x}^{\pm }})$; one of the two possible choices for a two-particle bound state is illustrated by the lime-green curves in figure 15. This leads us to identify the regions $|{{x}^{\pm }}|\gt 1$ and ${\rm Im}({{x}^{\pm }})\lt 0$ as the physical regions of the string and mirror theory respectively⁷⁰ .

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We can cover each of these regions with a simple parametrization in terms of what we will call a rapidity u by introducing the Zhukowski maps

$$\begin{eqnarray}&&x(u)=\frac{1}{2}(u-i\sqrt{4-{{u}^{2}}})\;,\end{eqnarray} \tag{ 3.33 }$$

and

$$\begin{eqnarray}&&{{x}_{s}}(u)=\frac{u}{2}(1+\sqrt{1-\frac{4}{{{u}^{2}}}})\;,\end{eqnarray} \tag{ 3.34 }$$

which are the explicit solutions to the dispersion relation (3.3) upon identifying ${{x}^{\pm }}$ with $x(u\pm i/g)$ or ${{x}_{s}}(u\pm i/g)$ when restricting to ${\rm Im}({{x}^{\pm }})\lt 0$ or $|{{x}^{\pm }}|\gt 1$ respectively. These functions satisfy

$$\begin{eqnarray}&&{{x}_{(s)}}(u)+\frac{1}{{{x}_{(s)}}(u)}=u\;.\end{eqnarray} \tag{ 3.35 }$$

Note that x_s has a branch cut on the interval $(-2,2)$ while x has the complementary one on the real line, and that apart from these $|{{x}_{s}}|\gt 1$ and ${\rm Im}(x)\lt 0$ on the complex u plane. Moreover these functions are equal on the lower half plane and inverse on the upper half plane; they are each others' analytic continuation through their respective branch cuts. We would like to emphasize that the rapidity variable u is more similar to the rapidity variable of a magnon than that of a relativistic particle.

In terms of this rapidity then, the mirror bound state condition ${{x}^{-}}({{\tilde{p}}_{i}})={{x}^{+}}({{\tilde{p}}_{i+1}})$ becomes the condition

$$\begin{eqnarray}&&x({{u}_{i}}-i/g)=x({{u}_{i+1}}+i/g)\;\Rightarrow \;\;{{u}_{i}}={{u}_{i+1}}+2i/g\;,\end{eqnarray} \tag{ 3.36 }$$

exactly taking the form of the Bethe strings of section 2.5.2! Of course, up to an interchange of u_i and ${{u}_{i+1}}$ string bound states also take this form in terms of x_s(u). The energy, momentum, and S-matrices for these bound states can naturally be found by fusion as discussed in section 2. Concretely for example the energy and momentum of Q-particle string and mirror bound states are given by

$$\begin{eqnarray}&&{{\mathcal{E}}_{Q}}(u)=ig({{x}_{s}}(u-iQ/g)-{{x}_{s}}(u+iQ/g))-Q\;,\;\;\;\;\;\;{{p}_{Q}}(u)=-i{\rm log} \frac{{{x}_{s}}(u+iQ/g)}{{{x}_{s}}(u-iQ/g)}\;\end{eqnarray} \tag{ 3.37 }$$

and

$$\begin{eqnarray}&&{{\tilde{\mathcal{E}}}_{Q}}(u)={\rm log} \frac{x(u+iQ/g)}{x(u-iQ/g)}\;,\;\;\;\;\;\;{{\tilde{p}}_{Q}}(u)=g(x(u-iQ/g)-x(u+iQ/g))+i\;\end{eqnarray} \tag{ 3.38 }$$

respectively. The explicit mapping between the rapidity and the physical momentum p is given by

$$\begin{eqnarray}&&u=\frac{1}{g}{\rm cot} \frac{{{p}_{Q}}}{2}\;\sqrt{{{Q}^{2}}+4{{g}^{2}}{{{\rm sin} }^{2}}\frac{{{p}_{Q}}}{2}}\;.\end{eqnarray} \tag{ 3.39 }$$

Within the physical regions we have defined above we can try to define an analogue of the (two-particle) physical strip. The physical string and mirror region overlap on the torus, as do the string and anti-mirror region and vice versa⁷¹ . Clearly, this isolates a smaller region within the string and mirror regions which does not overlap with the (anti-)mirror or (anti-)string regions respectively. These regions correspond to the strip ${\rm Im}(u)\in (-1/g,1/g)$ on the u-plane⁷² . Furthermore, as we will see below the analogue of the shift by $\pm i$ in the relativistic TBA of section 2 is played by a shift of u by $\pm i/g$. For these reasons the strip ${\rm Im}(u)\in (-1/g,1/g)$ deserves a special name, and we will call it our physical or analyticity strip. We will see more of its relevance especially in section 6.

Now that we have understood the scattering theory of the mirror model we can finally analyze its thermodynamics.

Based on this string hypothesis, following the procedure described in section 2 we get the following TBA [32–34] (we work in the conventions of [32])

$$\begin{eqnarray}\begin{array}{rcl} {\rm log} {{Y}_{Q}} & = & -J\;{{\tilde{\mathcal{E}}}_{Q}}+{\rm log} (1+{{Y}_{M}})\star K_{\mathfrak{s}\mathfrak{l}(2)}^{MQ} \\ {} & {} & +\mathop{\sum }\limits_{a}{\rm log} \left( 1+\frac{1}{Y_{M|vw}^{(a)}} \right)\star K_{vwx}^{MQ}+{\rm log} \left( 1-\frac{1}{Y_{\beta }^{(a)}} \right)\hat{\star }K_{\beta }^{yQ}\;, \\ \end{array}\end{eqnarray} \tag{ 3.43 }$$

$$\begin{eqnarray}\begin{array}{rcl} {\rm log} Y_{M|vw}^{(a)} & = & {\rm log} \left( 1+\frac{1}{Y_{N|vw}^{(a)}} \right)\star {{K}_{NM}}+{\rm log} \frac{1-\frac{1}{Y_{-}^{(a)}}}{1-\frac{1}{Y_{+}^{(a)}}}\hat{\star }{{K}_{M}} \\ {} & {} & -{\rm log} (1+{{Y}_{Q}})\star K_{xv}^{QM}\;, \\ \end{array}\end{eqnarray} \tag{ 3.44 }$$

$$\begin{eqnarray}&&\begin{array}{l} {\rm log} Y_{M|w}^{(a)}={\rm log} \left( 1+\frac{1}{Y_{N|w}^{(a)}} \right)\star {{K}_{NM}}+{\rm log} \frac{1-\frac{1}{Y_{-}^{(a)}}}{1-\frac{1}{Y_{+}^{(a)}}}\hat{\star }{{K}_{M}}\;, \\ \end{array}\end{eqnarray} \tag{ 3.45 }$$

$$\begin{eqnarray}&&{\rm log} Y_{\pm }^{(a)}=-{\rm log} \left( 1+{{Y}_{Q}} \right)\star K_{\pm }^{Qy}+{\rm log} \frac{1+\frac{1}{Y_{M|vw}^{(a)}}}{1+\frac{1}{Y_{M|w}^{(a)}}}\star {{K}_{M}}\;,\end{eqnarray} \tag{ 3.46 }$$

where there is an implicit sum over $\beta =\pm$ in the first equation and we note that the fermionic y-particles have chemical potential $i\pi$ (by convention absorbed in the definition of their Y-functions) because the original physical fermions are periodic for for the ground state which has no winding (cf the discussion of section 2.5.1), as well as that the counting function for the y-particles is defined via the inverse of their Bethe equations (3.23)⁷³ . Moreover, just as we defined ${{Y}_{0}}=Y_{f}^{-1}$ in section 2, for future convenience the Y_Q functions are defined inversely from the standard way Y-functions were introduced in section 2. The convolutions in the above equations are defined as

$$\begin{eqnarray}&&f\star h(u,v)=\int _{-\infty }^{\infty }\;{\rm d}t\;f(u,t)h(t,v)\;,\end{eqnarray} \tag{ 3.47 }$$

$$\begin{eqnarray}&&f\;\hat{\star }\;h(u,v)=\int _{-2}^{2}\;{\rm d}t\;f(u,t)h(t,v),\end{eqnarray} \tag{ 3.48 }$$

where the $\hat{\star }$ convolution naturally appears since y-particles only exist on the interval $(-2,2)$. Let us also define the complementary convolution

$$\begin{eqnarray}&&f\;\check{\star }\;h(u,v)=\int _{-\infty }^{-2}\;{\rm d}t\;f(u,t)h(t,v)+\int _{2}^{\infty }\;{\rm d}t\;f(u,t)h(t,v)\;.\end{eqnarray} \tag{ 3.49 }$$

The free of the mirror model gives us the ground state energy of the string as

$$\begin{eqnarray}&&E(L)=-\int {\rm d}u\;\mathop{\sum }\limits_{Q=1}^{\infty }\frac{1}{2\pi }\frac{{\rm d}{{\tilde{p}}^{Q}}}{{\rm d}u}{\rm log} \left( 1+{{Y}_{Q}} \right).\end{eqnarray} \tag{ 3.50 }$$

We can directly read off the kernels and associated S-matrices appearing here from the Bethe–Yang equations just like we did in section 2.5.2; they are concretely listed in appendix A.3.2. By applying various identities involving these kernels summarized in this appendix, these equations can be simplified resulting in the set of equations

$$\begin{eqnarray}&&{\rm log} {{Y}_{1}}=\mathop{\sum }\limits_{a}{\rm log} \left( 1-\frac{1}{Y_{-}^{(a)}} \right)\hat{\star }s-{\rm log} \left( 1+\frac{1}{{{Y}_{2}}} \right)\star s-\check{\Delta }\check{\star }s\;,\end{eqnarray} \tag{ 3.51 }$$

$$\begin{eqnarray}&&{\rm log} {{Y}_{Q}}={\rm log} \frac{{{Y}_{Q+1}}{{Y}_{Q-1}}}{(1+{{Y}_{Q-1}})(1+{{Y}_{Q+1}})}\star s+\mathop{\sum }\limits_{a}{\rm log} \left( 1+\frac{1}{Y_{Q-1|vw}^{(a)}} \right)\star s\;\quad {\rm for}\;Q\gt 1\;,\end{eqnarray} \tag{ 3.52 }$$

$$\begin{eqnarray}\begin{array}{rcl} {\rm log} Y_{M|vw}^{(a)} & = & {\rm log} (1+Y_{M+1|vw}^{(a)})(1+Y_{M-1|vw}^{(a)})\star s-{\rm log} \left( 1+{{Y}_{M+1}} \right)\star s \\ {} & {} & +{{\delta }_{M,1}}{\rm log} \frac{1-Y_{-}^{(a)}}{1-Y_{+}^{(a)}}\hat{\star }s \\ \end{array}\end{eqnarray} \tag{ 3.53 }$$

$$\begin{eqnarray}&&{\rm log} Y_{M|w}^{(a)}={\rm log} (1+Y_{M+1|w}^{(a)})(1+Y_{M-1|w}^{(a)})\star s+{{\delta }_{M,1}}{\rm log} \frac{1-{{\left( Y_{-}^{(a)} \right)}^{-1}}}{1-{{\left( Y_{+}^{(a)} \right)}^{-1}}}\hat{\star }s\;,\end{eqnarray} \tag{ 3.54 }$$

where

$$\begin{eqnarray}\begin{array}{rcl} \check{\Delta } & = & L\check{\mathcal{E}}+\mathop{\sum }\limits_{a}{\rm log} \left( 1-\frac{1}{Y_{-}^{(a)}} \right)\left( 1-\frac{1}{Y_{+}^{(a)}} \right)\hat{\star }\check{K}+2{\rm log} (1+{{Y}_{Q}})\star \check{K}_{Q}^{\Sigma } \\ {} & {} & +\mathop{\sum }\limits_{a}{\rm log} \left( 1-\frac{1}{Y_{M|vw}^{(a)}} \right)\star {{\check{K}}_{M}}+\mathop{\sum }\limits_{a}{\rm log} \left( 1-Y_{k-1|vw}^{(a)} \right)\star {{\check{K}}_{k-1}}\;. \\ \end{array}\end{eqnarray} \tag{ 3.55 }$$

The equations for y-particles cannot be directly put in truly simplified form. However, we can remove the infinite sums of $(v)w$-strings similarly to how we did this for Y_f in section 2.5.2 (see equations (2.138–2.141)). Doing so we obtain their equations in so-called hybrid form [35]

$$\begin{eqnarray}&&{\rm log} \frac{Y_{+}^{(a)}}{Y_{-}^{(a)}}={\rm log} (1+{{Y}_{Q}})\star {{K}_{Qy}}\;,\end{eqnarray} \tag{ 3.56 }$$

$$\begin{eqnarray}&&{\rm log} Y_{-}^{(a)}Y_{+}^{(a)}=-{\rm log} (1+{{Y}_{Q}})\star {{K}_{Q}}+2{\rm log} (1+{{Y}_{Q}})\star K_{xv}^{Q1}\star s+2{\rm log} \frac{1+Y_{1|vw}^{(a)}}{1+Y_{1|w}^{(a)}}\star s\;,\end{eqnarray} \tag{ 3.57 }$$

Let us for completeness note that we can similarly find a set of hybrid equations for Q-particles given by

$$\begin{eqnarray}\begin{array}{rcl} {\rm log} {{Y}_{Q}} & = & \;-L{{\tilde{\mathcal{E}}}_{Q}}+{\rm log} (1+{{Y}_{P}})\star \left( K_{\mathfrak{s}\mathfrak{l}(2)}^{PQ}+2s\star K_{vx}^{P-1,Q} \right) \\ {} & {} & +\mathop{\sum }\limits_{a}\left[ {\rm log} (1+Y_{1|vw}^{(a)})\star s\;\hat{\star }{{K}_{yQ}}+{\rm log} (1+Y_{Q-1|vw}^{(a)})\star s \right. \\ {} & {} & -{\rm log} \frac{1-Y_{-}^{(a)}}{1-Y_{+}^{(a)}}\hat{\star }s\star K_{vwx}^{1Q}+\frac{1}{2}{\rm log} \frac{1-\frac{1}{Y_{-}^{(a)}}}{1-\frac{1}{Y_{+}^{(a)}}}\hat{\star }{{K}_{Q}} \\ {} & {} & \left. +\frac{1}{2}{\rm log} (1-\frac{1}{Y_{-}^{(a)}})(1-\frac{1}{Y_{+}^{(a)}})\hat{\star }{{K}_{yQ}} \right]\;, \\ \end{array}\end{eqnarray} \tag{ 3.58 }$$

which will be useful in particular cases we will encounter later.

To derive the Y-system we should act on the simplified equations with ${{s}^{-1}}$. However, since $\check{\Delta }$ has internal convolutions involving Y-functions, there is no true Y-system for Y₁ outside the interval $(-2,2)$. Explicitly we get the following Y-system

• Q-particles
  $$\begin{eqnarray}\begin{array}{rcl} \frac{Y_{1}^{+}Y_{1}^{-}}{{{Y}_{2}}} & = & \frac{\mathop{\prod }\limits_{a}\left( 1-\frac{1}{Y_{-}^{(a)}} \right)}{1+{{Y}_{2}}}\;{\rm for}\;|u|\lt 2, \\ \frac{Y_{Q}^{+}Y_{Q}^{-}}{{{Y}_{Q+1}}{{Y}_{Q-1}}} & = & \frac{\mathop{\prod }\limits_{a}\left( 1+\frac{1}{Y_{Q-1|vw}^{(a)}} \right)}{(1+{{Y}_{Q-1}})(1+{{Y}_{Q+1}})}, \\ \end{array}\end{eqnarray} \tag{ 3.59 }$$
• vw-strings
  $$\begin{eqnarray}\begin{array}{rcl} Y_{1|vw}^{+}Y_{1|vw}^{-} & = & \;\frac{1+{{Y}_{2|vw}}}{1+{{Y}_{2}}}{{\left( \frac{1-{{Y}_{-}}}{1-{{Y}_{+}}} \right)}^{\theta (2-|u|)}}\;, \\ Y_{M|vw}^{+}Y_{M|vw}^{-} & = & \;\frac{(1+{{Y}_{M-1|vw}})(1+{{Y}_{M+1|vw}})}{1+{{Y}_{M+1}}}\;, \\ \end{array}\end{eqnarray} \tag{ 3.60 }$$
• w-strings
  $$\begin{eqnarray}\begin{array}{rcl} Y_{1|w}^{+}Y_{1|w}^{-} & = & (1+{{Y}_{2|w}}){{\left( \frac{1-Y_{-}^{-1}}{1-Y_{+}^{-1}} \right)}^{\theta (2-|u|)}}, \\ Y_{M|w}^{+}Y_{M|w}^{-} & = & (1+{{Y}_{M-1|w}})(1+{{Y}_{M+1|w}})\;, \\ \end{array}\end{eqnarray} \tag{ 3.61 }$$
• y-particles
  $$\begin{eqnarray}&&Y_{-}^{+}Y_{-}^{-}=\frac{1+{{Y}_{1|vw}}}{1+{{Y}_{1|w}}}\frac{1}{1+{{Y}_{1}}}\;.\end{eqnarray} \tag{ 3.62 }$$

We can find this equation by acting with ${{s}^{-1}}$ on the hybrid equations (3.56) and (3.57) and using the identity⁷⁴

$$\begin{eqnarray}&&({{K}_{Qy}}+{{K}_{Q}})\circ {{s}^{-1}}=2K_{xv}^{Q1}+2{{\delta }_{Q1}}\;.\end{eqnarray} \tag{ 3.63 }$$

It is not possible to reduce the TBA equations for ${{Y}_{+}}$ to a local Y-system form [32]. Still there is something interesting to be said about ${{Y}_{+}}$. From the TBA equations we can easily deduce that the Y-functions have cuts on the u-plane by analyzing the cut structure of the integration kernels. In particular the fermionic Y-functions ${{Y}_{\pm }}$ have cuts on the real line on the complement of the interval $(-2,2)$. At the same time from equation (3.57) we can see that their product has no cut on the real line, while by equation (3.56) their ratio has precisely the same cut structure and flips sign across the cut; put together we see that ${{Y}_{+}}$ and ${{Y}_{-}}$ are each others analytic continuation through this cut.

We should note that before the ground state TBA equations were found, there was already a conjecture for the Y-system that should describe the spectrum of the light-cone superstring [114]. This Y-system is of the usual universal form

$$\begin{eqnarray}&&Y_{a,s}^{+}Y_{a,s}^{-}=\frac{(1+{{Y}_{a,s+1}})(1+{{Y}_{a,s-1}})}{\left( 1+\frac{1}{{{Y}_{a+1,s}}} \right)\left( 1+\frac{1}{{{Y}_{a-1,s}}} \right)}\;,\end{eqnarray} \tag{ 3.64 }$$

and the Y-functions lie on the so-called T-hook lattice [114] illustrated in figure 16, which precisely corresponds to the structure of the TBA equations above. Indeed, provided we make the identification

$$\begin{eqnarray}&&\begin{array}{l} Y_{M|vw}^{(l)}=1/{{Y}_{M+1,-1}}\;,\quad {{Y}_{Q}}={{Y}_{Q,0}}\quad Y_{M|vw}^{(r)}=1/{{Y}_{M+1,1}}\;, \\ Y_{+}^{(l)}={{\left( -1 \right)}^{m+1}}{{Y}_{2,-2}}\;,\quad Y_{-}^{(l)}={{\left( -1 \right)}^{m+1}}/{{Y}_{1,-1}}\;,\quad Y_{+}^{(r)}={{\left( -1 \right)}^{m+1}}{{Y}_{2,2}}\;, \\ Y_{M|w}^{(l)}={{Y}_{1,M+1}}\;,\quad Y_{-}^{(r)}={{\left( -1 \right)}^{m+1}}/{{Y}_{1,1}}\;,\quad Y_{M|w}^{(r)}={{Y}_{1,-(M+1)}}\;, \\ \end{array}\end{eqnarray} \tag{ 3.65 }$$

we find that the Y-system derived from our ground state TBA equations agrees with the conjectured Y-system of [114] for $u\in (-2,2)$, and with the exception that the $Y_{+}^{(a)}$ (${{{\rm Y}}_{2,\pm 2}}$) functions have no local Y-system equation at all.

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From the ground state TBA equations we can determine that for $u\in (-\infty ,-2)\cup (2,\infty )$ the Y-functions have branch cuts at

$$\begin{eqnarray}&&\begin{array}{l} Y_{\pm }^{(a)}(u+i2m/g),\quad m\in \mathbb{Z}\;, \\ {{Y}_{M}}(u\pm i(M+2n)/g)\;,\;Y_{M|vw}^{(a)}(u\pm i(M+2n)/g)\;,\;Y_{M|w}^{(a)}(u\pm i(M+2n)/g)\;,n\in \mathbb{N}\;, \\ \end{array}\end{eqnarray} \tag{ 3.66 }$$

and find their discontinuities. In other words, the Y-functions have branch cuts spaced by $2i/g$ outward from the real line, starting from $\pm iM/g$, where M is their index which we can define as 0 for ${{Y}_{\pm }}$. The precise discontinuities of the Y-functions across these cuts are explicitly given in [115], where it was also shown that (only) when supplemented with these discontinuity relations and conditions on the asymptotics of Y₁ the Y-system is equivalent to the TBA equations and can be explicitly 'integrated' to find them⁷⁵ .

It will also be useful for us to map the Y-system equations (3.64) to the so-called Hirota equations, or T-system,

$$\begin{eqnarray}&&T_{a,s}^{+}T_{a,s}^{-}={{T}_{a+1,s}}{{T}_{a-1,s}}+{{T}_{a,s+1}}{{T}_{a,s-1}}\;,\end{eqnarray} \tag{ 3.67 }$$

which we can do via the identification [89, 116, 117]

$$\begin{eqnarray}&&{{Y}_{a,s}}=\frac{{{T}_{a,s+1}}{{T}_{a,s-1}}}{{{T}_{a+1,s}}{{T}_{a-1,s}}}\;.\end{eqnarray} \tag{ 3.68 }$$

However, we should note that the T-system has a 'gauge' invariance

$$\begin{eqnarray}&&{{T}_{a,s}}\to g_{1}^{[a+s]}g_{2}^{[a-s]}g_{3}^{[-a+s]}g_{4}^{[-a-s]}{{T}_{a,s}}\;,\end{eqnarray} \tag{ 3.69 }$$

where ${{f}^{[m]}}(u)=f(u+im/g)$, while the Y-functions are gauge invariant quantities. In terms of our parametrization, the mapping between T-functions and Y-functions is

$$\begin{eqnarray}\begin{array}{rcl} Y_{M|vw}^{(l)}=\frac{{{T}_{M+2,-1}}{{T}_{M,-1}}}{{{T}_{M+1,0}}{{T}_{M+1,-2}}}\;, & {{Y}_{Q}}=\frac{{{T}_{Q,1}}{{T}_{Q,-1}}}{{{T}_{Q+1,0}}{{T}_{Q-1,0}}}\;, & Y_{M|vw}^{(r)}=\frac{{{T}_{M+2,1}}{{T}_{M,1}}}{{{T}_{M+1,0}}{{T}_{M+1,2}}}\;, \\ Y_{+}^{(l)}=-\frac{{{T}_{2,-1}}{{T}_{2,-3}}}{{{T}_{1,-2}}{{T}_{1,-3}}}\;, & Y_{-}^{(l)}=-\frac{{{T}_{0,-1}}{{T}_{2,-1}}}{{{T}_{1,0}}{{T}_{1,-2}}}\;, & Y_{+}^{(r)}=-\frac{{{T}_{2,1}}{{T}_{2,3}}}{{{T}_{1,2}}{{T}_{1,3}}}\;, \\ Y_{M|w}^{(l)}=\frac{{{T}_{1,-(M+2)}}{{T}_{1,-M}}}{{{T}_{0,-(M+1)}}{{T}_{2,-(M+1)}}}\;, & Y_{-}^{(r)}=-\frac{{{T}_{0,1}}{{T}_{2,1}}}{{{T}_{1,0}}{{T}_{1,2}}}\;, & Y_{M|w}^{(r)}=\frac{{{T}_{1,M+2}}{{T}_{1,M}}}{{{T}_{0,M+1}}{{T}_{2,M+1}}}\;. \\ \end{array}\end{eqnarray} \tag{ 3.70 }$$

These T-functions live on the gray fat T-hook of figure 16. Up to first non-trivial order at large string tension, this T-system has been derived directly from the string sigma model as well, by identifying the T-functions directly as transfer matrices and using the pure spinor formalism [118].

The TBA equations of this section allow us to compute the ground state energy of the light-cone superstring. Since the ground state is a half-BPS state its energy is protected and in fact zero. In light of the energy formula (3.50) the corresponding solution of the TBA equations should have Y_Q = 0 [119]; we will discuss the corresponding solution in the generalized setting of twisted theories in the next section. The next natural question is how to generalize these equations to excited states, as we did not go through all these details just to confirm that our ground state indeed has zero energy. Because of the complicated structure of these equations and lack of simple analytic results to compare to it does not seem very promising to attempt direct analytic continuation in the spirit of section 2.5.3. Fortunately, by the time these equations were developed there already existed a natural candidate solution for these equations in the limit ${{Y}_{Q}}\sim 0$.

Let us now come to the TBA description of excited states. We argued in section 2.5.3 that the excited state TBA equations should differ from the ground state ones by a set of driving terms, but should otherwise be of the exact same form. Considering the energy formula (3.50) in this light, we realize that at weak coupling or very large J the Y_Q functions should be small due to the $-J\tilde{\mathcal{E}}$ term in their canonical TBA equations. Expanding the energy formula for small Y_Q and comparing this to the leading weak coupling correction (3.76) it is natural to identify

$$\begin{eqnarray}&&Y_{Q}^{o}={{\Upsilon }_{Q}}\;T_{Q,1}^{(l)}(v|\{{{u}_{j}}\})T_{Q,1}^{(r)}(v|\{{{u}_{j}}\})\;,\end{eqnarray} \tag{ 3.80 }$$

where

$$\begin{eqnarray}&&{{\Upsilon }_{Q}}\equiv {{{\rm e}}^{-J{{\tilde{\mathcal{E}}}_{Q}}}}\mathop{\prod }\limits_{i=1}^{{{K}^{{\rm I}}}}S_{\mathfrak{s}\mathfrak{l}(2)}^{Q{{1}_{*}}}(v,{{u}_{i}}),\end{eqnarray} \tag{ 3.81 }$$

and we now use an asterix * to denote that the corresponding rapidity lies in the string region to avoid confusion. We have put a superscript o on Y_Q to indicate this holds for small Y_Q. This naturally fits with the fact that the Y-system in the limit of small Y_Q naturally decouples in two wings which each have a solution in terms of the $\mathfrak{p}\mathfrak{s}{{\mathfrak{u}}_{c.e.}}(2|2)$ transfer matrices [114], and in addition tells us we should take the 'string-mirror' transfer matrix as the appropriate solution rather than for example the 'string-string' transfer matrix. Concretely, from equation (3.80) we can find that the solution of the Y-system in the small Y_Q limit is [114, 123]

$$\begin{eqnarray}&&Y_{M|vw}^{(a)o}=\frac{T_{M+2,1}^{(a)}T_{M,1}^{(a)}}{T_{M+1,2}^{(a)}}\;,\quad Y_{M|w}^{(a)o}=\frac{T_{1,M+2}^{(a)}T_{1,M}^{(a)}}{T_{2,M+1}^{(a)}},\end{eqnarray} \tag{ 3.82 }$$

$$\begin{eqnarray}&&Y_{+}^{(a)o}=-\frac{T_{2,1}^{(a)}T_{2,3}^{(a)}}{T_{1,2}^{(a)}T_{1,3}^{(a)}}\;,Y_{-}^{(a)o}=-\frac{T_{2,1}^{(a)}}{T_{1,2}^{(a)}}\;.\end{eqnarray} \tag{ 3.83 }$$

where the $T_{Q,M}^{(a)}$ are given in terms of the transfer matrix $T_{Q,1}^{(a)}$ via the Bazhanov–Reshetikhin formula [124]⁸⁰

$$\begin{eqnarray}&&{{T}_{a,s}}(u)=\begin{array}{ccccccccccccccc} {\rm det} \\ 1\leqslant m,n\leqslant s \\ \end{array}{{T}_{a+m-n,1}}(u+i(s+1-m-n)/g).\end{eqnarray} \tag{ 3.84 }$$

In fact by a redefinition of the general T functions we can explicitly put a factor of ${{\Upsilon }_{Q}}$ in the parametrization (3.70) for Y_Q without changing the parametrization of the other Y-functions⁸¹ . Doing so in the asymptotic limit the ${{T}_{Q,\pm 1}}$ directly reduce to the transfer matrices $T_{Q,1}^{(r/l)}$ while the ${{T}_{Q,0}}$ reduce to one.

Strictly speaking we have now argued what the leading order weak coupling and leading order large volume expression for the Y-functions is⁸² . However, in weak coupling applications it is natural to practically consider the above equations without expanding them in g as long as the resulting Y_Q functions are still numerically small. One reason for doing so is the following. Based on experience in other integrable models it is natural to expect that on the exact solution of the TBA equations ${{Y}_{1}}=-1$ when evaluated at the exact rapidity of a particle in the excited state under consideration. In this case this means we should analytically continue Y₁ which has its argument in the mirror region, to the string region where the physical rapidity lies. This means that the proper finite volume analogue of the Bethe–Yang equation for an excited state should be the exact Bethe equation

$$\begin{eqnarray}&&Y_{1}^{*}({{u}_{j}})=-1\;.\end{eqnarray} \tag{ 3.85 }$$

If we consider this at the level of the asymptotic formula (3.80) we see [114] that this is indeed precisely equivalent to the Bethe–Yang equations (cf the general equation (2.34))

$$\begin{eqnarray}&&Y_{1}^{*}({{u}_{j}})=-1\leftrightarrow {{{\rm e}}^{i{{p}_{j}}J}}\Lambda ({{u}_{j}}|\{{{u}_{i}}\})=-1\;,\end{eqnarray} \tag{ 3.86 }$$

at any value of g.

In fact equation (3.85) is very natural indeed from the point of view of section 2.5.3; the points where ${{Y}_{1}}=-1$ are singular points that can lead to driving terms, and including these driving terms appropriately precisely results in an energy formula of the form

$$\begin{eqnarray}&&E=\mathop{\sum }\limits_{i=1}^{{{K}^{{\rm I}}}}\mathcal{E}({{p}_{i}})-\int {\rm d}u\;\mathop{\sum }\limits_{Q=1}^{\infty }\frac{1}{2\pi }\frac{{\rm d}{{\tilde{p}}^{Q}}}{{\rm d}u}{\rm log} \left( 1+{{Y}_{Q}} \right),\end{eqnarray} \tag{ 3.87 }$$

where $\mathcal{E}({{p}_{i}})$ is the asymptotic energy of the ith particle (recall that $\tilde{p}$ evaluated on string momenta is just $-i\mathcal{E}$). Of course there can be further modifications of this energy formula as we will see in section 6.

Let us now briefly come back to the contour deformation trick described in section 2.5.3. Essentially the generalized Lüscher formulae of [41] tell us what form solution of the excited state TBA equations in the asymptotic limit ${{Y}_{Q}}\sim 0$ is supposed to take. Concretely we can readily construct the asymptotic Y-functions of equations (3.80) and (3.82) via the eigenvalues of the transfer matrix (3.78). As discussed we can naturally assume that the excited state TBA equations differ from the ground state ones only in their integration contours, and by substituting the asymptotic Y-functions in the asymptotic limit of the TBA equations we can find particular contours for a given excited state such that the equations are satisfied. As a net effect, upon taking the integration contours back to the real line we find a set of driving terms in the TBA equations which are such that they are solved by the asymptotic Y-functions in the asymptotic limit. If we finally assume that the solution to these equations can be smoothly deformed away from the asymptotic one, we end up with a concrete set of excited state TBA equations as well as a concrete starting point for their solution at weak coupling.

The driving terms we introduce this way depend on the location of the singular points typically specified by $Y=-1$,⁸³ which can in principle exist for any of the Y-functions, not just the explicit example of Y₁ at the physical rapidities. This means that beyond the asymptotic limit where we can explicitly determine these numerically from the asymptotic solution, we should impose the 'quantization conditions' or exact Bethe equations

$$\begin{eqnarray}&&Y(r)=-1\;,\end{eqnarray} \tag{ 3.88 }$$

on any relevant Y-function with corresponding root r, through its TBA equation. Of course these roots generically move as we change the coupling. In particular there is the possibility that such roots (whether having already induced a driving term or not) can move through the integration contour of our equations at some finite value of g, leading to pictures very similar to figure 13 of the original analytic continuation story. If this happens, the explicit form of the driving terms (now for a fixed excited state) clearly changes as the coupling crosses this so-called critical value. We can of course explicitly investigate whether the roots of the asymptotic solution show this behavior as we increase the coupling constant, but since the asymptotic solution becomes less and less appropriate as we do so it is not clear that this gives a conclusive answer. Still, if we take the asymptotic properties to be truly representative of the exact solution this should give the qualitative answer⁸⁴ .

We should mention that as an alternative to the contour deformation trick we could explicitly 'integrate' the Y-system for a excited states similar to how it is done for the ground state. We can do this by assuming that the discontinuity relations of [115] universally apply to any state in our theory, and also here assuming that the analytic properties of the asymptotic Y-functions are representative of the exact solution. Carefully taking into account the effect of singular points, this approach explicitly reproduces the excited state TBA equations found by the contour deformation trick, at least in the $\mathfrak{s}\mathfrak{l}(2)$ sector [128]. Given the typical types of singular points of Y-functions in the $\mathfrak{s}\mathfrak{l}(2)$ sector the two approaches are basically identical at a practical level. For more complicated states however, this is no longer clearly the case.

As mentioned in the introduction, this whole procedure has most notably been successfully applied to the Konishi state. We will discuss this procedure for various other states in sections 5 and 6. While we are using the TBA to find scaling dimensions of trace operators, we should also mention that more recently the TBA approach has also been applied to Wilson loops in $\mathcal{N}=4$ SYM [129, 130], and to baryonic (determinant-type) operators [131, 132]. In both cases the basic structure of the TBA equations is essentially identical to the one we are describing here, while the source terms are complicated.

In section 2 we mentioned 'nonlinear integral equations' [89, 93–97] as a possible alternative to the TBA approach. Of course, even if it is not possible to derive a set of functional relations required as input for this approach (such as a Y-system) directly, we may derive it from the TBA equations and use this as a starting point. This way we may be able to reexpress the TBA equations (Y-system) in a way that is practically very useful⁸⁵ . Although quite involved this has been worked out for the superstring TBA [134, 135] (see also [125, 136] and [137]), at least for states in the $\mathfrak{s}\mathfrak{l}(2)$ sector⁸⁶ . Thus far these equations have been used particularly efficiently at weak coupling to find the anomalous dimension of the Konishi operator analytically up to eight-loop order [139, 140], matching exactly with generalized Lüscher results available up to seven loops [141]⁸⁷ . Very recently this approach was refined further, resulting in a so-called quantum spectral curve or $P\mu$-system [142], reformulating the problem as a Riemann–Hilbert-type problem. With a tiny bit of external help, with this approach even the nine-loop anomalous dimension for the Konishi operator can be computed in the space of a single talk [143].

As the final point of this section we should address the symmetries of our model, and how they arise at the level of the spectrum. The superstring originally has ${\rm PSU}(2,2|4)$ symmetry, of which a manifest ${\rm PSU}{{(2|2)}^{2}}$ is left at the level of the light-cone S-matrix and hence the above TBA equations. Of course the spectrum should be ${\rm PSU}(2,2|4)$ invariant and so the missing symmetry should be hiding somewhere in our equations. In fact, they are built into the TBA equations by construction [123]. We already discussed in section 2 that the Bethe ansatz describes highest weight states, or in other words describes whole multiplets at a time. In the ${\rm Ad}{{{\rm S}}_{5}}\times {{{\rm S}}^{5}}$ context this means that the Bethe–Yang equations describe entire superconformal multiplets in one go. Now since we build our excited state TBA equations based on the asymptotic solution which has this extra symmetry, they will necessarily have this same symmetry. One curious consequence of this fact is that because supersymmetry transformations can change the J-charge of a state, if we happen to consider a descendant instead of a superconformal primary, the length J that enters the TBA equations of the descendant is not its J-charge, but rather the maximal J-charge within its multiplet. Still, we can construct the asymptotic solution based on any state in the multiplet as the change simply corresponds to a gauge transformation on the asymptotic T-functions (transfer matrices) and leaves the Y-functions invariant⁸⁸ .

In the next section we will consider various deformations of our superstring background that result in theories with reduced (super)symmetry. In line with the above discussion, the reduction of symmetry is of course appropriately reflected in the asymptotic Bethe equations and asymptotic Y-functions, and consequently the TBA.

Starting with a (super-)string theory on (super-)space-time $\mathcal{M}$ and the action of a discrete group $\Gamma$ on $\mathcal{M}$, we can orbifold the original string theory by $\Gamma$ and obtain a string theory on $\mathcal{M}/\Gamma$ [144]. The orbifolded background is described in terms of the fields $X\in \mathcal{M}$ of the original string theory as

$$\begin{eqnarray}&&X(\tau ,\sigma )\simeq gX(\tau ,\sigma ),\;g\in \Gamma \;.\end{eqnarray} \tag{ 4.1 }$$

Were we considering point particles on this space, we would take the Hilbert space of the original model, and project it onto its $\Gamma$ invariant subspace. However in the case of strings we have more structure; because of the orbifold equivalence (4.1) the string does not have to close on itself. Rather it can also close modulo an orbifold group element

$$\begin{eqnarray}&&X(\tau ,2\pi )\simeq hX(\tau ,0)\;,\;\;h\in \Gamma \;.\end{eqnarray} \tag{ 4.2 }$$

In the orbifolded theory, we need to consider all independent sectors of this type, which are known as twisted sectors. These twisted sectors are in one to one correspondence with the conjugacy classes of $\Gamma$, as follows from (4.1), (4.2). As such, in each twisted sector we should project onto its $\Gamma -[g]$ invariant subspace, where $[g]$ is the corresponding conjugacy class. In other words, we should project on the subspace invariant under the stabilizer of $g$, Γ_g.

For the ${\rm Ad}{{{\rm S}}_{5}}\times {{{\rm S}}^{5}}$ superstring, this means we want to consider strings on ${\rm Ad}{{{\rm S}}_{5}}/{{\Gamma }^{a}}\times {{{\rm S}}^{5}}/{{\Gamma }^{s}}$ by following these considerations. Let us start by discussing the boundary conditions obtained by orbifolding the sphere.

The possible orbifolds of the sphere are obtained by the action of the discrete subgroups of ${\rm SU}(4)$. Fortunately enough, these have been classified completely in [145]. Of course, all elements of the abelian subgroups can be simultaneously diagonalized, and hence immediately put into correspondence with elements of the Cartan subgroup of ${\rm SU}(4)$. This simply corresponds to one and the same field redefinition in every twisted sector; explicitly

$$\begin{eqnarray}&&X(\tau ,\sigma )\to \tilde{X}(\tau ,\sigma )=gX(\tau ,\sigma ),\;\;g\in {\rm SU}(4)\;,\end{eqnarray} \tag{ 4.3 }$$

such that each of the representatives h_k of the conjugacy classes is diagonalized

$$\begin{eqnarray}&&\tilde{X}(\tau ,2\pi )\simeq {{d}_{k}}\tilde{X}(\tau ,0)\;,\end{eqnarray} \tag{ 4.4 }$$

where d_k is in the Cartan subgroup of ${\rm SU}(4)$. Simultaneous diagonalization is not possible within a non-abelian subgroup; still we can diagonalize the representative conjugacy class element in each twisted sector by an independent field redefinition, as already used in this context in [146]. Picking up a different basis in each twisted sector is not a problem since each sector is closed⁹⁰ . In short, in every twisted sector we can restrict ourselves to boundary conditions corresponding to elements of the Cartan subgroup of ${\rm SU}(4)$; boundary conditions on the isometry angles of the sphere. Compared to abelian orbifolds, the only complication for non-abelian orbifolds is in the bookkeeping of the twisted sectors and corresponding orbifold invariance as clearly explained in [146]; there also the quiver structure of these orbifolds is carefully worked out.

The string theory constructed in this fashion is dual to an orbifolded version of $\mathcal{N}=4$ SYM, with field content $\mathcal{G}/\Gamma$ where $\mathcal{G}$ denotes the field content of $\mathcal{N}=4$ SYM. Correspondingly, $\Gamma$ is a discrete subgroup of the $R$-symmetry group of the field theory, meaning that these orbifolds will break a certain amount of supersymmetry. The amount of supersymmetry that is preserved after orbifolding can be found by considering the embedding of the orbifold in ${\rm SU}(4)$, in other words finding the residual $R$-symmetry; we obtain an $\mathcal{N}=2$ theory if $\Gamma \subset {\rm SU}(2)$, whereas $\Gamma \subset {\rm SU}(3)$ gives us an $\mathcal{N}=1$ theory. Of course, taking $\Gamma \subset {\rm SU}(4)$ gives a non-supersymmetric theory.

On the string theory side we can expect tachyons to be present in the non-supersymmetric theories, and as a result to suffer from instabilities [147–149] related to tachyon condensation⁹¹ . The corresponding vacuum expectation values of the dual twisted operators break the quantum orbifold symmetry [149, 150]⁹² . Apart from the instabilities themselves, this also presents us with a practical problem; without orbifold symmetry the Hilbert space of the theory formally no longer necessarily decomposes over the various twisted sectors. For non-abelian orbifolds this would greatly complicate the problem by preventing us from choosing a different field basis in each twisted sector. We might wish to disregard such theories and focus on orbifolds with $\Gamma \subset {\rm SU}(3)$ at most. However, there are no practical obstacles in studying non-supersymmetric abelian orbifolds. In line with this, non-supersymmetric abelian orbifolds of $\mathcal{N}=4$ SYM have been considered before [151], while non-abelian ones have not [146]. Moreover, it may in fact be possible to use the exact description of the spectrum we will develop below to get some concrete (quantitative) insight into the presence and effects of tachyons in these orbifold theories. As we already mentioned, in certain cases part of the supersymmetry can be restored by simultaneously orbifolding ${\rm Ad}{{{\rm S}}_{5}}$.

Exactly as we did for the sphere we can consider orbifolds of ${\rm Ad}{{{\rm S}}_{5}}$. However, since ${\rm Ad}{{{\rm S}}_{5}}$ contains the time-like direction of our background space-time we should proceed cautiously to avoid possible time-like orbifolds. With this restriction in mind, we are left to consider orbifolds by $\Gamma \subset {\rm SU}(2)\times {\rm SU}(2)\subset {\rm SU}(2,2)$. For these orbifolds we can proceed as we did for the sphere, again reducing the boundary conditions to those corresponding to diagonal elements of ${\rm SU}(2)\times {\rm SU}(2)$ in each twisted sector. Rather than breaking supersymmetry, now we break part of the conformal symmetry. From the point of view of integrability of the string theory this is no real objection as we will see below⁹³ . While the details of the corresponding dual field theories have not been worked out in full detail, at the level of the Bethe ansatz describing the spectrum of the dilatation operator such orbifolds can also be readily accounted for [151]. Naturally it is possible to simultaneously orbifold ${\rm Ad}{{{\rm S}}_{5}}$ and ${{{\rm S}}^{5}}$ in any of the ways we discussed.

The second type of deformation we can apply to our background is a sequence of so-called TsT transformations which can be applied to backgrounds with more than one commuting ${\rm U}(1)$ isometries. A TsT transformation takes a pair of angles corresponding to two isometry directions, say $({{\phi }_{1}},{{\phi }_{2}})$, and acts on them via a $T$-duality along the ϕ₁ direction, followed by a shift in the ϕ₂ direction⁹⁴ , ${{\phi }_{2}}\to {{\phi }_{2}}+\hat{\gamma }{{\tilde{\phi }}_{1}}$,⁹⁵ and finally $T$-dualizing back in the ${{\tilde{\phi }}_{1}}$ direction, where ${{\tilde{\phi }}_{1}}$ is the $T$-dual variable to ϕ₁. By sequentially applying TsT transformations on a background with $d$ commuting ${\rm U}(1)$ isometries we can obtain a $d(d-1)/2$ parametric deformation. Note that these transformations require all other fields to be uncharged under the relevant ${\rm U}(1)$s, specifically in this context the fermions. Fortunately they can be discharged by a field redefinition [152].

Very important for our present considerations, the resulting equations of motion are in one-to-one correspondence with the untransformed equations of motion with twisted boundary conditions imposed on the ${\rm U}(1)$ isometry fields [153]. In fact, since by definition these ${\rm U}(1)$ isometries are nothing but elements of the Cartan subgroup of the isometry group of ${\rm Ad}{{{\rm S}}_{5}}\times {{{\rm S}}^{5}}$, it should not be surprising that the resulting boundary conditions can be put into correspondence with elements of the Cartan subgroup⁹⁶ , exactly as we could for orbifolds. Let us now briefly discuss the various possible TsT transformations.

Deformations of the sphere are very sensible from the point of view of AdS/CFT. The primary example of a deformation that can be achieved through TsT transformations is the Lunin–Maldacena background [154] for real β, with β-deformed SYM as its field theory dual. This deformation preserves $\mathcal{N}=1$ supersymmetry. Restricted to the sphere, a generic sequence of TsT transformations gives a three parameter deformed background [153] (see also [155, 156]), which we will refer to as a γ-deformed background. The corresponding dual field theories in general have no supersymmetry.

To describe string propagation on these backgrounds through the original theory, we have to implement the following boundary conditions on the three angles on the sphere

$$\begin{eqnarray}&&{{\phi }_{i}}(2\pi )-{{\phi }_{i}}(0)=2\pi ({{n}_{i}}-{{\epsilon }_{ijk}}{{\gamma }_{j}}{{J}_{k}}),\;\;\;{{n}_{i}}\in \mathbb{Z}\;.\end{eqnarray} \tag{ 4.5 }$$

Here the J_k are the angular momenta of S⁵ and γ_j the three parameters of the deformation.

Just as we discussed for orbifolds above, without supersymmetry various subtleties can enter the game, and it appears they do; conformal symmetry of the generically γ-deformed non-supersymmetric proposed dual field theories [153, 156] is broken at the quantum level [157], despite previous results [158]. From the point of view of AdS/CFT this can have various interpretations briefly discussed in [157]. One interesting possibility closely related to the situation for non-supersymmetric orbifolds is that the resulting non-supersymmetric string background is unstable due to the presence of tachyons (which are at least manifestly present in string theory on γ-deformed flat space [159]). As in the orbifold case, here we may similarly hope to get some concrete insight with an exact description of the string spectrum on these backgrounds. Also, the effects of this non-conformality appears to be limited to particular subsets of states in the theory [157, 160], outside of which we can still talk sensibly of scaling dimensions. Indeed, outside of these subsets, explicit results [161, 162] obtained in the string theory picture we are describing here can again be successfully matched with perturbative gauge theory [160]. Finally, as far as adapting the boundary conditions of our integrable string is concerned there are of course no immediate issues.

Even when we preserve some supersymmetry there is still a subtlety to be mentioned. The asymptotic Bethe ansatz for β-deformed SYM [156, 163] is constructed based on the dilatation operator density for the ${\rm U}(N)$ gauge theory. In the case of $\mathcal{N}=4$ there is no difference between the dilatation operators of the ${\rm U}(N)$ and ${\rm SU}(N)$ theories, but for β-deformations there is a difference for certain short operators [156, 164, 165]⁹⁷ . This turns out to make the asymptotic Bethe ansatz more asymptotic than in $\mathcal{N}=4$ SYM, meaning there exists what has been dubbed a pre-wrapping effect [164] accounting for the difference between the ${\rm U}(N)$ and ${\rm SU}(N)$ theory, entering one loop order before conventional wrapping corrections. Since only the ${\rm SU}(N)$ theory is conformal [167], we may hope that the dual string theory knows about this effect as it is expected to describe the ${\rm SU}(N)$ theory [156]. We will briefly come back to this point in the next section, when we consider the operator ${\rm Tr}(XZ)$ which is affected by pre-wrapping.

Less studied but clearly possible from the point of view of string theory, we can do TsT transformations on the three commuting ${\rm U}(1)$ isometries of anti-de Sitter space [168, 169]. However, as always we should be proceed cautiously when time-like directions are involved. Also, while they give sensible string backgrounds these deformations have a less clear interpretation from the point of view of AdS/CFT.

Let us briefly address the issues regarding TsT transformations involving ${\rm Ad}{{{\rm S}}_{5}}$, starting with issues regarding TsT transformations involving time-like directions. First of all, strictly speaking our string theory is defined on the universal cover of ${\rm Ad}{{{\rm S}}_{5}}$ which does not have a compact time direction, such that $T$-dualizing in this direction is not an option. Secondly, if we do a TsT transformation involving the time direction we will generate space-time directions with mixed signature. To circumvent the first problem we could initially consider TsT transformations on the space-time, and only subsequently pass to its universal cover. Also, the second problem can pragmatically be avoided by restricting study of the corresponding string theory to regions of the geometry where the signature is the same as the parent theory [168]. In fact, we could proceed in the spirit of [168] and formally apply a general sequence of TsT transformations, not necessarily viewing the resulting geometry as a deformation of any particular parent geometry but rather studying it at face value.

Furthermore, any TsT transformation involving an angle from ${\rm Ad}{{{\rm S}}_{5}}$ necessarily breaks part of the conformal symmetry. The corresponding free string theory should be sensible, even integrable, but again the effect of breaking conformal symmetry on the dual field theory is not obvious. By analogy to how β-deformed SYM can be obtained by introducing a ⋆-product related to the deformation in the $R$-symmetry group, the duals are expected to be certain non-commutative dual field theories [163, 168]. Despite these difficulties, TsT transformations of this type have been investigated in the setting of AdS/CFT. In fact as far as integrability is concerned we can readily implement any of these deformations, so while keeping the above concerns in mind we will discuss generic TsT transformations of ${\rm Ad}{{{\rm S}}_{5}}\times {{{\rm S}}^{5}}$. As we can expect, the interesting features of these deformations turn to rather peculiar ones as soon as time-like directions are involved.

The boundary conditions corresponding to the most generic sequence of TsT transformations are given by

$$\begin{eqnarray}&&{{\phi }_{i}}(2\pi )-{{\phi }_{i}}(0)=2\pi ({{n}_{i}}+{{\gamma }_{ik}}{{J}_{k}})\;,\;\;\;{{n}_{i}}\in \mathbb{Z}\;\end{eqnarray} \tag{ 4.6 }$$

where $i$ is a generalized index labeling the six ${\rm U}(1)$ isometry fields of ${\rm Ad}{{{\rm S}}_{5}}\times {{{\rm S}}^{5}}$, J_i the corresponding angular momentum and γ_ij an antisymmetric six by six matrix containing the 15 deformations parameters. Clearly we can also combine orbifolds and TsT transformations [151].

We have seen that strings on both orbifolds of, and TsT transformed ${\rm Ad}{{{\rm S}}_{5}}\times {{{\rm S}}^{5}}$ can be directly described through the original string theory with modified boundary conditions. Let us parametrize a completely generic deformation of the above type by the boundary conditions

$$\begin{eqnarray}&&{{\psi }_{i}}(2\pi )={{\psi }_{i}}(0)+{{\alpha }_{i}}\;,\;\;\;{{\phi }_{j}}(2\pi )={{\phi }_{j}}(0)+{{\vartheta }_{i}}\;,\end{eqnarray} \tag{ 4.7 }$$

where ${{\psi }_{1}}\equiv t$ and ${{\phi }_{1}}\equiv \phi$ with associated Noether charges $E$ and $J$ respectively, and ${{\psi }_{2,3}}$ and ${{\phi }_{2,3}}$ are the remaining ${\rm U}(1)$ isometry fields in ${\rm Ad}{{{\rm S}}_{5}}$ and ${{{\rm S}}^{5}}$ respectively. The boundary condition on $t$ in the form of α₁ is a formal bookkeeping device and is not meant to be interpreted in a direct sense; we introduce it for uniformity since it can appear for general TsT transformed backgrounds cf (4.6) with all their caveats, for orbifolds it is absent. We would like to explicitly mention here that the parameters α_i and ϑ_i can be taken between zero and $2\pi$ since only their values modulo $2\pi$ have physical relevance.

Up to now the boundary conditions on the isometry fields were all treated on an equal footing, but this changes when fixing a light-cone gauge. Fixing a light-cone gauge breaks the manifest ${\rm SU}(2,2)\times {\rm SU}(4)$ bosonic symmetry of the superstring to ${\rm SU}{{(2)}^{4}}$ by fixing the combination ${{{\rm x}}^{+}}\equiv t+\phi$ to be equal to the proper time parametrizing the propagation of the superstring. Naturally, the information on the non-trivial boundary conditions on $t$ and ϕ is far from lost; we get the so-called level matching condition from the boundary conditions on ${{{\rm x}}^{-}}\equiv t-\phi$

$$\begin{eqnarray}&&{{p}_{ws}}=\int _{0}^{2\pi }{\rm d}\sigma {{x}^{-^{\prime} }}={{\alpha }_{1}}-{{\vartheta }_{1}}+2\pi m\;,\end{eqnarray} \tag{ 4.8 }$$

where $m$ is the winding number of the string⁹⁸ . We would like to emphasize that p_ws is the world-sheet momentum of the light-cone gauge fixed superstring with quasi-periodic boundary conditions, not of the superstring on a deformed background. Deformations in the ${{{\rm x}}^{+}}$ direction are simply world sheet diffeomorphisms.

Next, we should recall that when fixing a light-cone gauge we need to redefine the fermions so that they are not charged under isometries in the $t$ and ϕ directions. In the undeformed model this turns the periodic fermionic fields into anti-periodic fields in the odd winding sectors [31]

$$\begin{eqnarray}&&\eta (\sigma +2\pi )={{(-1)}^{m}}\eta (\sigma )\;,\theta (\sigma +2\pi )={{(-1)}^{m}}\theta (\sigma )\;.\end{eqnarray} \tag{ 4.9 }$$

Here $m$ is the winding number of the string and η and θ denote the fermionic fields of the light-cone gauge fixed coset model [49]. Precisely because of this uncharging the light-cone fermions are insensitive to deformations in the $t$ and ϕ₁ directions, and only care about the winding number in this regard. As a net result, deformations in the $t$ and ϕ₁ directions enter our model only through the level matching condition (4.8).

The remaining boundary conditions corresponding to ${{\alpha }_{2,3}}$ and ${{\vartheta }_{2,3}}$ can now be put in correspondence with the four remaining Cartan generators of ${\rm SU}{{(2)}^{4}}$. The easiest way to make this link is to consider the generators of shifts of the angles ${{\psi }_{2,3}}$ and ${{\phi }_{2,3}}$ in the fundamental of ${\rm SU}(2,2)$ and ${\rm SU}(4)$ [49]

$$\begin{eqnarray}&&{{C}_{2}}={\rm diag}(-1,1,-1,1)\;,\;\;\;{{C}_{3}}={\rm diag}(-1,1,1,-1)\end{eqnarray} \tag{ 4.10 }$$

and the diagonal embedding of the Cartan generators of the four unbroken ${\rm SU}(2){\rm S}$ in ${\rm SU}(2,2)$ and ${\rm SU}(4)$, which are each just the usual

$$\begin{eqnarray}&&C={\rm diag}(1,-1)\;.\end{eqnarray} \tag{ 4.11 }$$

If we then parametrize the four diagonal elements of ${\rm SU}(2)$ by α and $\dot{\alpha }$ for anti-de Sitter and ϑ and $\dot{\vartheta }$ for the sphere we get

$$\begin{eqnarray}&&\alpha =-\frac{{{\alpha }_{2}}+{{\alpha }_{3}}}{2},\;\;\dot{\alpha }=\frac{{{\alpha }_{3}}-{{\alpha }_{2}}}{2}\;,\end{eqnarray} \tag{ 4.12 }$$

$$\begin{eqnarray}&&\vartheta =-\frac{{{\vartheta }_{2}}+{{\vartheta }_{3}}}{2},\;\;\dot{\vartheta }=\frac{{{\vartheta }_{3}}-{{\vartheta }_{2}}}{2}\;.\end{eqnarray} \tag{ 4.13 }$$

Let us introduce $\varphi \equiv {{\alpha }_{1}}-{{\vartheta }_{1}}$ to parametrize the level matching condition (4.8). Now we are ready to implement these boundary conditions in the integrable structure of the superstring.

As we discussed in the previous section, the S-matrix of the ${\rm Ad}{{{\rm S}}_{5}}\times {{{\rm S}}^{5}}$ superstring consists of two copies of the centrally extended $\mathfrak{s}\mathfrak{u}(2|2)$ invariant S-matrix and a scalar factor. Up to this scalar factor, the transfer matrix is then the product of two transfer matrices each based on a $\mathfrak{s}\mathfrak{u}(2|2)$ invariant S-matrix. Naturally, the twists we want to implement on the light-cone fields are precisely compatible with this structure.

In general the centrally extended $\mathfrak{s}\mathfrak{u}(2|2)$ invariant S-matrix commutes with elements from ${\rm SU}(2)\times {\rm SU}(2)$

$$\begin{eqnarray}&&[\mathbb{S},G\otimes G]=0\;,\;\;G\in {\rm SU}(2)\times {\rm SU}(2),\end{eqnarray} \tag{ 4.14 }$$

so that for any $G\in SU(2)\times SU(2)$ we can define the following twisted transfer matrix

$$\begin{eqnarray}&&{{T}^{G}}(q)={\rm st}{{{\rm r}}_{a}}\;{{G}_{a}}\;{{\mathbb{S}}_{a1}}(q,{{p}_{1}})\ldots \;{{\mathbb{S}}_{aK}}(q,{{p}_{{{K}^{{\rm I}}}}}),\end{eqnarray} \tag{ 4.15 }$$

We should note that for TsT transformations our twist element $G$ depends on the excitation numbers of the state under consideration so that in general $G$ acts also in the physical space. Within each eigenspace however, the action of $G$ is purely in the auxiliary space. To implement the boundary conditions we just discussed, cf equations (4.12) and (4.13) we should choose

$$\begin{eqnarray}&&G={\rm diag}({{{\rm e}}^{{\rm I}\alpha }},{{{\rm e}}^{-{\rm I}\alpha }},{{{\rm e}}^{{\rm I}\vartheta }},{{{\rm e}}^{-{\rm I}\vartheta }})\in {\rm SU}(2)\times {\rm SU}(2),\end{eqnarray} \tag{ 4.16 }$$

for the left copy of the transfer matrix, and the same $G$ with dotted angles for the right one⁹⁹ . These boundary conditions are already diagonal by the way we set up the problem, but we would like to note that considering arbitrary elements in ${\rm SU}(2)\times {\rm SU}(2)$ is no generalization because of the ${\rm SU}(2)\times {\rm SU}(2)$ symmetry of the S-matrix. To see this, we first write a general ${{G}_{1}}\in {\rm SU}(2)$ as

$$\begin{eqnarray}{{G}_{1}}=UH{{U}^{-1}}\;,\;\;{\rm where}\;\;U\in {\rm SU}(2),\;\;H=\left( \begin{array}{ccccccccccccccc} {{{\rm e}}^{{\rm I}\phi }} & 0 \\ 0 & {{{\rm e}}^{-{\rm I}\phi }} \\ \end{array} \right)\in {\rm SU}(2).\end{eqnarray} \tag{ 4.17 }$$

Now since $U\in SU(2)$ it is easy to see that

$$\begin{eqnarray}&&{{T}^{G}}(q)={{\vec{U}}^{-1}}\;{{T}^{H}}(q)\vec{U}\;,\;\;{\rm where}\;\;\vec{U}={{U}_{1}}\ldots {{U}_{{{K}^{{\rm I}}}}}\;,\end{eqnarray} \tag{ 4.18 }$$

thanks to equation (4.14). In other words, any twisted transfer matrix is related to a transfer matrix twisted by an element of the Cartan subgroup via a simple basis transformation on the physical space. Consequently, we see that the eigenvalues of T^G and T^H coincide.

We already encountered a simple twisted transfer matrix in section 2. There we saw explicitly that the effect of a twist is very simple since it does not affect the fundamental commutation relations but only the eigenvalues of the transfer matrix and the associated Bethe equations. In particular, the different terms which sum up to give the eigenvalue of the transfer matrix are typically eigenstates of the twist element, just with different eigenvalues. For example in equation (2.52) we saw that for the XXX spin chain the eigenvalue of $A$ got twisted by ${{{\rm e}}^{{\rm i}\alpha }}$ while that of $D$ got twisted by ${{{\rm e}}^{-{\rm i}\alpha }}$. We see that in general we need to carefully keep track of the 'charge' of the various terms in the transfer matrix under the twist, but apart from this there is no change in the derivation.

If we look at the derivation of the $\mathfrak{s}\mathfrak{u}(2|2)$ transfer matrix [104, 121], we readily realize that the effect of the ϑ twist on the transfer matrix is very simple while the effect of the α type twist is slightly more involved. The ϑ twist just directly multiplies two terms in the full eigenvalue, while the α twist modifies the derivation at the auxiliary level of this $\mathfrak{s}\mathfrak{u}(2|2)$ problem, turning an XXX spin chain hiding there into a twisted XXX spin chain. Carefully keeping track of the phase factors the undeformed eigenvalue of equation (3.78) becomes¹⁰⁰

$$\begin{eqnarray}\begin{array}{rcl} {{T}_{a,1}}(v\;|\;\vec{u}) & = & \mathop{\prod }\limits_{{{K}^{{\rm II}}}}^{i=1}\frac{{{y}_{i}}-{{x}^{-}}}{{{y}_{i}}-{{x}^{+}}}\sqrt{\frac{{{x}^{+}}}{{{x}^{-}}}}{\rm \ [\ }{{{\rm e}}^{{\rm i}a\alpha }}+{{{\rm e}}^{-{\rm i}a\alpha }}\mathop{\prod }\limits_{{{K}^{{\rm II}}}}^{i=1}\frac{v-{{\nu }_{i}}+\frac{ia}{g}}{v-{{\nu }_{i}}-\frac{ia}{g}} \\ {} & \times & \mathop{\prod }\limits_{{{K}^{{\rm i}}}}^{i=1}\frac{({{x}^{-}}-x_{i}^{-})(1-{{x}^{-}}x_{i}^{+}){{x}^{+}}}{({{x}^{+}}-x_{i}^{-})(1-{{x}^{+}}x_{i}^{+}){{x}^{-}}} \\ {} & {} & +\mathop{\sum }\limits_{a-1}^{k=1}{{{\rm e}}^{{\rm i}(a-2k)\alpha }}\mathop{\prod }\limits_{{{K}^{{\rm II}}}}^{i=1}\frac{v-{{\nu }_{i}}+\frac{ia}{g}}{v-{{\nu }_{i}}+\frac{i(a-2k)}{g}} \\ {} & \times & \left\{ \mathop{\prod }\limits_{{{K}^{{\rm i}}}}^{i=1}{{\lambda }_{+}}(v,{{u}_{i}},k)+ \right.\left. \mathop{\prod }\limits_{{{K}^{{\rm i}}}}^{i=1}{{\lambda }_{-}}(v,{{u}_{i}},k) \right\} \\ {} & {} & -\mathop{\sum }\limits_{a-1}^{k=0}{{{\rm e}}^{{\rm i}(a-2k-1)\alpha }}\mathop{\prod }\limits_{{{K}^{{\rm II}}}}^{i=1}\frac{v-{{\nu }_{i}}+\frac{ia}{g}}{v-{{\nu }_{i}}+\frac{i(a-2k)}{g}} \\ {} & \times & \mathop{\prod }\limits_{{{K}^{{\rm i}}}}^{i=1}\frac{{{x}^{+}}-x_{i}^{+}}{{{x}^{+}}-x_{i}^{-}}\sqrt{\frac{x_{i}^{-}}{x_{i}^{+}}}\left[ 1-\frac{\frac{2ik}{g}}{v-{{u}_{i}}+\frac{i(a-1)}{g}} \right] \\ {} & {} & \times \left\{ {{{\rm e}}^{{\rm i}\vartheta }}\mathop{\prod }\limits_{{{K}^{{\rm III}}}}^{i=1}\frac{{{w}_{i}}-v-\frac{i(a-2k+1)}{g}}{{{w}_{i}}-v-\frac{i(a-2k-1)}{g}}+{{{\rm e}}^{-{\rm i}\vartheta }}\mathop{\prod }\limits_{{{K}^{{\rm II}}}}^{i=1}\frac{v-{{\nu }_{i}}+\frac{i(a-2k)}{g}}{v-{{\nu }_{i}}+\frac{i(a-2k-2)}{g}} \right. \\ {} & {} & \times \left. \left. \mathop{\prod }\limits_{{{K}^{{\rm III}}}}^{i=1}\frac{{{w}_{i}}-v-\frac{i(a-2k-3)}{g}}{{{w}_{i}}-v-\frac{i(a-2k-1)}{g}} \right\} \right], \\ \end{array}\end{eqnarray} \tag{ 4.19 }$$

where the auxiliary roots $y$, and $w$ satisfy the Bethe equation (4.21), (4.22) given below. As in the previous section the rapidities ν are related to $y$ via $\nu =y+1/y$, and the parameters ${{\lambda }_{\pm }}$ are not affected by the twists and still given by equation (3.79).

To get the Bethe–Yang equations we should recall that the complete S-matrix consists of two copies of the $\mathfrak{s}\mathfrak{u}(2|2)$ invariant S-matrix along with a scalar factor. As usual many terms in (the eigenvalue of) the transfer matrix cancel on solutions of the auxiliary Bethe equations below, and the general Bethe–Yang equation (2.34) in this case becomes

$$\begin{eqnarray}&&1={{{\rm e}}^{{\rm i}(\alpha +\dot{\alpha })}}{{\left( \frac{x_{k}^{+}}{x_{k}^{-}} \right)}^{J}}\mathop{\prod }\limits_{l=1,l\ne k}^{{{K}^{{\rm i}}}}{{S}_{\mathfrak{s}\mathfrak{l}(2)}}({{p}_{k}},{{p}_{l}})\mathop{\prod }\limits_{a}\mathop{\prod }\limits_{l=1}^{K_{(a)}^{{\rm II}}}\frac{x_{k}^{-}-y_{l}^{(l)}}{x_{k}^{+}-y_{l}^{(l)}}\sqrt{\frac{x_{k}^{+}}{x_{k}^{-}}}\;.\end{eqnarray} \tag{ 4.20 }$$

The auxiliary Bethe equations are given by

$$\begin{eqnarray}&&1={{{\rm e}}^{{\rm i}(\vartheta -\alpha )}}\mathop{\prod }\limits_{l=1}^{{{K}^{{\rm i}}}}\frac{{{y}_{k}}-x_{l}^{+}}{{{y}_{k}}-x_{l}^{-}}\sqrt{\frac{x_{l}^{-}}{x_{l}^{+}}}\mathop{\prod }\limits_{l=1}^{K_{(l)}^{{\rm III}}}\frac{{{y}_{k}}+\frac{1}{{{y}_{k}}}-w_{l}^{(l)}+\frac{i}{g}}{{{y}_{k}}+\frac{1}{{{y}_{k}}}-w_{l}^{(l)}-\frac{i}{g}}\end{eqnarray} \tag{ 4.21 }$$

$$\begin{eqnarray}&&1={{{\rm e}}^{-2{\rm i}\vartheta }}\mathop{\prod }\limits_{l=1}^{K_{(l)}^{{\rm II}}}\frac{w_{k}^{(l)}-y_{l}^{(l)}-\frac{1}{y_{l}^{(l)}}+\frac{i}{g}}{w_{k}^{(l)}-y_{l}^{(l)}-\frac{1}{y_{l}^{(l)}}-\frac{i}{g}}\mathop{\prod }\limits_{l\ne k}^{K_{(l)}^{{\rm III}}}\frac{w_{k}^{(l)}-w_{l}^{(l)}-\frac{2i}{g}}{w_{k}^{(l)}-w_{l}^{(l)}+\frac{2i}{g}},\end{eqnarray} \tag{ 4.22 }$$

along with a copy for ${{y}_{(r)}}$ and ${{w}_{(r)}}$ with dotted angles.

The above Bethe–Yang equations can be used to find the asymptotic energy of any state upon specifying the corresponding excitation numbers. Since we chose the $\mathfrak{s}\mathfrak{l}(2)$ vacuum to diagonalize our transfer matrix over, the corresponding equations are simplest when considering states from the $\mathfrak{s}\mathfrak{l}(2)$ sector; there are no auxiliary excitations. Had we instead diagonalized our transfer matrix over the $\mathfrak{s}\mathfrak{u}(2)$ vacuum, the description of states in this sector would have been simpler. From the point of view of the $\mathfrak{s}\mathfrak{l}(2)$ sector such states are described by states with $K_{(a)}^{{\rm II}}={{K}^{{\rm i}}}$ and $K_{(a)}^{{\rm III}}=0$, so in this case a description over the $\mathfrak{s}\mathfrak{u}(2)$ vacuum is clearly easier¹⁰¹ . The corresponding eigenvalue of the transfer matrix is given by

$$\begin{eqnarray*}\begin{array}{rcl} T_{a,1}^{\mathfrak{s}\mathfrak{u}(2)}(v|\vec{u}) & = & \;\frac{{\rm sin} (a+1)\alpha }{{\rm sin} \alpha }\mathop{\prod }\limits_{{{K}^{{\rm I}}}}^{i=1}\frac{{{x}^{-}}-x_{i}^{-}}{{{x}^{+}}-x_{i}^{-}}\sqrt{\frac{{{x}^{+}}}{{{x}^{-}}}} \\ {} & {} & +\frac{{\rm sin} (a-1)\alpha }{{\rm sin} \alpha }\mathop{\prod }\limits_{{{K}^{{\rm I}}}}^{i=1}\frac{{{x}^{-}}-x_{i}^{+}}{{{x}^{+}}-x_{i}^{-}}\frac{x_{i}^{-}-\frac{1}{{{x}^{+}}}}{x_{i}^{+}-\frac{1}{{{x}^{+}}}}\sqrt{\frac{{{x}^{+}}}{{{x}^{-}}}} \\ {} & {} & -\frac{{\rm sin} a\;\alpha }{{\rm sin} \alpha }\left[ {{{\rm e}}^{{\rm I}\vartheta }}\mathop{\prod }\limits_{{{K}^{{\rm I}}}}^{i=1}\frac{{{x}^{-}}-x_{i}^{+}}{{{x}^{+}}-x_{i}^{-}}\sqrt{\frac{{{x}^{+}}x_{i}^{-}}{{{x}^{-}}x_{i}^{+}}} \right. \\ {} & {} & +\left. {{{\rm e}}^{-{\rm I}\vartheta }}\mathop{\prod }\limits_{{{K}^{{\rm I}}}}^{i=1}\frac{{{x}^{-}}-x_{i}^{-}}{{{x}^{+}}-x_{i}^{-}}\frac{x_{i}^{-}-\frac{1}{{{x}^{+}}}}{x_{i}^{+}-\frac{1}{{{x}^{+}}}}\sqrt{\frac{{{x}^{+}}x_{i}^{+}}{{{x}^{-}}x_{i}^{-}}} \right]. \\ \end{array}\end{eqnarray*}$$

and the Bethe–Yang equations are

$$\begin{eqnarray}&&1={{{\rm e}}^{{\rm i}(\vartheta +\dot{\vartheta })}}{{\left( \frac{x_{k}^{+}}{x_{k}^{-}} \right)}^{J}}\mathop{\prod }\limits_{l=1,l\ne k}^{{{K}^{{\rm i}}}}{{S}_{\mathfrak{s}\mathfrak{u}(2)}}({{p}_{k}},{{p}_{l}}),\end{eqnarray} \tag{ 4.23 }$$

where we reexpressed the $\mathfrak{s}\mathfrak{l}(2)$ S-matrix (3.25) in terms of the $\mathfrak{s}\mathfrak{u}(2)$ S-matrix (3.12). We can obtain this eigenvalue by explicitly acting with the transfer matrix on the $\mathfrak{s}\mathfrak{u}(2)$ vacuum, or by 'dualizing' the $\mathfrak{s}\mathfrak{l}(2)$ Bethe–Yang equations and transfer matrix¹⁰² . Of course, we can do the same for the generic eigenvalue of the $\mathfrak{s}\mathfrak{l}(2)$ transfer matrix and the full set of Bethe–Yang equations, but let us not write yet more formulae.

In order to compare the twisted Bethe equations for our models with [151, 163] we will rewrite them in terms of seven equations corresponding to the underlying $\mathfrak{p}\mathfrak{s}\mathfrak{u}(2,2|4)$ Dynkin diagram. However, as was shown in [104, 105] the equations given in [9] agree with Bethe equations derived from string theory for P = 0, while in general they can differ by factors of e^iP. This is not an issue for physical states in the untwisted case, but here these factors have to be taken carefully into account. Let us follow the procedure outlined in [104] to bring the equations (4.20–4.22) to the form presented in [9]. Our equations are in the $\mathfrak{s}\mathfrak{l}(2)$ grading and hence we will bring them to the form where $\eta =-1$. First, we identify the labels

$$\begin{eqnarray}&&{{K}^{{\rm i}}}={{K}_{4}},\;\;K_{(l)}^{{\rm II}}={{K}_{1}}+{{K}_{3}},\;\;K_{(r)}^{{\rm II}}={{K}_{5}}+{{K}_{7}},\;\;K_{(l)}^{{\rm III}}={{K}_{2}},\;\;K_{(r)}^{{\rm III}}={{K}_{6}},\end{eqnarray} \tag{ 4.24 }$$

and we split the products accordingly, i.e.,

$$\begin{eqnarray}&&\mathop{\prod }\limits_{j=1}^{K_{(l)}^{{\rm II}}}\to \mathop{\prod }\limits_{j=1}^{{{K}_{1}}}\ \ \mathop{\prod }\limits_{j=1}^{{{K}_{3}}},\qquad \;\mathop{\prod }\limits_{j=1}^{K_{(r)}^{{\rm II}}}\to \mathop{\prod }\limits_{j=1}^{{{K}_{5}}}\ \ \mathop{\prod }\limits_{j=1}^{{{K}_{7}}}.\end{eqnarray} \tag{ 4.25 }$$

Next we relabel the parameters in the following way

$$\begin{eqnarray}x_{k}^{\pm }=\frac{{{x}_{4,k}}}{g},\;\;{{y}_{k}}=\left\{ \begin{array}{ccccccccccccccc} {{x}_{3,k}}/g & \ k=1,\ldots {{K}_{3}} \\ g/{{x}_{1,k}} & \ k=1,\ldots {{K}_{1}} \\ \end{array} \right.\quad \;{{\dot{y}}_{k}}=\left\{ \begin{array}{ccccccccccccccc} {{x}_{5,k}}/g & \ k=1,\ldots {{K}_{5}} \\ g/{{x}_{7,k}} & \ k=1,\ldots {{K}_{7}} \\ \end{array} \right.\end{eqnarray} \tag{ 4.26 }$$

Under this map, the main Bethe equation (4.20) becomes

$$\begin{eqnarray*}\begin{array}{rcl} 1 & = & {{{\rm e}}^{-{\rm i}({{p}_{k}}L+\alpha +\dot{\alpha })}}\mathop{\prod }\limits_{j\ne k}^{{{K}_{4}}}{{S}_{\mathfrak{s}\mathfrak{l}(2)}}({{p}_{j}},{{p}_{k}})\mathop{\prod }\limits_{j=1}^{{{K}_{3}}}\frac{x_{4,k}^{+}-{{x}_{3,j}}}{x_{4,k}^{-}-{{x}_{3,j}}}\mathop{\prod }\limits_{j=1}^{{{K}_{1}}}\frac{1-\frac{{{g}^{2}}}{x_{4,k}^{+}{{x}_{1,j}}}}{1-\frac{{{g}^{2}}}{x_{4,k}^{-}{{x}_{1,j}}}} \\ {} & {} & \times \mathop{\prod }\limits_{j=1}^{{{K}_{5}}}\frac{x_{4,k}^{+}-{{x}_{5,j}}}{x_{4,k}^{-}-{{x}_{5,j}}}\mathop{\prod }\limits_{j=1}^{{{K}_{7}}}\frac{1-\frac{{{g}^{2}}}{x_{4,k}^{+}{{x}_{7,j}}}}{1-\frac{{{g}^{2}}}{x_{4,k}^{-}{{x}_{7,j}}}}\;, \\ \end{array}\end{eqnarray*}$$

where $L=J-\frac{{{K}_{1}}-{{K}_{3}}-{{K}_{5}}+{{K}_{7}}}{2}$. To address the auxiliary Bethe equations we need to introduce the rapidities

$$\begin{eqnarray}&&w_{k}^{(l)}={{u}_{2,k}}/g,\quad w_{k}^{(r)}={{u}_{6,k}}/g,\quad {{u}_{i,j}}\equiv {{x}_{i,j}}+\frac{{{g}^{2}}}{{{x}_{i,j}}}.\end{eqnarray} \tag{ 4.27 }$$

Then (4.22) is straightforwardly rewritten as (the dotted version is obtained by switching labels ${{K}_{1,2,3}}\to {{K}_{7,6,5}}$ and parameters similarly)

$$\begin{eqnarray}&&1={{{\rm e}}^{2{\rm i}\vartheta }}\mathop{\prod }\limits_{j\ne k}^{{{K}_{2}}}\frac{{{u}_{2,k}}-{{u}_{2,j}}+2i}{{{u}_{2,k}}-{{u}_{2,j}}-2i}\mathop{\prod }\limits_{j}^{{{K}_{1}}}\frac{{{u}_{2,k}}-{{u}_{1,j}}-i}{{{u}_{2,k}}-{{u}_{1,j}}+i}\mathop{\prod }\limits_{j}^{{{K}_{3}}}\frac{{{u}_{2,k}}-{{u}_{1,j}}-i}{{{u}_{2,k}}-{{u}_{1,j}}+i},\end{eqnarray} \tag{ 4.28 }$$

while (4.21) splits into two sets of equations for roots of type 1 and 3 (we have used the level matching condition $\mathop{\prod }\limits_{k}\frac{x_{k}^{+}}{x_{k}^{-}}={{e}^{{\rm i}\varphi }}$)

$$\begin{eqnarray}&&1={{{\rm e}}^{{\rm i}(\alpha -\vartheta -\varphi /2)}}\mathop{\prod }\limits_{j=1}^{{{K}_{4}}}\frac{1-\frac{{{g}^{2}}}{x_{4,j}^{-}{{x}_{1,k}}}}{1-\frac{{{g}^{2}}}{x_{4,j}^{+}{{x}_{1,k}}}}\mathop{\prod }\limits_{j=1}^{{{K}_{2}}}\frac{{{u}_{1,k}}-{{u}_{2,j}}-i}{{{u}_{1,k}}-{{u}_{2,j}}+i},\;\;k=1,\ldots {{K}_{1}},\end{eqnarray} \tag{ 4.29 }$$

$$\begin{eqnarray}&&1={{{\rm e}}^{{\rm i}(\alpha -\vartheta +\varphi /2)}}\mathop{\prod }\limits_{j=1}^{{{K}_{4}}}\frac{x_{4,j}^{-}-{{x}_{3,k}}}{x_{4,j}^{+}-{{x}_{3,k}}}\mathop{\prod }\limits_{j=1}^{{{K}_{2}}}\frac{{{u}_{3,k}}-{{u}_{2,j}}-i}{{{u}_{3,k}}-{{u}_{2,j}}+i},\;\;k=1,\ldots {{K}_{3}},\end{eqnarray} \tag{ 4.30 }$$

again with similar equations for the dotted indices. We see that written this way our equations are exactly of the form of [9] up to the twisting factors. These factors are precisely what was added to the Bethe equations in [151, 163]; next we will show explicit agreement with their twists for the various deformations.

Abelian orbifolds

The twist corresponding to a generic abelian ${{\mathbb{Z}}_{S}}$ orbifold is represented by an element

$$\begin{eqnarray}&&{\rm diag}({{{\rm e}}^{-2\pi {\rm i}T{{t}_{1}}/S}},{{{\rm e}}^{2\pi {\rm i}({{t}_{1}}-{{t}_{2}})T/S}},{{{\rm e}}^{2\pi {\rm i}({{t}_{2}}-{{t}_{3}})T/S}},{{{\rm e}}^{2\pi {\rm i}{{t}_{3}}T/S}}),\end{eqnarray} \tag{ 4.31 }$$

where $T$ labels the twisted sector. From this we deduce that the angles transform as

$$\begin{eqnarray}&&(\delta {{\phi }_{1}},\delta {{\phi }_{2}},\delta {{\phi }_{3}})=\left( -\frac{2\pi {{t}_{2}}T}{S},-\frac{2\pi ({{t}_{1}}-{{t}_{2}}+{{t}_{3}})T}{S},-\frac{2\pi ({{t}_{1}}-{{t}_{3}})T}{S} \right),\end{eqnarray} \tag{ 4.32 }$$

resulting in the twist

$$\begin{eqnarray}&&\begin{array}{l} \alpha =\dot{\alpha }=0,\;\;\varphi =\frac{2\pi {{t}_{2}}T}{S}\;, \\ \vartheta =\frac{\pi (2{{t}_{1}}-{{t}_{2}})T}{S},\;\;\dot{\vartheta }=\frac{\pi (2{{t}_{3}}-{{t}_{2}})T}{S}. \\ \end{array}\end{eqnarray} \tag{ 4.33 }$$

These twists are uniquely fixed by the geometry through the boundary conditions on the fields and in turn fix our Bethe equations (4.20–4.22). These should then fully agree with [151] in the one-loop limit. Actually their result is readily generalized to all loop equations by supplementing [9] with the appropriate phase factors.

Following [151] we should introduce phase factors ${{{\rm e}}^{2\pi {\rm i}\frac{T{{s}_{k}}}{S}}}$ in front of the equations for roots of type $k$. In the $\mathfrak{s}\mathfrak{l}(2)$ grading these twists are of the form [171]

$$\begin{eqnarray*}\begin{array}{rclcl} {{s}_{1}} & = & -{{t}_{1}},\;\;{{s}_{2}}=2{{t}_{1}}-{{t}_{2}},\;\;{{s}_{3}}={{t}_{2}}-{{t}_{1}},\;\;{{s}_{4}}=0,\;\;{{s}_{5}}={{t}_{2}}-{{t}_{3}},\quad {{s}_{6}} & = & 2{{t}_{3}}-{{t}_{1}},\;\;{{s}_{7}}=-{{t}_{3}}. \\ \end{array}\end{eqnarray*}$$

Our Bethe equations already contain phase factors, and comparing the above against our general twisted Bethe equations we find agreement exactly when

$$\begin{eqnarray}&&-\vartheta -\varphi /2=2\pi \frac{T{{s}_{1}}}{S},\;\;2\vartheta =2\pi \frac{T{{s}_{2}}}{S},\;-\vartheta +\varphi /2=2\pi \frac{T{{s}_{3}}}{S},\end{eqnarray} \tag{ 4.34 }$$

with similar expressions for the dotted versions. Comparing this against (4.33) we see that they are in perfect agreement.

Since we are considering an orbifold, we should not forget to project onto the orbifold invariant part of the spectrum. This means we should consider states which satisfy [151]

$$\begin{eqnarray}&&{{t}_{2}}\;p+{{t}_{1}}\;{{q}_{2}}+{{t}_{3}}\;{{q}_{1}}=0{\rm mod} S.\end{eqnarray} \tag{ 4.35 }$$

Naturally this constraint can be phrased in more geometric terms via the relation of the $\mathfrak{s}\mathfrak{u}(4)$ weights $[{{q}_{1}},p,{{q}_{2}}]$ to the angular momenta of the sphere [123]

$$\begin{eqnarray}&&J={{J}_{1}}=\frac{{{q}_{1}}+2p+{{q}_{2}}}{2},\;\;{{J}_{2}}=\frac{{{q}_{1}}+{{q}_{2}}}{2},\;\;{{J}_{3}}=\frac{{{q}_{2}}-{{q}_{1}}}{2}\;,\end{eqnarray} \tag{ 4.36 }$$

so that (4.35) becomes

$$\begin{eqnarray}&&\mathop{\sum }\limits_{i=1}^{3}{{J}_{i}}\;\delta {{\phi }_{i}}{{|}_{T=1}}=0{\rm mod} 2\pi .\end{eqnarray} \tag{ 4.37 }$$

As discussed, TsT transformations of the sphere give rise to γ-deformed theories. Recalling that the twisted boundary conditions are parametrized by three deformation parameters ${{\gamma }_{1,2,3}}$ as

$$\begin{eqnarray}&&\delta {{\phi }_{i}}={{\phi }_{i}}(2\pi )-{{\phi }_{i}}(0)=-2\pi {{\epsilon }_{ijk}}{{\gamma }_{j}}{{J}_{k}},\end{eqnarray} \tag{ 4.38 }$$

where the J_i are the Noether charges corresponding to the shift isometries in the ϕ_i direction on the sphere, these boundary conditions correspond to the twists

$$\begin{eqnarray}&&\begin{array}{l} \alpha =\dot{\alpha }=0,\;\;\varphi =2\pi ({{\gamma }_{2}}{{J}_{3}}-{{\gamma }_{3}}{{J}_{2}}), \\ \vartheta =\pi ({{\gamma }_{1}}({{J}_{2}}-{{J}_{3}})+{{J}_{1}}({{\gamma }_{3}}-{{\gamma }_{2}})),\;\;\dot{\vartheta }=\pi ({{J}_{1}}({{\gamma }_{2}}+{{\gamma }_{3}})-{{\gamma }_{1}}({{J}_{2}}+{{J}_{3}})), \\ \end{array}\end{eqnarray} \tag{ 4.39 }$$

For ${{\gamma }_{1}}={{\gamma }_{2}}={{\gamma }_{3}}=\beta$ this reduces to the β-deformed theory for real β.

We will now compare the Bethe equations following from this twist to those derived in [163] or equivalently to [172]. After a duality transformation to the $\mathfrak{s}\mathfrak{l}(2)$ grading we find the following set of equations (see also the appendix of [172] for the explicit $\mathfrak{s}\mathfrak{l}(2)$ grading)

$$\begin{eqnarray} &&\left(\begin{array}{c} -\varphi\\ \alpha-\vartheta-\varphi/2\\ 2\vartheta\\ \alpha-\vartheta+\varphi/2\\ -\alpha-\dot{\alpha}\\ \dot{\alpha}-\dot{\vartheta}+\varphi/2\\ 2\dot{\vartheta}\\ \dot{\alpha}-\dot{\vartheta}-\varphi/2 \end{array}\right) = \left(\begin{array}{c} \delta_1 (K_5-2 K_6+K_7)-\delta_3 (K_1-2 K_2+K_3) \\ \delta_3 (K_0+K_2-K_3-K_5+K_6)-\delta_2 (K_5-2 K_6+K_7) \\ (\delta_1+2 \delta_2)(K_5-2 K_6+K_7) + \delta_3(2 K_5-2 K_6-2 K_0-K_1+K_3) \\ \delta_3(K_0+K_1-K_2-K_5+K_6)-(\delta_1 + \delta_2)(K_5-2 K_6+K_7) \\ 0 \\ (\delta_2+\delta _3)(K_1-2 K_2+K_3)-\delta_1(K_0+K_2-K_3-K_6+K_7) \\ \delta_1(2 K_0+2 K_2-2 K_3-K_5+K_7)-(2\delta_2 + \delta_3)(K_1-2 K_2+K_3) \\ \delta_2(K_1-2 K_2+K_3) - \delta_1(K_0+K_2-K_3-K_5+K_6) \end{array}\right), \end{eqnarray} \tag{ 4.40 }$$

where K_i are the excitation labels from the Bethe equations and ${{K}_{0}}\equiv L$. The minus sign in the first term of the left-hand side is due to the fact that $P=-{{(AK)}_{0}}$ in the notation of [163].

This over-determined system of equations indeed has a unique solution. In order to make contact with the twist discussed above we relate the labels to the conserved charges J_i via (4.36), and use their relation to the excitation numbers $K$ (see also (4.24)) as

$$\begin{eqnarray}&&\begin{array}{l} {{q}_{1}}=K_{(r)}^{{\rm II}}-2K_{(r)}^{{\rm III}},\;\;{{q}_{2}}=K_{(l)}^{{\rm II}}-2K_{(l)}^{{\rm III}},\;\;p=J-\frac{1}{2}(K_{(l)}^{{\rm II}}+K_{(r)}^{{\rm II}})+K_{(l)}^{{\rm III}}+K_{(r)}^{{\rm III}}, \\ {{s}_{1}}={{K}^{{\rm i}}}-K_{(r)}^{{\rm II}},\;\;{{s}_{2}}={{K}^{{\rm i}}}-K_{(l)}^{{\rm II}}. \\ \end{array}\end{eqnarray} \tag{ 4.41 }$$

The parameters δ_i are related to the parameters γ_i as

$$\begin{eqnarray}&&{{\delta }_{1}}=\pi ({{\gamma }_{2}}+{{\gamma }_{3}}),\;\;{{\delta }_{2}}=\pi ({{\gamma }_{1}}-{{\gamma }_{2}}),\;\;{{\delta }_{3}}=\pi ({{\gamma }_{2}}-{{\gamma }_{3}}).\end{eqnarray} \tag{ 4.42 }$$

Putting this together correctly reproduces our twist.

Let us now consider models involving the most generic set of boundary conditions.

The twist for a general TsT transformation can be immediately read off from (4.6) together with (4.12) and (4.13). It is worthwhile to note that if the TsT transformation involves the time direction that the twist will become energy dependent. In other words the asymptotic Bethe equations pick up phases that depend on E(p). This results in a very involved coupled system of equations.

We can also consider the generic orbifold ${\rm Ad}{{{\rm S}}_{5}}/{{\Gamma }^{a}}\times {{{\rm S}}^{5}}/{{\Gamma }^{s}}$, with ${{\Gamma }^{a}}={{\mathbb{Z}}_{R}}$ and ${{\Gamma }^{s}}={{\mathbb{Z}}_{S}}$. Since we do not orbifold the $t$ direction, φ is unaffected by Γ^a and consequently the twists $\vartheta ,\dot{\vartheta }$ and φ are again given by (4.33). We now quickly find the expressions for $\alpha ,\dot{\alpha }$ by taking analogous expressions to those for $\vartheta ,\dot{\vartheta }$ with the equivalent of t₂ put to zero. Labeling the twisted sectors for ${{\mathbb{Z}}_{R}}$ and ${{\mathbb{Z}}_{S}}$ by $\tilde{T}$ and $T$ respectively, this gives

$$\begin{eqnarray}\begin{array}{lcl} \alpha & = & \frac{2\pi \;{{r}_{1}}\tilde{T}}{R},\qquad \qquad \ \ \dot{\alpha }=\frac{2\pi \;{{r}_{3}}\tilde{T}}{R}, \\ \vartheta & = & \frac{\pi (2{{t}_{1}}-{{t}_{2}})T}{S},\qquad \dot{\vartheta }=\frac{\pi (2{{t}_{3}}-{{t}_{2}})T}{S}, \\ \varphi & = & \frac{2\pi {{t}_{2}}T}{S}. \\ \end{array}\end{eqnarray} \tag{ 4.43 }$$

Orbifold invariance now requires

$$\begin{eqnarray}&&{{t}_{2}}\;p+{{t}_{1}}\;{{q}_{2}}+{{t}_{3}}\;{{q}_{1}}+{{r}_{1}}\;{{s}_{2}}+{{r}_{3}}\ {{s}_{1}}=0\;{\rm mod} \;{\rm LCM}(S,R),\end{eqnarray} \tag{ 4.44 }$$

where ${\rm LCM}(S,R)$ denotes the least common multiple of $S$ and $R$ and s₁ and s₂ are the two spins of the conformal group. As mentioned earlier it is also entirely possible to describe non-abelian orbifolds in this framework. In each twisted sector we still get a twist of the form (4.43). The only complication is in determining the physical states we should study, i.e. in the analogue of (4.44). We refer the interested reader to [146] where some specific examples are discussed in detail.

The canonical ground state TBA equations with non-zero chemical potentials are given by

$$\begin{eqnarray}&&{\rm log} Y_{M|w}^{(a)}=-\mu _{M|w}^{(a)}+{\rm log} \left( 1+\frac{1}{Y_{N|w}^{(a)}} \right)\star {{K}_{NM}}+{\rm log} \frac{1-\frac{1}{Y_{-}^{(a)}}}{1-\frac{1}{Y_{+}^{(a)}}}\hat{\star }{{K}_{M}}\;,\end{eqnarray} \tag{ 4.45 }$$

$$\begin{eqnarray}\begin{array}{rcl} {\rm log} Y_{M|vw}^{(a)} & = & -\mu _{M|vw}^{(a)}+{\rm log} \left( 1+\frac{1}{Y_{N|vw}^{(a)}} \right)\star {{K}_{NM}}+{\rm log} \frac{1-\frac{1}{Y_{-}^{(a)}}}{1-\frac{1}{Y_{+}^{(a)}}}\hat{\star }{{K}_{M}} \\ {} & {} & -{\rm log} (1+{{Y}_{Q}})\star K_{xv}^{QM}\;, \\ \end{array}\end{eqnarray} \tag{ 4.46 }$$

$$\begin{eqnarray}&&{\rm log} Y_{\pm }^{(a)}=-\mu _{\pm }^{(a)}-{\rm log} \left( 1+{{Y}_{Q}} \right)\star K_{\pm }^{Qy}+{\rm log} \frac{1+\frac{1}{Y_{M|vw}^{(a)}}}{1+\frac{1}{Y_{M|w}^{(a)}}}\star {{K}_{M}}\;,\end{eqnarray} \tag{ 4.47 }$$

$$\begin{eqnarray}\begin{array}{rcl} {\rm log} {{Y}_{Q}} & = & \;-{{\mu }_{Q}}-J\;{{\tilde{\mathcal{E}}}_{Q}}+{\rm log} \left( 1+{{Y}_{M}} \right)\star K_{\mathfrak{s}\mathfrak{l}(2)}^{MQ} \\ {} & {} & +\mathop{\sum }\limits_{(a)=\pm }{\rm log} \left( 1+\frac{1}{Y_{M|vw}^{(a)}} \right)\star K_{vwx}^{MQ}+{\rm log} \left( 1-\frac{1}{Y_{\pm }^{(a)}} \right)\hat{\star }K_{\pm }^{yQ}\;, \\ \end{array}\end{eqnarray} \tag{ 4.48 }$$

where the ${{\mu }_{\chi }}$ are given in table 1 keeping in mind the implicit regulator. The ground state energy is given by

$$\begin{eqnarray}&&E=-\frac{1}{2\pi }\int \;{\rm d}u\;\frac{{\rm d}{{\tilde{p}}_{Q}}}{{\rm d}u}{\rm log} (1+{{Y}_{Q}}).\end{eqnarray} \tag{ 4.49 }$$

As we will see below, it is useful to transform the canonical TBA equations to an alternate form. In particular we can readily simplify the equations for $w$ and $vw$-strings by application of the kernel ${{(K+1)}^{-1}}$.¹⁰⁵ The chemical potentials are in the null-space of this operator, and hence the simplified equations do not depend explicitly on them, meaning they are just given by their untwisted versions (3.53) and (3.54). These equations together with the canonical TBA equations allow us to find the asymptotic behavior of the ${{Y}_{M|w}}$ and ${{Y}_{M|vw}}$ functions as we will show shortly. Then, with this asymptotic behavior and the simplified equations we can eliminate the infinite sums over $w$ and $vw$-strings from the equations for $y$ and $Q$-particles to bring them to their hybrid form. The only modification in the derivation of the hybrid TBA equations is the presence of chemical potentials and we discuss the resulting contribution in appendix A.4.1. The upshot is the following set of equations

$$\begin{eqnarray}&&{\rm log} Y_{M|w}^{(a)}={{I}_{MN}}{\rm log} (1+Y_{N|w}^{(a)})\star s+{{\delta }_{M1}}{\rm log} \frac{1-\frac{1}{Y_{-}^{(a)}}}{1-\frac{1}{Y_{+}^{(a)}}}\hat{\star }s\;,\end{eqnarray} \tag{ 4.50 }$$

$$\begin{eqnarray}&&{\rm log} Y_{M|vw}^{(a)}=-{\rm log} (1+{{Y}_{M+1}})\star s+{{I}_{MN}}{\rm log} (1+Y_{N|vw}^{(a)})\star s+{{\delta }_{M1}}{\rm log} \frac{1-Y_{-}^{(a)}}{1-Y_{+}^{(a)}}\hat{\star }s\;,\end{eqnarray} \tag{ 4.51 }$$

$$\begin{eqnarray}&&{\rm log} \frac{Y_{+}^{(a)}}{Y_{-}^{(a)}}={\rm log} (1+{{Y}_{Q}})\star {{K}_{Qy}}\;,\end{eqnarray} \tag{ 4.52 }$$

$$\begin{eqnarray}\begin{array}{rcl} {\rm log} Y_{-}^{(a)}Y_{+}^{(a)} & = & -{\rm log} (1+{{Y}_{Q}})\star {{K}_{Q}}+2{\rm log} (1+{{Y}_{Q}})\star K_{xv}^{Q1}\star s \\ {} & {} & +2{\rm log} \frac{1+Y_{1|vw}^{(a)}}{1+Y_{1|w}^{(a)}}\star s\;, \\ \end{array}\end{eqnarray} \tag{ 4.53 }$$

$$\begin{eqnarray}\begin{array}{rcl} {\rm log} {{Y}_{Q}} & = & -L{{\tilde{\mathcal{E}}}_{Q}}+{\rm log} (1+{{Y}_{{{Q}^{\prime }}}})\star \left( K_{sl(2)}^{{{Q}^{\prime }}Q}+2s\star K_{vx}^{{{Q}^{\prime }}-1,Q} \right) \\ {} & {} & +\mathop{\sum }\limits_{(a)=\pm }\left[ {\rm log} \left( 1+Y_{1|vw}^{(a)} \right)\star s\;\hat{\star }{{K}_{yQ}}+{\rm log} (1+Y_{Q-1|vw}^{(a)})\star s \right. \\ {} & {} & -{\rm log} \frac{1-Y_{-}^{(a)}}{1-Y_{+}^{(a)}}\hat{\star }s\star K_{vwx}^{1Q}+\frac{1}{2}{\rm log} \frac{1-\frac{1}{Y_{-}^{(a)}}}{1-\frac{1}{Y_{+}^{(a)}}}\hat{\star }{{K}_{Q}} \\ {} & {} & +\left. \frac{1}{2}{\rm log} (1-\frac{1}{Y_{-}^{(a)}})(1-\frac{1}{Y_{+}^{(a)}})\hat{\star }{{K}_{yQ}} \right]. \\ \end{array}\end{eqnarray} \tag{ 4.54 }$$

From these equations and the large $u$ asymptotics for the Y-functions for $w$ and $vw$-strings we can directly read off the large $u$ asymptotics of all Y-functions¹⁰⁶ . We see that the chemical potentials have disappeared from this form of the TBA equations, meaning they are unchanged from the original model. Consequently also the $Y$-system is unchanged for these deformations. This also follows from the fact that the chemical potentials satisfy the conditions to obtain an undeformed Y-system, formulated in [115]. We would like to emphasize again that while the chemical potentials disappear from the simplified TBA and Y-system equations, the Y-functions must satisfy the canonical TBA equations, which translates to a set of large $u$ asymptotics on them.

The first concrete state we need to consider in a generically deformed theory is the ground state. The Y-functions for this state will give us the concrete large $u$ asymptotics of the Y-functions for any excited state with the same chemical potentials. We start by considering the TBA equations in the asymptotic limit. In this limit the Y_Q functions are exponentially small, and from the canonical TBA equations it follows that ${{Y}_{+}}={{Y}_{-}}$ since their chemical potentials are equal. Then then simplified TBA equations for $w$ and $vw$-strings (3.53), (3.54), become simple recursion relations for constant Y-functions

$$\begin{eqnarray}&&Y_{M+1|w}^{{}^\circ }=\frac{Y_{M|w}^{{}^\circ 2}}{1+Y{{{}^\circ }_{M-1|w}}}-1,\end{eqnarray} \tag{ 4.55 }$$

with an identical equation for $vw$-strings, and we note that $Y_{0|(v)w}^{{}^\circ }$ is zero by definition. By this recursion relation all $Y_{M|w}^{{}^\circ }$ are uniquely fixed in terms of the value of $Y_{1|w}^{{}^\circ }$. The constant solution of this equation which also satisfies the canonical TBA equations with zero chemical potentials is given by [119]

$$\begin{eqnarray}&&Y_{M|w}^{{}^\circ }=M(M+2),\end{eqnarray} \tag{ 4.56 }$$

which satisfies the recursion relation for the simple reason that ${{M}^{2}}=(M+1)(M-1)+1$. A clear generalization of this solution is

$$\begin{eqnarray}&&Y_{M|w}^{{}^\circ +}={{[M]}_{q}}{{[M+2]}_{q}},\end{eqnarray} \tag{ 4.57 }$$

where we have introduced $q$-numbers ${{[n]}_{q}}$ as

$$\begin{eqnarray}&&{{[n]}_{q}}=\frac{{{q}^{n}}-{{q}^{-n}}}{q-{{q}^{-1}}},\end{eqnarray} \tag{ 4.58 }$$

since $q$-numbers retain the property $[M]_{q}^{2}={{[M+1]}_{q}}{{[M-1]}_{q}}+1$ for any $q\in \mathbb{C}$. By picking $q$ appropriately, we can obtain any desired constant value for $Y_{1|w}^{{}^\circ }$, and hence this is the general constant solution of the simplified TBA equations. What remains is to determine the value of $q$ such that these Y-functions also satisfy their canonical TBA equations. We do this in appendix A.4.1, and by substituting the result in the hybrid (4.52), (4.53), (4.54) we find the following asymptotic auxiliary Y-functions

$$\begin{eqnarray}&&Y_{M||w}^{{}^\circ +}={{[M]}_{{{q}_{\vartheta }}}}{{[M+2]}_{{{q}_{\vartheta }}}},Y_{M|w}^{{}^\circ -}={{[M]}_{{{q}_{{\dot{\vartheta }}}}}}{{[M+2]}_{{{q}_{{\dot{\vartheta }}}}}},\end{eqnarray} \tag{ 4.59 }$$

$$\begin{eqnarray}&&Y_{M|vw}^{{}^\circ +}={{[M]}_{{{q}_{\alpha }}}}{{[M+2]}_{{{q}_{\alpha }}}},Y_{M|vw}^{{}^\circ -}={{[M]}_{{{q}_{{\dot{\alpha }}}}}}{{[M+2]}_{{{q}_{{\dot{\alpha }}}}}},\end{eqnarray} \tag{ 4.60 }$$

$$\begin{eqnarray}&&Y_{\pm }^{{}^\circ +}={{[2]}_{{{q}_{\alpha }}}}/{{[2]}_{{{q}_{\vartheta }}}},\qquad \ \ \;Y_{\pm }^{{}^\circ -}={{[2]}_{{{q}_{{\dot{\alpha }}}}}}/{{[2]}_{{{q}_{{\dot{\vartheta }}}}}},\end{eqnarray} \tag{ 4.61 }$$

while the main Y-functions are

$$\begin{eqnarray}&&Y_{Q}^{{}^\circ }=({{[2]}_{{{q}_{\alpha }}}}-{{[2]}_{{{q}_{\vartheta }}}})({{[2]}_{{{q}_{{\dot{\alpha }}}}}}-{{[2]}_{{{q}_{{\dot{\vartheta }}}}}}){{[Q]}_{{{q}_{\alpha }}}}{{[Q]}_{{{q}_{{\dot{\alpha }}}}}}{{{\rm e}}^{-J{{\tilde{\mathcal{E}}}_{Q}}(\tilde{p})}},\end{eqnarray} \tag{ 4.62 }$$

where the $q$ have become phases denoted ${{q}_{\theta }}\equiv {{{\rm e}}^{{\rm i}\theta }}$. This asymptotic solution is a generalization of the asymptotic solution presented in [161] and [162] for deformations of the sphere. Of course, this asymptotic solution also follows directly from the twisted transfer matrix. For example, the asymptotic Y_Q-function is directly given by

$$\begin{eqnarray}&&Y_{Q}^{{}^\circ }=\chi (\pi (G))\chi (\pi (\dot{G})){{{\rm e}}^{-J{{\tilde{\mathcal{E}}}_{Q}}}},\end{eqnarray} \tag{ 4.63 }$$

where $\chi (\pi (G))={\rm Tr}(\pi (G))$ is the character of the twist element $G$ in the $Q$-particle bound state representation in the transfer matrix (4.15); the twisted transfer matrix for zero particles. Naturally this agrees with (4.62). Just as for the chemical potential, here we should keep in mind that the charges entering the twisted transfer matrix should be those of the ground state.

In the large $u$ limit, the convolutions on the right-hand side of the canonical TBA equations become insensitive to fluctuations of the Y-functions around the origin, and only their constant 'background' values play a role. It follows that for any state with a given set of chemical potentials, these constant values should be given by the asymptotic ground state solution. In other words

$$\begin{eqnarray}&&\mathop{{\rm lim} }\limits_{u\to \pm \infty }{{Y}_{\chi }}(u)=Y_{\chi }^{{}^\circ }\;,\end{eqnarray} \tag{ 4.64 }$$

where $Y_{\chi }^{{}^\circ }$ denotes the asymptotic ground state values of the Y-function of type χ, as given in equations (4.59–4.62). Note that while the regulator plays an essential role in the canonical TBA equations, the regulator can be taken away by describing the spectral problem through the simplified TBA equations and the asymptotics (4.64)¹⁰⁷ .

Coming to excited states, as we saw above the twisted transfer matrix can be used to construct the asymptotic Y-functions for the ground state. The same is true for excited states and allows us to apply the contour deformation trick [35, 123] to obtain excited state TBA equations from the ground state equations. The differences in analytic structure that distinguish the different excited states, i.e. the number and location of poles and zeros of the Y-functions, can be determined from the asymptotic Y-functions constructed through the twisted transfer matrix. This idea has already been used to find the excited state TBA equations for the $\mathfrak{s}\mathfrak{l}(2)$ descendant of the Konishi operator upon orbifolding the sphere [161]. We would like to point out that for orbifolds the asymptotics (4.64) are the same for any state in the same model, but this is not the case for theories obtained by TsT transformations as the chemical potentials depend on the state under consideration.

The simplified ground state TBA equations naturally have real Y-functions as solutions, and indeed the asymptotic solution (4.59)–(4.62) is real. Reality of the asymptotic Y-functions for excited states follows by the way the twist parameters enter the twisted transfer matrix (4.19); the terms in the undeformed transfer matrix that are related by conjugation are now multiplied by conjugate phases. The exact Y-functions that solve the excited state TBA equations are therefore also real, since the equations are obtained by the contour deformation trick with a real asymptotic solution.

In [119] the authors discussed the vanishing of the ground state energy of the undeformed theory at any finite size. However they observed a curious divergence of the ground state energy at J = 2. This divergence was not surprisingly also found subsequently in the case of orbifolds [161] and γ-deformations [162].

In order to show vanishing of the ground state energy, we would basically like to show that the asymptotic solution (here for zero twist) is exact, i.e. that vanishing Y_Q functions are a solution of the TBA equations. In order to show this the authors of [119] introduced a regularization in the form of a small chemical potential for the fermionic $y$-particles, denoted $h$, which made the Y_Q functions $O({{h}^{2}})$ such that they vanish in the $h\to 0$ limit. However for J = 2 this limit does not commute with the infinite sum in the energy formula leading to a divergent energy even though the Y_Q functions vanish. The chemical potential $h$ introduced there directly corresponds to an infinitesimal twist $\vartheta =\dot{\vartheta }=h$, and this divergence is indeed reproduced by our energy formula (4.66) which diverges as $\alpha ,\dot{\alpha }\to 0$ for non-zero ϑ and $\dot{\vartheta }$. However (4.66) does not diverge as $\alpha ,\dot{\alpha }\to 0$ for $\vartheta =\dot{\vartheta }=0$, meaning that a chemical potential in the ${\rm AdS}$ sector regularizes the ground state solution for the undeformed theory, showing that the ground state energy vanishes without any divergence as it should.

While the construction above provides us with a way to regularize the ground state energy of the undeformed theory, the interpretation or possible resolution of the divergence of the ground state energy¹⁰⁹ at J = 2 for certain orbifolds and γ-deformed theories remains an open question. Of course, as we discussed earlier these deformations break all supersymmetry and may result in broken conformal invariance in the 'dual' gauge theory. In particular, it is interesting to note that for γ-deformed SYM the operator ${\rm Tr}({{Z}^{J}})$ is insensitive to the non-conformality of the theory for $J\gt 2$, with anomalous dimensions precisely matching the energies of equation (4.67) above, while for J = 2 the result is scheme dependent [160].

As we saw above, the asymptotic ground state solution is an exact solution of the TBA equations when the corresponding Y_Q functions vanish in some appropriately regularized sense. In addition to the undeformed superstring, this situation arises when we deform parts of anti-de Sitter space and the sphere in an identical fashion. By this we mean a twist where at least two group elements are identical, i.e., $\alpha =\pm \vartheta$ or $\dot{\alpha }=\pm \dot{\vartheta }$. Indeed, in this limit the asymptotic $Y_{Q}^{{}^\circ }$ functions (4.62) and correspondingly the ground state energy (4.66) vanish. In other words, just as for the undeformed ground state, the corresponding asymptotic solution is an exact solution of the TBA equations. Naturally this is not an accident but owes to the residual supersymmetry left in this situation.

In the undeformed theory the ground state is a half-BPS state, meaning it is annihilated by a set of supercharges

$$\begin{eqnarray}&&Q_{\gamma }^{1,2}|0\rangle =\bar{Q}_{{\dot{\gamma }}}^{3,4}|0\rangle =0,\;\;\;\gamma =1,2.\end{eqnarray} \tag{ 4.73 }$$

Combined with the fact that it is a highest weight state this allows us to derive a relation between the scaling dimension and other weights of the state from the superconformal algebra in the usual fashion, in particular by considering the anti-commutator $\{Q,S\}$ [174]

$$\begin{eqnarray}&&\{Q_{\gamma }^{i},S_{j}^{\delta }\}=4\delta _{i}^{j}(M_{\gamma }^{\delta }-\frac{i}{2}D)-4\delta _{\gamma }^{\delta }R_{j}^{i},\end{eqnarray} \tag{ 4.74 }$$

where the $M_{\gamma }^{\delta }$ are the generators of the conformal group corresponding to the ${\rm SU}(2)$ subgroup with spin s₁ and the Rⁱ_j are the generators of ${\rm SU}(4)$ and $D$ is the dilatation operator. Indeed, since the left-hand side annihilates a highest weight half-BPS state for i = 1, 2 and the action of the Rⁱ_i is expressible via the corresponding weights, we get a relation between the scaling dimension and the $\mathfrak{s}\mathfrak{u}(4)$ weights $[{{q}_{1}},p,{{q}_{2}}]$. As a result the state cannot have an anomalous dimension; its scaling dimension is protected.

In a deformed theory, the ground state can still be considered half-BPS in the sense that it is annihilated by certain operators, but those operators have lost their meaning and relation to the scaling dimension since the superconformal algebra is broken. This means there are generically no protected operators. However, precisely deformations of the type $\alpha =\pm \vartheta$ or $\dot{\alpha }=\pm \dot{\vartheta }$ preserve part of the superconformal algebra. In particular some of the relations (4.74) which protect the scaling dimension of the ground state remain. Concretely we preserve the relations (4.74) for $i=j=1,2$, $\gamma =\delta =2,1,$ if we twist with $\alpha =\vartheta$ and for $i=j=\gamma =\delta =1,2,$ if we twist with $\alpha =-\vartheta$. The relations are similar for the $\bar{Q}{\rm s}$ and $\bar{S}{\rm s}$ with dotted twists. As a result the ground state energy is still a protected quantity in such deformed models.

In addition to the above, there is another class of deformed models with a ground state energy which does not receive corrections. The most interesting representative of this class is γ-deformed theory with ${{\gamma }_{2}}={{\gamma }_{3}}=0$. This feature arises because the ground state solution for generic γ-deformed theories is independent of γ₁ as immediately follows from our twisted transfer matrix since the ground state has only charge $J={{J}_{1}}$. This matches with the results of [162] since in the twisted $S$-matrix approach the ground state energy is independent of the Drinfeld–Reshetikhin twist, which in this case carries all dependence on γ₁. Since the Y_Q functions and ground state energy vanish in the limit ${{\gamma }_{2,3}}\to 0$, we have an exact solution of the TBA equations with zero energy. However, except for the cases $\{{{\gamma }_{1}},{{\gamma }_{2}},{{\gamma }_{3}}\}=\{\gamma ,\pm \gamma ,\pm \gamma \}$ (with all possible choices of signs) γ-deformations break all supersymmetry. Therefore we have no immediate explanation for the apparent 'protection' of the ground state energy, and it would be interesting to understand how this arises from the point of view of the deformed theory¹¹⁰ . Other models of this type would be generic TsT transformed backgrounds, with the parameters that couple to $J$ put to zero.

A single magnon in the $\mathfrak{s}\mathfrak{u}(2)$ sector of β-deformed theory has the following Bethe equation (cf equations (4.23) and (4.39))

$$\begin{eqnarray}&&1={{{\rm e}}^{{\rm I}(p+2\pi \beta )J}}\;\;\;\;p=-2\pi \beta +\frac{2\pi n}{J}\;,\;\;n=0,\ldots \left[ \frac{J}{2} \right]\;.\end{eqnarray} \tag{ 5.1 }$$

Level matching requires $p=-2\pi \beta$ so that a physical magnon has n = 0. Its asymptotic energy is

$$\begin{eqnarray}&&E=J+\sqrt{1+4{{g}^{2}}{{{\rm sin} }^{2}}\pi \beta }.\end{eqnarray} \tag{ 5.2 }$$

On the gauge theory side the corresponding operator is of the form ${\rm Tr}(Z\ldots ZXZ\ldots Z)$ with $J$ $Z{\rm s}$.

We can find the leading order wrapping correction to the energy from the weak coupling expansion of the exact energy formula (3.87), giving¹¹²

$$\begin{eqnarray}&&{{E}_{{\rm LO}}}=-\frac{1}{2\pi }\mathop{\sum }\limits_{Q=1}^{\infty }\int {\rm d}v\frac{{\rm d}{{\tilde{p}}^{Q}}}{{\rm d}v}Y_{Q}^{{}^\circ }(v).\end{eqnarray} \tag{ 5.3 }$$

To compute this we need to expand the twisted Y_Q-functions to leading order in $g$. From equations (4.39) we see that the two transfer matrices entering in Y_Q^o should be twisted by $\vartheta =\pi \beta$ and $\dot{\vartheta }=(2J-1)\pi \beta$ respectively. Denoting these generically as $T(v|u;\vartheta )$, we obtain the following Y-functions

$$\begin{eqnarray}&&{{Y}_{Q}}(v)=\frac{{{g}^{2J}}}{{{\left( {{Q}^{2}}+{{v}^{2}} \right)}^{J}}}\frac{T_{Q,1}^{\mathfrak{s}\mathfrak{u}(2)\ell }(v|u;\beta )T_{Q,1}^{\mathfrak{s}\mathfrak{u}(2)r}(v|u;(2J-1)\beta )}{{{S}_{0}}(u,v)},\end{eqnarray} \tag{ 5.4 }$$

where the coefficient ${{S}_{0}}(u,v)$ is the leading order expansion of the $\mathfrak{s}\mathfrak{l}(2)$ S-matrix given in appendix A.5. The leading order twisted transfer matrix in the $\mathfrak{s}\mathfrak{u}(2)$ sector is

$$\begin{eqnarray}\begin{array}{rcl} T_{Q,1}^{\mathfrak{s}\mathfrak{u}(2)}(v|u;\vartheta ) & = & \frac{g}{\sqrt{{{Q}^{2}}+{{v}^{2}}}}\left[ \frac{(Q-1)\left( {{\left( Q+1 \right)}^{2}}+{{\left( v-u \right)}^{2}} \right)}{(i(Q-1)+v-u)(i-u)} \right. \\ {} & {} & +\frac{Q{{{\rm e}}^{-{\rm I}\vartheta }}\sqrt{\frac{u+i}{u-i}}({{Q}^{2}}+{{\left( u-v+i \right)}^{2}})}{(i(1-Q)-v+u)(i-u)} \\ {} & {} & \left. +\frac{Q{{{\rm e}}^{{\rm I}\vartheta }}(i(Q+1)-v+u)}{(i-u)\sqrt{\frac{u+i}{u-i}}}+\frac{(Q+1)(i(1-Q)+v-u)}{i-u} \right]. \\ \end{array}\end{eqnarray} \tag{ 5.5 }$$

This transfer matrix starts at order $g$ so that our Y-function and associated corrections to energy and momentum will be of order ${{g}^{2J+2}}$. We can now straightforwardly compute the first wrapping correction to the energy by integrating and summing these Y-functions. The result is a sum of ζ-functions¹¹³

$$\begin{eqnarray}&&E_{{\rm LO}}^{\beta }(J)={{g}^{2J+2}}\mathop{\sum }\limits_{n=1}^{\left \lfloor \frac{J+1}{2}\right \rfloor }16{{\left( -1 \right)}^{n}}\frac{\Gamma (J-n+1)\Gamma \left( J-n+\frac{3}{2} \right)}{\sqrt{\pi }\;\Gamma (J+1)\Gamma (J-2n+3)}B_{n}^{\beta }(J)\zeta (2J-2n+1),\end{eqnarray} \tag{ 5.6 }$$

where the coefficients are given by the following expressions

$$\begin{eqnarray}\begin{array}{rcl} B_{1}^{\beta }(J) & = & -\frac{J}{2}{\rm sin} (J\pi \beta ){{{\rm sin} }^{2}}(\pi \beta ){\rm sin} ((J-2)\pi \beta ), \\ B_{n\gt 1}^{\beta }(J) & = & {\rm sin} (J\pi \beta ){{{\rm sin} }^{2n}}(\pi \beta )\left[ (n-1){\rm sin} ((J-2n)\pi \beta ) \right. \\ {} & {} & -\left. J{\rm cos} (\pi \beta ){\rm sin} ((J-2n+1)\pi \beta ) \right]. \\ \end{array}\end{eqnarray} \tag{ 5.7 }$$

This correction perfectly agrees with the field-theoretic result obtained in [187, 188], as well as the results of [189] obtained by arbitrarily twisting the asymptotic solution of the Y-system and fixing the twists by comparison with the Bethe equations.

We should pay particular attention when $\beta =\frac{n}{J}$ with $n=0,1,\ldots ,J-1$. For these values of the deformation parameter, the naive lowest order in the weak-coupling expansion of the transfer matrix $T_{Q,1}^{\mathfrak{s}\mathfrak{u}(2)}(v|u;(2J-1)\beta )$ vanishes and the expansion really starts two powers of $g$ higher at ${{g}^{2J+4}}$. Bearing in mind that the rapidity $u$ is $g$-dependent through equation (3.39) with momentum $p=-2\pi \beta$, we can easily check that the functions Y_Q for these special values of β are absolutely the same as for generic β provided we shift $J\to J+1$. Hence we find

$$\begin{eqnarray}&&E_{{\rm LO}}^{\beta =n/J}(J)={{g}^{2}}{{\left. E_{{\rm LO}}^{\beta }(J+1) \right|}_{\beta =\frac{n}{J}}}.\end{eqnarray} \tag{ 5.8 }$$

In section 4.1.2 of the previous section we discussed that the asymptotic Bethe ansatz describes ${\rm U}(N)$ β-deformed SYM rather than ${\rm SU}(N)$ β-deformed SYM [156]. We mentioned that the operator ${\rm Tr}(XZ)$ (J = 1) is affected by a so-called pre-wrapping effect which distinguishes between ${\rm U}(N)$ and ${\rm SU}(N)$.¹¹⁴ ${\rm Tr}(XZ)$ is in fact protected in the ${\rm SU}(N)$ theory [165], at least up to two loops [166], while it acquires a non-zero anomalous dimension in the ${\rm U}(N)$ theory. As our string theory is supposed to describe the ${\rm SU}(N)$ rather than the ${\rm U}(N)$ theory [156] this raises an immediate question because the asymptotic Bethe ansatz for our string predicts a non-zero anomalous dimension

$$\begin{eqnarray}&&E=1+2{{{\rm sin} }^{2}}(\pi \beta ){{g}^{2}}-2{{{\rm sin} }^{4}}(\pi \beta ){{g}^{4}}+\ldots \;.\end{eqnarray} \tag{ 5.9 }$$

Finite-size effects can and will of course correct this energy, but the conventional wrapping correction discussed above gives a two loop correction for J = 1, meaning it cannot cancel the one loop contribution of the asymptotic Bethe ansatz. At the same time, taking a look at the above energy correction we see that the leading transcendentality term is $\zeta (2J-1)$. Extending this result to J = 1 would suggest that the corresponding wrapping correction is divergent, raising questions of its own.

Naively extending our computation to J = 1 we indeed find a divergent term as in equation (5.6), but also a finite piece

$$\begin{eqnarray}&&E_{{\rm LO}}^{\beta }(1)={{g}^{4}}(4-4\zeta (1)){{{\rm sin} }^{4}}\pi \beta \;.\end{eqnarray} \tag{ 5.10 }$$

This divergence is similar (in form) to that of the leading order wrapping correction for ${\rm Tr}({{Z}^{2}})$ in the γ-deformed theory. This divergence clearly signals some kind of problem, and its interpretation is an interesting question.

Let us first consider that our string theory really describes the ${\rm SU}(N)$ theory, meaning that finite-size effects should contain the pre-wrapping effects that cancel the one and two-loop anomalous dimension. It then becomes tempting to consider that there is a technical problem with the naive computation we just considered, and that resolving this would result in the desired cancellations. Of course, we also cannot exclude that for some reason the generalized Lüscher formula should be adapted in this particular case, but at the moment it would be unclear how. In case our string theory¹¹⁵ is really 'dual' to the ${\rm U}(N)$ theory rather than the ${\rm SU}(N)$ one, this divergence may be related to the lack of conformality of the former, and could link it to the divergence for ${\rm Tr}({{Z}^{2}})$ in γ-deformed or non-supersymmetric orbifolded theories mentioned in the previous section (see also the discussion in [157]). Still, β-deformed ${\rm U}(N)$ SYM has the ${\rm SU}(N)$ theory as its fixed point while the γ-deformed and non-supersymmetric orbifolded theories have no such fixed point.

From a purely technical point of view we should note that the divergent wrapping corrections we have seen here and in the previous section arise for operators of length two, and that it seems unlikely that the asymptotic Y-functions for longer operators would result in divergent energies. Hence, by considering longer operators which are effected by pre-wrapping [164] we should be able to gain valuable insight into part of the questions raised here, while side-stepping complicating divergences. Unfortunately these longer operators lie outside the simple $\mathfrak{s}\mathfrak{u}(1|1)$, $\mathfrak{s}\mathfrak{u}(2)$, and $\mathfrak{s}\mathfrak{l}(2)$ sectors and would require venturing into the $\mathfrak{s}\mathfrak{u}(2|3)$ sector. Provided the corresponding energy corrections are non-divergent, comparing these to anomalous dimensions in the dual gauge theory would presumably give concrete indications regarding the gauge group as well. This point clearly deserves further investigation.

Next we would like to reproduce the wrapping energy correction found in [187] for the Konishi state in the $\mathfrak{s}\mathfrak{u}(2)$ sector of β-deformed theory. The perturbative computation of the anomalous dimension of β-deformed $\mathfrak{s}\mathfrak{u}(2)$ Konishi-like states [187] agrees perfectly with results coming from the asymptotic Bethe equations up to three loops. At four loops there is a discrepancy arising due to wrapping effects, which we should be able to explain via Lüscher formulae. In [190] a modified S-matrix was used to obtain an expression for the wrapping correction to the energy, which is in agreement with the explicit perturbative results. Here we would like to approach the problem via our twisted transfer matrix. However, as the result of [190] fits in the framework of the subsequently fully developed twisted S-matrix approach discussed at the end of the last section, our twisted transfer matrix gives equivalent input and the result is immediate. The energy correction is given by

$$\begin{eqnarray*}\begin{array}{rcl} E_{{\rm Konishi}}^{\beta } & = & {{g}^{8}}\left[ -54{{\left( 1+\Delta \right)}^{3}}(-5+3\Delta )\zeta (3)-360{{\left( 1+\Delta \right)}^{2}}\zeta (5) \right. \\ {} & {} & +\left. \frac{81{{\left( 1-3\Delta \right)}^{2}}{{\left( 1+\Delta \right)}^{4}}}{{{\left( 1+3\Delta \right)}^{2}}} \right], \\ \end{array}\end{eqnarray*}$$

where $\Delta =\frac{\sqrt{5+4{\rm cos} 4\pi \beta }}{3}$. Of course explicit calculations based on the twisted transfer matrix gives the same result. We can compute wrapping corrections to many other states in this way, some showing interesting quantitative features. Explicit results can be found in [170, 191].

We will take the approach advocated in sections 2 and 3 and find the TBA equations for our orbifolded Konishi state by means of the contour deformation trick. For this we need to understand the analytic properties of the asymptotic solution, so we will start there.

Naturally we should expect that the analytic properties of the Y-functions depend rather directly on the twist while their finer details depend on the coupling constant $g$. Firstly, in line with the general discussion in section 4.4.2 the asymptotics of ${{Y}_{M|vw}}$ on the mirror line are $M(M+2)$

$$\begin{eqnarray}&&\mathop{{\rm lim} }\limits_{v\to \infty }{{Y}_{M|vw}}(v)=M(M+2),\end{eqnarray} \tag{ 5.13 }$$

while the asymptotics of ${{Y}_{M|w}}$ and ${{Y}_{\pm }}$ depend on the twist and indicate interesting behavior as the asymptote can be both positive and negative. The asymptotic values of both ${{Y}_{\pm }}$ are given by

$$\begin{eqnarray}&&\mathop{{\rm lim} }\limits_{v\to \infty }\;{{Y}_{\pm }}(v)={\rm sec} \alpha ,\end{eqnarray} \tag{ 5.14 }$$

while those of ${{Y}_{M|w}}$ are given by

$$\begin{eqnarray}&&\mathop{{\rm lim} }\limits_{v\to \infty }\;{{Y}_{M|w}}(v)=M+2\mathop{\sum }\limits_{Q=0}^{M}(M-Q){\rm cos} (2Q+2)\alpha .\end{eqnarray} \tag{ 5.15 }$$

We have plotted the asymptotics of ${{Y}_{-}}$, ${{Y}_{1|w}}$, ${{Y}_{2|w}}$, and ${{Y}_{3|w}}$ in figure 18. It is interesting to note that the asymptotics in the zero twist limit of course smoothly reduce to the known values of 1 and $M(M+2)$ respectively, but that this also almost happens when the twist parameter is equal to π,

$$\begin{eqnarray}&&\mathop{{\rm lim} }\limits_{v\to \infty }\;Y_{M|w}^{\alpha =0,\pi }(v)=M(M+2),\end{eqnarray} \tag{ 5.16 }$$

$$\begin{eqnarray}&&\mathop{{\rm lim} }\limits_{v\to \infty }\;Y_{\pm }^{\alpha =0}(v)=-\mathop{{\rm lim} }\limits_{v\to \infty }Y_{\pm }^{\alpha =\pi }(v)=1.\end{eqnarray} \tag{ 5.17 }$$

This is of course a very particular value, but still we clearly see that the asymptotic solution is really different from the untwisted case and corresponds to a ${{\mathbb{Z}}_{2}}$ orbifold.

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The fact that the asymptotic values of the ${{Y}_{M|w}}$-functions can change from positive to negative means that these functions must have roots on the real mirror line for certain values of the twist and coupling, roots which are not present for the untwisted Konishi state [35]. However, these roots will always become relevant at finite values of $g$. As the twist is varied this finite value of $g$ can approach zero, cf figure 18. At the same time these roots move away to infinity, so that at weak coupling they do not play a role. Now let us concretely discuss the analytic properties of the Y-functions.

Let us start by recalling the construction of the asymptotic Y-functions in terms of the transfer matrices,

$$\begin{eqnarray}\begin{array}{rcl} {{Y}_{M|w}} & = & \frac{{{T}_{1,M}}{{T}_{1,M+2}}}{{{T}_{2,M+1}}},\;\;\;\;\;{{Y}_{-}}=-\frac{{{T}_{2,1}}}{{{T}_{1,2}}},\;\;\;\;\;{{Y}_{+}}=-\frac{{{T}_{2,3}}{{T}_{2,1}}}{{{T}_{1,2}}{{T}_{3,2}}}, \\ {{Y}_{M|vw}} & = & \frac{{{T}_{M,1}}{{T}_{M+2,1}}}{{{T}_{M+1,2}}}=\frac{{{T}_{M,1}}{{T}_{M+2,1}}}{T_{M+1,1}^{-}T_{M+1,1}^{+}-{{T}_{M,1}}{{T}_{M+2,1}}}. \\ \end{array}\end{eqnarray} \tag{ 5.18 }$$

Here $T_{M,Q}^{\pm }$ denotes ${{T}_{M,Q}}$ with its argument shifted by $\pm i/g$. Roots of ${{Y}_{M|w}}$ arise from the roots of ${{T}_{1,M}}$ and ${{T}_{1,M+2}}$, while roots of ${{Y}_{M|vw}}$ arise from the roots of ${{T}_{M,1}}$ and ${{T}_{M+2,1}}$. For ${{Y}_{M|vw}}$ it is immediately clear that its roots shifted by $\pm i/g$ give roots of $1+{{Y}_{M-1|vw}}$ and $1+{{Y}_{M+1|vw}}$,

$$\begin{eqnarray}&&1+{{Y}_{M|vw}}(r_{M}^{\pm })=0,\;\;{\rm where}\;\;\;{{T}_{M,1}}({{r}_{M-1}})=0\;.\end{eqnarray} \tag{ 5.19 }$$

While not obvious from their expression, a similar relation holds for ${{Y}_{M|w}}$,

$$\begin{eqnarray}&&1+{{Y}_{M|w}}(\rho _{M}^{\pm })=0,\;\;{\rm where}\;\;\;{{T}_{1,M}}({{\rho }_{M-1}})=0.\end{eqnarray} \tag{ 5.20 }$$

The analytic properties for the orbifolded Konishi state are quite involved, in the sense that a large number of roots appear and play a role in the TBA, with many of them displaying critical behavior of a nature that depends on the specific value of the twist. Let us also mention here that this criticality is naturally related to the roots of the ${{Y}_{M|w}}$-functions on the real mirror line, indicated above. In the main text we will refrain from discussing these technical complications and focus on the TBA equations at small coupling. A short discussion on the finer details of the analytic properties can be found in appendix A.5.1 ¹¹⁸ .

At small coupling, the ${{Y}_{M|vw}}$-functions have four roots, which we will denote by $\pm {{{\bf r}}_{M\pm 1}}$ ¹¹⁹ . These roots were not observed for the true Konishi state, which agrees with our findings since these roots move away towards infinity as the twist is removed, see figure 19. As these roots are real, their shifted counterparts, the roots of $1+{{Y}_{M\pm 1|vw}}$, will always appear in the TBA for orbifolded Konishi. Just as in the Konishi case, ${{Y}_{+}}$ has poles at $u_{i}^{\pm }$, which lead to the appearance of additional driving terms in the TBA equations. While not discussing criticality here, let us briefly note that we reproduce the critical behavior found in [35] as the smooth $\vartheta \to 0$ limit of our general story, see appendix A.5.1 for more details.

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To summarize, at small coupling the following points are important in the simplified TBA equations,

$$\begin{eqnarray}&&{{Y}_{+}}(u_{i}^{\pm })=\infty \;,{{Y}_{M|vw}}({\bf r}_{M}^{\pm })={{Y}_{M|vw}}(-{\bf r}_{M}^{\pm })=-1\;.\end{eqnarray} \tag{ 5.21 }$$

Following the contour deformation trick, in order for the asymptotic solution to be a solution we take the ground state TBA equations in simplified (and hybrid) form and define the integration contour such that it goes slightly below the line $-i/g$, i.e. such that it encloses the poles of ${{Y}_{+}}$ at $u_{i}^{-}$ and the roots of $1+{{Y}_{M|vw}}$ at $\pm {\bf r}_{M}^{-}$ between itself and the real line. By taking the integration contour back to the real line we get our explicit TBA equations.