The Nicolai-map approach to supersymmetry

In 1980 Hermann Nicolai proposed a characterization of supersymmetric theories that became known as the Nicolai map. This is a particular nonlocal and nonlinear field transformation, whose perturbative expansion is given by fermion-line trees with bosonic leaves. Quantum correlation functions can by evaluated using the inversely transformed fields in the free theory. After initial promise and excitement (fuelling the author’s PhD work!), the subject all but fell dormant for 35 years. Recently however, technical progress in the construction as well as a deeper insight into the nature of the map have been achieved, from quantum mechanics to super Yang–Mills in various dimensions. I will present the Nicolai map from this modern perspective and touch on some of the current developments.

1.A question raised in 1980 (by Nicolai) and answered until 1984 (but not fully ...) The key idea is best illustrated by an example.Let us look at the Wess-Zumino model in 3+1 dimensional Minkowski space, consisting of a complex scalar ϕ, a Weyl fermion ψ and a complex auxiliary F , characterized by a superpotential W (ϕ) and defined by the off-shell lagrangian 1 where (σ µ ) = (1, ⃗ σ) and (σ µ ) = (1, −⃗ σ) with Pauli matrices ⃗ σ.Integrating out the auxiliary fields yields F * = −W ′ (ϕ) and Integrating out the fermions (ψ, ψ) produces a functional determinant det M = exp{ i ℏ •(−iℏ tr ln M )} so that the action becomes Here, g denotes some coupling constant(s) or parameter(s) inside the superpotential W (ϕ). The objects of desire are quantum correlators for any bosonic (local or nonlocal) functional Y .
1 A multi-field generalization is straightforward.

arXiv:2309.00481v1 [hep-th] 1 Sep 2023
The path integral in (4) describes a purely bosonic nonlocal field theory.What is characteristic of its supersymmetric origin?In other words: given such a nonlocal action S g , how could one infer its hidden supersymmetric roots?This question was answered in 1980 by Hermann Nicolai [1,2,3]: Such hiddenly supersymmetric theories admit a nonlocal and nonlinear invertible map relating correlators in the interacting theory (g̸ =0) to (more complicated) correlators in the free theory (g=0).For the path integrals, this is equivalent to Separating powers of ℏ in the exponent, this splits into two properties, "free action condition" , (7a) "determinant matching condition" .
Every Nicolai map has to fulfil these two conditions, which originally were taken as its definition.
The reason for the name of (7b) is that its exponentiation gives an equality of the functional fermion determinant det M with the Jacobian of the transformation (the first term is a constant since S f 0 does not depend on ϕ).From now on we put ℏ=1.
The Nicolai map provides an alternative characterization of supersymmetry or, more colloquially, "supersymmetry without fermions".Here is a sketch of its early history: In 1984, the author derived (for his dissertation) an infinitesimal version [4,5,6,7] of the Nicolai map by considering the g-derivative of (5), with a "coupling flow operator"2 representing a functional differential operator derived from T g .Nothing is gained, however, by these formal considerations, unless we can reverse the logic and somehow obtain R g and exponentiate it in order to create a finite coupling flow T g from g ′ =0 to g ′ =g, by inverting  As we shall see in a moment, however, there exists a more direct construction of T g .In any case, we have to establish the existence of the coupling flow operator R g and find an explicit expression for it.We shall do this now for the exemplary case of scalar theories (gauge theories will be treated in the following section).If supersymmetry is realized off-shell on the action S then there exists a functional ∆α [ϕ, ψ, F ] such that for the supersymmetry transformations δ α , where α denotes a Majorana spinor index.Integrating out the auxiliary F one has that for the on-shell action S SUSY = dx L SUSY with an anticommuting functional ∆ α .For our Wess-Zumino model example, it reads The construction of R g employs the supersymmetry Ward identity, Integrating out the fermions contracts bilinears to produce fermion propagators ψ ψ (in the ϕ background), hence [4] For a simple example of the Wess-Zumino model with (massless where the subscript on the curly brace indicates a spin trace.It is instructive to develop a diagrammatical shorthand notation.For the sake of illustration, here we oversimplify (ϕ, ϕ * ) ∼ ϕ and display the coupling flow operator as in Figure 1 with graphical rules [8] outlined in Figure 2. The linear tree for R g exponentiates to a series of branched trees for T g ϕ, as exhibited in Figure 3, and likewise for the inverse T −1 g ϕ.Inserting the latter into (5) and performing the free-theory bosonic contractions, one obtains an alternative "Nicolai" perturbation series for correlators, as shown in Figure 4 for the twopoint function.Notably, the multiple action of R g produces multiple spin traces (graphically separated by dots).The supersymmetric cancellation of the leading UV divergencies is automatically built in, as pure fermion loops and boson tadpoles are absent by construction.A more universal answer in 2021 Let us briefly focus on two important properties of the Nicolai map.Firstly, R g is a derivation, and hence Secondly, by moving the map "to the other side", choosing ∂ g Y = 0 and differentiating with respect to g, we learn that for any (not explicitly g-dependent) functional Y , and therefore necessarily This "fixpoint property" of the Nicolai map under the infinitesimal coupling flow allows us to directly construct T g ϕ from R g without invoking the inverse first [7].Indeed, as had been missed in 1984, ( 19) is formally solved by a path-ordered exponential, providing a "universal formula" for the Nicolai map in terms of the infinitesimal coupling flow [9].It is often useful to expand the flow operator in powers of the coupling, from which one easily computes a power series expansion for the map itself, where 1 ≤ s ≤ n and the n=0 term is the identity.The numerical coefficients are computed as (23) and related to the Stirling numbers of the second kind.Writing out the first few terms and suppressing the functional argument ϕ of r k , the perturbative Nicolai map reads [9] T ( For computing correlation functions à la (5) we need the inverse map.It possesses an analogous universal representation in terms of an anti-path-ordered exponential, which gives rise to a different power series expansion, (25) whose first terms are Comparing this with (24) it is clear how the two sets of coefficents are related.This universal formula naturally generalizes to multiple couplings [10].Collecting a number k of couplings in a formal vector, ⃗ g = g (i) = g (1) , g (2) , . . ., g (k) , the individual flow equations read and define a formal vector ⃗ R ⃗ g := R (1) ⃗ g , . . ., R of flow operators (depending on all couplings).The universal formula now operates in a k-dimensional coupling space, thus the path-ordered exponential requires choosing a path With these preliminaries, we obtain [10] which upon Taylor-expanding ⃗ R ⃗ h(t i ) in powers of g (j) produces a generalization of (22).We note that this map in general depends on the chosen path h in coupling space, despite ⃗ g [ϕ] ⃗ g = 0 .(31)

Figure 5. Graphical Nicolai map expansion for supersymmetric quantum mechanics with theta term
This "flatness condition" in coupling space is only valid "on the average".Hence, generically we have a functional family of Nicolai maps, yet all members yield the same correlation functions.It is of course possible to consider paths that keep some couplings fixed, which then are "spectating parameters" (giving partial Nicolai maps) or to connect two finite-coupling values by flowing along some path leading from one to the other.In special cases, however, the map may be path-independent, and then it is unique!In such a situation with more than one coupling, all powers in g (i) beyond the first cancel out, and the power series truncates to a linear map, This is however not so for single-coupling flows, since these are path independent and in general nonpolynomial.Yet, even a single-coupling flow collapses to a linear map if only [10] where "no branch" means omitting all branched trees in the expansion.This is not an empty condition but can happen for special fixed-coupling values.An example is supersymmetric quantum mechanics, say in one dimension for a bosonic trajectory x(t) and its Grassmann-valued fermionic partner ψ(t).
With a cubic superpotential W (x) = 1 2 mx 2 + 1 3 gx 3 and the inclusion of a total-derivative "theta term", the (on-shell) lagrangian reads ( Let us keep the mass m and the topological parameter θ fixed and flow in the coupling g only.Fourier-transforming from the time (t) to the frequency (ω) domain, the fermion propagator in ω-space, depending on its direction, is denoted as Depicting the x insertions by wavy lines as usual, with energy conservation at each vertex in frequency space, the diagrammatic representation of the Nicolai map up to order g 3 [10] takes the form of Figure 5, from which it is evident that the power series collapses for θ = ±1.

The case of gauge theories
Suprsymmetric gauge theories present additional challenges for the Nicolai map.Firstly, one has to deal with the gauge redundancy necessitating a gauge fixing which in a Wess-Zumino gauge breaks supersymmetry and, secondly, the g-derivative of the supersymmetric action cannot easily be expressed as a supervariation but also needs a BRST variation.Hence, Ward identities for BRST as well as broken supersymmetry will be required [11,12].The fermion matching condition for the map now equates its Jacobian with the product of the fermion and the Faddeev-Popov determinant, and the gauge-fixing condition should not be altered by the map.We allow for a topological theta term (only in D=4) with θ ′ = g 2 θ 8π 2 , use a local gauge-fixing functional G to fix a gauge G(A)=0 with a parameter ξ and include the corresponding ghost fields to formulate a BRST-invariant action The trace refers to the color degrees of freedom.We allow for various spacetime dimensionalities D by letting the fields live on R 1,D−1 so that Lorentz indices µ, ν, . . .= 0, 1, . . ., D−1 and Majorana indices α = 1, . . ., r, where r is the complex dimension of the corresponding Majorana representation, i.e. λ A ∈ C r .It essentially grows exponentially with D.
We know of two different Nicolai-map constructions, an off-shell one and an on-shell one.In both cases, a gauge-fixing breaks supersymmetry.The off-shell variant [4,13,14] requires off-shell supersymmetry, thus works only (for a finite number of auxiliary fields) in four or less spacetime dimensions.It parallels the construction for chiral multiplets, employing gauge superfields, and admits any choice of gauge fixing.The on-shell version [8,12] works in higher dimensions but relies on an ansatz for the coupling flow operator, which however does its job only partially, Here, a multiplicative contribution Z g destroys the derivation property of R g and hence the distributivity of T g , which is not acceptable.A somewhat lengthy computation reveals, however, that in the Landau gauge, G=∂ µ A µ with ξ→∞, the obstacle may be overcome because Amazingly, these are precisely the "critial spacetime dimensions" which admit super Yang-Mills theory to exist [15], demonstrating that the Nicolai map knows about them [11]!In both versions, the most simple form of the coupling flow operator and thus the Nicolai map arises in the Landau gauge on the gauge hypersurface with the non-Abelian transversal projector [12] forcing the coupling flow onto the gauge surface: R g G ∼ G.We have reversed the direction of the derivatives since acting towards the left is more convenient for the graphical representation.Due to the identity D ν . . .ν = 0, the three pieces of P ν µ named above are of order g 0 , g 1 and g 2 , respectively.The upshot is that our explicit construction formula (20) carries over to gauge theory, for D≤4 in any gauge and for D=6 and 10 in the Landau gauge [14], Here, we have expanded the coupling flow operator into homogeneous pieces, where the Taylor coefficients r k decompose according to (40), The diagrammatics in the Landau gauge looks fairly simple [8,12].With the solid line representing the free fermion propagator (i / ∂) −1 and the dashed line standing for the free ghost propagator □ −1 , we obtain for ← − R g [A] a linear tree expansion [12] as in Figure 6.Iterating this functional differential operator in the universal formula (41) produces (with rules analogous to the scalar case) the graphical representation of the Nicolai map [12] is given in Figure 7. where the color structure follows the graphical one, and we have suppressed the Lorentz and spinor indices.In fact, performing the spin traces creates various contractions of Lorentz indices on the gauge-field legs and on the propagators due to (i / ∂) −1 = iγ µ ∂ µ □ −1 , so that the number of terms at O(g n ) grows rapidly with n.Nevertheless, the expansion is algorithmic and may be implemented on a computer.
For θ=0 the map has been evaluated to order g 3 in 2020 [12] and pushed to order g 4 one year later [13].Let us go beyond and allow for the topological term in D=4.Abbreviating we obtain (now r=4) Reminiscent of the map collapse for supersymmetric quantum mechanics in the previous section, we anticipate simplifications for the magical values θ ′ = ±i, where at least the second-order contribution cancels out!And indeed, for these two choices the square bracket in (39) becomes 1 ∓ γ 5 , which is (twice) the chiral spin projector.In this special case, the Fierz identity tr Y 1 1979 the map for scalar theories, existence proof (incomplete) 1980 refinement of the proof but a gap remains (Golterman 1982) 1980 extension to gauge theories, construction to O(g 2 ) [to O(g 3 ) in 2020, O(g 4 ) in 2021!] 1982 examples of linear maps in D≤2, zero-mode obstruction for super Yang-Mills ← Witten index) 1984 linear maps for D = 4 & 6 super Yang-Mills, doubtful due to "Euclidean light-cone gauge" 1983/84 constructive (perturbative) proof via coupling flow, requires off-shell supersymmetry

Figure 1 .
Figure 1.Graphical representation of the coupling flow operator

Figure 3 .
Figure 3. Graphical representation of the Nicolai map to third order

Figure 4 .
Figure 4. "Nicolai diagrams" for the two-point function