Field Theory Equivalences as Spans of L∞-algebras

Semi-classically equivalent ﬁeld theories are related by a quasi-isomorphism bet-ween their underlying L 8 -algebras, but such a quasi-isomorphism is not necessarily a homotopy transfer. We demonstrate that all quasi-isomorphisms can be lifted to spans of L 8 -algebras in which the quasi-isomorphic L 8 -algebras are obtained from a correspondence L 8 -algebra by a homotopy transfer. Our construction is very useful: homotopy transfer is computationally tractable, and physically, it amounts to integrating out ﬁelds in a Feynman diagram expansion. Spans of L 8 -algebras appear naturally in many contexts within physics. As examples, we ﬁrst consider scalar ﬁeld theory with interaction vertices blown up in diﬀerent ways. We then show that (non-Abelian) T-duality can be seen as a span of L 8 -algebras, and we provide full details in the case of the principal chiral model. We also present the relevant span of L 8 -algebras for the Penrose– Ward transform in the context of self-dual Yang–Mills theory and Bogomolny monopoles.

More precisely, it is the Chevalley-Eilenberg algebra of the cyclic L 8 -algebra.An L 8algebra is a generalisation of a differential graded Lie algebra in which the Jacobi identity is violated up to homotopies, resulting in a tower of homotopy Jacobi identities.In physics, these homotopy Jacobi identities amount to closure of gauge transformations and gauge covariance of the equation of motion.
All perturbative ghosts, fields, anti-fields, anti-fields of ghosts, etc., arrange into a graded vector space, and the free or linear terms in the equations of motion of the BV action give rise to differentials, turning the graded vector space into a cochain complex.We note that the cohomology of this cochain complex is given by the free fields up to gauge transformations.The interaction terms in the equations of motion that are of order n in the fields define operations with n inputs and one output, which provide the higher products of the L 8 -algebra.The additional structures (inner products and integrals) contained in the BV action over its equations of motion induce a metric structure on the L 8 -algebra.These facts have been observed and explored further numerous times since the birth of L 8algebras in the context of closed string field theory [11], see e.g.[12] and [13] for important examples and [10,14] for a more complete list of references.Well-known is also [15], but in this paper the link to the BV formalism seems to have been made only partially.
Remarkably, the parallel between quantum field theory and homotopical algebra extends far beyond the equations of motion.Homotopy algebras come with a notion of quasi-isomorphism, extending the corresponding one from cochain complexes.Quasiisomorphisms between L 8 -algebras translate to semi-classically equivalent field theories, i.e. field theories with the same tree-level scattering amplitudes.Moreover, any homotopy algebra possesses a quasi-isomorphic minimal model, i.e. a homotopy algebraic structure on the cohomology of their underlying cochain complex, which is unique up to isomorphisms.In the case of L 8 -algebras of field theories, the minimal model encodes the tree-level scattering amplitudes.Furthermore, it is conveniently computed by the homological perturbation lemma [16][17][18][19], which encodes the usual tree-level Feynman diagram expansion in a geometric series.This geometric series gives rise to Berends-Giele recursion relations, see e.g.[20,21].As already implied in [11], much of this extends to the loop level, see [22,13] as well as the closely related work [23,24].
Generally, the homological perturbation lemma may be used to extend any homotopy retract (i.e. a weaker form of a homotopy equivalence) between the cochain complex of an L 8 -algebra and another cochain complex to a quasi-isomorphism of L 8 -algebras.This is known as homotopy transfer, see [25] for a detailed account.From a field theoretic perspective, a homotopy transfer translates a field theory on a field space to an equivalent field theory on another field space.If the latter space is embedded in the former, then a homotopy transfer amounts to integrating out fields, a well-known fact in BV quantisation 1 .
For a recent discussion and application of this fact, see [29,30].
However, not all equivalences between field theories or, equivalently, quasi-isomorphisms of L 8 -algebras can be captured by a homotopy transfer. 2From a field theoretic perspective, this observation is hardly surprising.For example, there is a quasi-isomorphism between the L 8 -algebra that describes the tree-level scattering amplitudes of a field theory and the L 8 -algebra that describes the action of this field theory.However, we clearly cannot reconstruct the complete form of a field theory from its tree-level scattering amplitudes. 3his, however, is unfortunate, as the perturbative expansions in terms of Feynman diagrams implied by homotopy transfer, together with the underlying recursion relations, can be very useful.
There are therefore many situations in which a physically equivalent field theory is constructed in a two-step-procedure, by first integrating in fields and then integrating out different fields.Notable examples are (non-Abelian) T-duality for sigma models and the Penrose-Ward transform.This naturally suggests the picture that for any two semiclassically equivalent field theories L p1q and L p2q , there is a 'correspondence theory' L pcq that is semi-classically equivalent to both L p1q and L p2q together with homotopy transfers between L pcq and L p1q and between L pcq and L p2q , respectively, amounting to integrating out fields in L pcq : L pcq L p1q L p2q (1.1) Mathematically speaking, one may therefore conjecture that for any pair of L 8 -algebras1 L p1q and L p2q , there is an L 8 -algebra L pcq quasi-isomorphic to both L p1q and L p2q , and where the quasi-isomorphism can be captured by a homotopy transfer.We note that the conjectured statement would also allow for a so far missing, clean definition of quasiisomorphisms of cyclic L 8 -algebras.
We shall prove that this is indeed the case in Section 2. We then work out the details for three distinct cases: firstly, an illustrative toy example of two quasi-isomorphic scalar field theories in Section 3; secondly, the much more intricate and interesting case of the principal chiral model and its (non-Abelian) T-dual in Section 4; and thirdly, the Penrose-Ward transforms between field theories allowing for a twistorial description and holomorphic Chern-Simons theory on the corresponding twistor spaces in Section 5. We note that further examples of such spans, but described from the dual, BV perspective, can be found in [32,33].

Quasi-isomorphisms and homotopy transfer
Quasi-isomorphisms of cochain complexes, such as de Rham complexes, are cochain maps that induce isomorphisms between the underlying cohomology groups.If one of the cochain complexes carries a homotopy algebra structure, then this structure can be transferred to the other cochain complex in a procedure called homotopy transfer, see e.g [34] and [25] for a comprehensive review.Explicit formulas for such a homotopy transfer are provided by the homological perturbation lemma [16][17][18], see also [19].
However, not all quasi-isomorphisms of homotopy algebras originate from a homotopy transfer [31].As we show below for the case of L 8 -algebras, however, each quasiisomorphism can be lifted to a span (or roof, or correspondence) of homotopy algebras, in which the projections are given by homotopy transfers.This perspective is very natural for a number of reasons.In particular, L 8 -algebras concentrated in non-positive degrees integrate to Lie 8-groupoids.Already for ordinary groupoids, fully faithful and essentially surjective functors can fail to be equivalences.In such situations, one usually switches to spans.
We will show that our new perspective gives rise to a natural definition of quasiisomorphism between cyclic L 8 -algebras, which so far existed only in special cases.
Moreover, it is useful in the context of perturbative quantum field theory, where quasiisomorphisms of L 8 -algebras amount to a semi-classical equivalence, and homotopy transfers amount to Feynman diagram expansions arising from integrating out fields.
An L 8 -algebra L is called cyclic, if there is an additional symmetric, non-degenerate, bilinear form x´, ´y : with K the ground field or ring such that We shall refer to this as a metric structure.
Morphisms of L 8 -algebras.Let pL p1q , µ p1q i q and pL p2q , µ p2q i q be two L 8 -algebras.Strict morphisms of L 8 -algebras ϕ : L p1q Ñ L p2q are cochain maps between the underlying cochain complexes that respect the higher brackets, for all i P N and a 1 , . . ., a i P L p1q .
More generally, L 8 -algebras can be regarded as codifferential graded cocommutative coalgebras, and (weak) morphisms of L 8 -algebras amount to morphisms between these.
For illustrative purposes, let us list the relation between the lowest three higher products explicitly, for all a 1 , . . ., a 3 P L p1q .Evidently, ϕ 1 is a cochain map on the underlying cochain complex.
Moreover, a weak morphism with ϕ i trivial for i ě 2 is a strict morphism.
A quasi-isomorphism between two L 8 -algebras is now a morphism of L 8 -algebras such that ϕ 1 induces an isomorphism between the cohomologies of the corresponding cochain complexes.Quasi-isomorphic L 8 -algebras can be regarded as equivalent for most intents and purposes.
Homotopy transfer.A deformation retract (see e.g.[25]) between two cochain complexes pL p1q , µ p1q 1 q and pL p2q , µ p2q 1 q is a pair of cochain maps p and e together with a linear map h of degree ´1 that fit into the diagram pL p1q , µ p1q 1 q pL p2q , µ It is thus a stricter form of a homotopy equivalence between cochain complexes.A deformation retract is called special if also the so-called annihilation or side conditions are satisfied, p ˝h " 0 , h ˝e " 0 , and h ˝h " 0 . (2.7) A deformation retract can always be turned into a special deformation retract, cf.[19] or [25], by performing the replacement h Ñ pid ´e ˝pq ˝h ˝pid ´e ˝pq ˝µp1q 1 ˝pid ´e ˝pq ˝h ˝pid ´e ˝pq . (2.8) In the following, we shall always work with special deformation retracts.
Given a special deformation retract as in (2.6) and an L 8 -algebra structure on L p1q , the homological perturbation lemma allows us to transfer this structure to the cochain complex L p2q , and detailed formulas 1 are found e.g. in [12,37].In principle, one can now consider the homological perturbation lemma for cyclic L 8 -algebras, as done e.g. in [13].For our purposes, however, it will be easier to work with morphisms of general L 8 -algebras and check that the result we obtain is cyclic.
Explicitly, the homological perturbation lemma provides a lift of the quasi-isomorphism of cochain complexes e to a quasi-isomorphism of L 8 -algebras E : L p2q Ñ L p1q .In particular, given an L 8 -structure on L p1q , it induces an L 8 -structure on L p2q via the component maps (2.9a) 1 albeit for A8-algebras, but readily translatable for all b 1 , . . ., b i P L p2q so that the induced higher products on L p2q are given by µ p2q 2 pb 1 , b 2 q :" ppµ p1q 2 pE 1 pb 1 q, E 1 pb 2 qq , . . .
for all b 1 , . . ., b i P L p2q with the sign ζ as defined in (2.3c).
Quasi-isomorphisms not originating from homotopy transfer.Upon comparing the formulas (2.9) with those for a general quasi-isomorphism (2.4), we can straightforwardly identify quasi-isomorphisms that do not originate from a homotopy transfer.
As an example, consider the trivial, one-element L 8 -algebra L p1q with the underlying cochain complex with no non-trivial higher products and the L 8 -algebra L p2q with the underlying cochain complex ChpL p2q q :" ´¨¨¨Ý Ý Ñ t0u Ý Ý Ñ g lo omo on for some Lie algebra g with the only non-trivial higher product pL p1q q and pL p2q q are both trivial, the trivial map from L p1q to L p2q is a quasi-isomorphism.The higher products on L p2q , however, clearly do not originate from a homotopy transfer of the (trivial) higher products on L p1q and so, this quasi-isomorphism L p1q Ñ L p2q is not given by a homotopy transfer.However, there certainly is a homotopy transfer L p2q Ñ L p1q .
Minimal model.Note that by the usual abstract Hodge-Kodaira decomposition, every L 8 -algebra L comes with a minimal model, that is, an L 8 -algebra structure on its cohomology H ‚ µ 1 pLq, cf.[38,12].This minimal model is obtained by homotopy transfer using the special deformation retract where e is an embedding of the cohomology H ‚ µ 1 into L, p is a corresponding projection, and h is the contracting homotopy [39,40].
where we have used that e and p are cochain maps.
We can invert the above observation to the following result.
We now have the following, useful corollary.
Corollary 2.2.A strict projection p of an L 8 -algebra L to a quasi-isomorphic L 8subalgebra1 L lifts to a homotopy transfer.
Proof.Because L is a subspace, we have besides the projection also an embedding: such that p ˝e " id L .We also have special deformation retracts from both L and L to the joint minimal model L ˝.By Proposition 2.1, we then also have a special deformation retract from L to L. In the resulting homotopy transfer, the maps E i defined in (2.9a) vanish for i ą 1: the μj close on the image of e, and hence hpμ k pE 1 pb 1 q, . . ., E 1 pb k qqq vanishes for all b i P L. Therefore, the homotopy transfer from L to L simply reproduces the higher brackets on L.

Spans of L 8 -algebras
It would certainly be useful if all computations of quasi-isomorphisms L 8 -algebras were encoded in homotopy transfers as for those, explicit and recursive formulas are provided by the homological perturbation lemma.Moreover, in many applications to perturbative quantum field theory, it is useful to turn computations into Feynman diagram expansions with all their combinatorial features, which is essentially what the homological perturbation lemma does.In this section, we shall show that every quasi-isomorphism of L 8 -algebras can indeed be encoded in pairs of homotopy transfers.
Pullbacks of L 8 -algebras.Given two L 8 -algebras L p1q and L p2q with surjections1 σ p1,2q to a third L 8 -algebra L pbq , then there is a fourth L 8 -algebra L ppq that fits into the pullback diagram with the usual universality of L ppq arising in pullbacks.Abstractly, this is a consequence of the existence of pullbacks for homotopy algebras, cf.[41, Theorem 4.1], and L ppq is called the pullback; see also [42] for the special case of L 8 -algebras concentrated in non-positive degrees.It remains to show that there exists an L 8 -algebra L pcq quasi-isomorphic to L ppq such that there are homotopy transfers L pcq Ñ L p1,2q .
Remark 2.3.One may be tempted to think that the pullback (2.23) should be regarded as a homotopy pullback.This is not the case, as the L 8 -algebras L p1,2,bq are all homotopically equivalent, and hence L ppq could be identified with L pbq .
Also, one may think that the existence of a suitable L pcq follows trivially from the decomposition theorem, which states that any L 8 -algebra L decomposes as L " L ˝' L lc into a minimal model and a linearly contractible 2 one.It is then tempting to try to identify L pcq " L p1q˝' L p1qlc ' L p2qlc , but generally, the quasi-isomorphisms to L p1,2q are not homotopy transfers.
Spans of L 8 -algebras.We have the following result.
Theorem 2.4.Consider a pair of quasi-isomorphic L 8 -algebras L p1q and L p2q .Then there is a span of L 8 -algebras, i.e. a third L 8 -algebra L pcq together with quasi-isomorphisms p p1,2q and e p1,2q that fit into the diagram e p1q e p2q (2.24) such that the higher products on L p1,2q and e p1,2q are obtained from a homotopy transfer and given by formulas (2.9).We call L pcq the correspondence L 8 -algebra.
We perform the proof in a number of steps.Firstly, by definition, L p1,2q have isomorphic minimal models L p1,2q˝, and we can choose them to be identical: L ˝" L p1q˝" L p2q˝.We then have the following diagram.
(2.25) At this point, it turns out convenient to switch to the dual, Chevalley-Eilenberg picture, because we can phrase arguments in a way familiar from the BV formalism.Recall that any L 8 -algebra L is dual to a semi-free differential graded commutative algebra, called its Chevalley-Eilenberg algebra where d ‚ V denotes the symmetric tensor algebra over a graded vector space V , and rks denotes the shift-isomorphism However, all of the arguments we give below can be dualised to the perhaps less familiar picture of codifferential coalgebras.In particular, the differential Q on CEpLq is the dual of the natural codifferential D :" µ 1 r1s `µ2 r1s `. . . on the codifferential coalgebra Ä ‚ Lr1s.
Therefore, the fact that the dualisation only exists in certain situations, e.g. for degree-wise finite-dimensional L 8 -algebras, is not a problem, and our formulation of our arguments in terms of Chevalley-Eilenberg algebras is indeed just for presentations sake.Also, since we only use this technology in this proof, we refrain from giving more details on Chevalley-Eilenberg algebras; for a detailed review in our conventions, see e.g.[10].
From this perspective, diagram (2.25) translates to and we need to construct the push-out A, which is given by A :" CEpL p1q q b CEpL ˝q CEpL p2q q " pCEpL p1q q b CEpL p2q qq{I (2.29) with I the ideal generated by expressions of the form for all a P CEpL p1q q, c P CEpL ˝q, and b P CEpL p2q q.Note that the ideal I is a differential ideal, and the differential on A is simply where Q p1,2q are the differentials on CEpL p1,2q q.
The algebra A is not yet the Chevalley-Eilenberg algebra of an L 8 -algebra, because it is not semi-free1 .To remedy this, we use a Koszul-Tate-type complex quasi-isomorphic to A, very analogously to the BV formalism, see e.g.[43]. 2o this end, we introduce the graded commutative algebra There is a natural algebra homomorphism where i : Ä ‚ pL ˝q˚Ñ Â is the evident inclusion.The algebra Â becomes differential graded by virtue of the following result.
Proposition 2.5.Consider the algebra Â freely generated by ξ α P pL p1q r1s ' L p2q r1sq ˚and There is a differential Q pcq on Â defined as where P i i 1 ¨¨¨in are power series (without constant terms) in the sβ i , and s is the shift isomorphism s " r´1s : pL ˝q˚Ñ pL ˝r1sq ˚.
Proof.We have to show the existence of suitable P i i 1 ¨¨¨in such that Q pcq squares to zero, which directly reduces to pQ pcq q 2 β i " 0. We compute (2.35) A construction for the P i i 1 ¨¨¨i k can be produced order by order in the1 β i .We start with the linear terms where κ is defined as the linear extension of and s ´1 is the inverse of s, continued as a derivation to products.The resulting Q pcq satisfies where we defined We can continue inductively.Say, we found the P i i 1 ...i k up to order n, so that Then we can choose the P i i 1 ¨¨¨i n`1 so that where we dropped terms that are of order different than n in the β i .At the next order, we therefore need to solve which always has a solution.A particular solution is given by as is readily seen by applying s to both sides and multiplying the results by Having defined a semi-free differential graded commutative algebra, we can convince ourselves that it is of the right size and that there are homotopy transfers from the L 8algebra defined by Â to either L p1,2q .
Proposition 2.6.The differential graded commutative algebra p Â, Q pcq q is the Chevalley-Eilenberg algebra of an L 8 -algebra L pcq quasi-isomorphic to both L p1q and L p2q .In fact, there are homotopy transfers from L pcq to L p1q and L p2q .
Proof.As a vector space, the L 8 -algebra L pcq is given by (2.45) Hence, the cochain complex underlying L pcq takes the form The cohomology of L p1q ' L p2q is L ˝' L ˝, but in L pcq , only a subspace isomorphic to L is in the kernel of the differential µ 1 .Therefore, it is clear that the cohomology of L pcq is isomorphic to L ˝.
In order to show that there are homotopy transfers from L pcq to both L p1q and L p2q , we adapt the argument that led to Corollary 2.2.Because the situation is symmetric, we can focus on L p1q .We have special deformation retracts from both L pcq and L p1q to L ˝, which gives us a special deformation retract from L pcq to L p1q by Proposition 2.1.In the resulting homotopy transfer, the maps E i defined in (2.9a) again vanish for i ą 1: for j ą 1, the µ pcq j close on the image of the embedding E 1 : L p1q Ñ L pcq , and hence hpµ pcq k pE 1 pb 1 q, . . ., E 1 pb k qqq vanishes for all b i P L p1q .Therefore, the homotopy transfer simply reproduces the higher brackets on L p1q .
With the last lemma, the proof of Theorem 2.4 is complete.
Lifting of homotopy transfers.As a trivial example, consider the lift of a quasiisomorphism arising from a homotopy transfer (2.47)Such a quasi-isomorphism trivially lifts to the span (2.48) More interesting examples, in particular with regards to applications in quantum field theory, are presented in Sections 3 to 5.
Composition of spans of L 8 -algebras.Given a pair of quasi-isomorphic L 8 -algebras, it is clear that, generically, there are several spans of L 8 -algebras between them.Their correspondence L 8 -algebras are related by a quasi-isomorphisms, which then also relate the various projection and embedding maps in an evident manner.
This ambiguity also induces an ambiguity in the composition of spans, which can simply be defined as spans between the correspondence L 8 -algebras.Therefore, L 8 -algebras as objects with spans of L 8 -algebras as morphisms do not form a category or groupoid (because the morphisms are evidently invertible) but can be regarded as a quasi-groupoid.
Alternatively, we can simply quotient by this ambiguity and regard different spans between the same pair of L 8 -algebra as equivalent, rendering composition again unique and associative.Since our interest in spans is mostly due to computational advantages, this distinction is largely irrelevant in the following.
Quasi-isomorphisms of cyclic L 8 -algebras.Before coming to the physical applications of spans of L 8 -algebras within field theories, let us briefly explore the mathematical uses.Recall that the definition of morphisms between cyclic L 8 -algebras is somewhat problematic.Essentially, the reason for the encountered problems is the fact that a cyclic structure corresponds to a symplectic structure on the underlying graded vector space, and hence, one is looking for a reasonable category of symplectic manifolds.In particular, we have seen in that the condition (2.5)only works for morphisms ϕ for which the cochain map ϕ 1 an injection.Our perspective solves this problem at least for quasi-isomorphisms, and we can make the following definition.
Definition 2.7.Given two cyclic L 8 -algebras pL p1,2q , x´, ´yp1,2q q, a metric or cyclic quasiisomorphism is a third cyclic L 8 -algebra pL pcq , x´, ´ypcq q such that we have a span of L 8algebras e p1q e p2q for all a 1,2 P L p1q and b 1,2 P L p2q .
Remark 2.8.We note that the full condition (2.5) for the injective quasi-isomorphisms e p1,2q is automatically satisfied if (2.49b) holds.Consider e.g. the Hodge-type decomposition 1 q, and C " imph p1q q of the initial special deformation retract between L pcq and L p1q before deformation.The metric structure on L pcq then is necessarily of the block matrix form for some block matrices ω HH , ω BC , and ω CB .The formulas for the maps e p1q k given in (2.9a) then show that we automatically have ÿ j`k"i j,kě1 xe p1q j pa 1 , . . ., a j q, e p1q k pa j`1 , . . ., a j`k qqy p2q " 0 , (2.52) the additional condition in (2.5).

Application to perturbative quantum field theory
In this section, we provide a very concise review of the dictionary that translates between perturbative field theories and L 8 -algebras, as well as the implications for our above results.
Observables in a classical field theory.The kinematical data of a perturbative classical field theory is the field space F, a vector space or module usually consisting of the sections of some vector bundle.The dynamics of the theory are governed by the equations of motion, which are the stationary points of an action functional S on the field space.Note that the equations of motion generate the ideal I in the ring of functions on field space which is generated by functions on F vanishing for classical solutions.
This description may contain redundancies known as gauge symmetries, i.e. an action of a group (of gauge transformations1 ) G ñ F such that the true kinematical data is given by the orbit space F{G.This evidently requires the action functional and the equations of motion to be invariant and covariant, respectively, under the group action.
The classical observables of a field theory are then identified with the quotient ring R{I, where R is the ring of functions on the orbit space F{G.
In this description, both the quotients by the group of gauge transformations G and the ideal I are replaced by a cochain complex whose cohomology is the actual quotient.Gauge transformations are dealt with the Chevalley-Eilenberg algebra of the corresponding gauge algebroid, introducing ghost fields.The equations of motion are divided out by introducing additional anti-fields, leading to a Koszul-Tate complex. 2 The resulting BV complex is a cochain complex with the differential encoding the equations of motion and the gauge transformations.The cochains form, in fact, a differential graded commutative algebra, which, as we mentioned above, is the dual of an L 8 -algebra.
Direct correspondence: Maurer-Cartan action.We can also establish the link between a field theory and an L 8 -algebra more directly.The data of any perturbative field theory 1 can be cast in the form of a cyclic L 8 -algebra L. Here, L 1 is the vector space or module of fields, while L 2 is the space of anti-fields.There is a metric structure of degree ´3, providing a non-degenerate pairing between L 2 and L 1 .If gauge symmetries (respectively, higher gauge symmetries) are present, we also have a non-trivial subspace L 0 (respectively, L 0 , L ´1, etc.), the space of (infinitesimal) gauge parameters or ghosts (respectively, ghosts, ghost-of-ghosts, etc.), as well as L 3 (respectively, L 3 , L 4 , etc.), the space of anti-ghosts (respectively, anti-ghosts, anti-ghosts-of-ghosts, etc.).
The cyclic higher products on L 1 are then uniquely defined by identifying the field theory's classical action with the homotopy Maurer-Cartan action of L, where a P L 1 .The remaining higher products for all elements in L are defined either via the gauge transformations of the fields or by writing the BV action in a particular way, cf.[10] (see also [46]).
Altogether, perturbative field theories with action principles are in one-to-one correspondence with L 8 -algebras endowed with a metric structure of degree ´3.
Perturbation theory.Interestingly, the way that tree-level perturbation theory is usually described within quantum field theory directly translates to the homological perturbation lemma.In order to compute a tree-level amplitude, we draw all relevant Feynman diagrams, amputate the external legs by the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula, replacing them with labels of gauge-fixed free fields.The construction of the minimal model via the special deformation retract (2.13) and the formulas (2.9) proceeds exactly in the same manner.The cohomology H ‚ µ 1 pLq describes the free fields up to gauge transformations, and the recursion relation for the E i produces the tree-level Feynman diagram expansion with the n-point tree-level scattering amplitudes themselves are identified with the expressions Apϕ 1 , . . ., ϕ n q " 1 n! xϕ 1 , µ n´1 pϕ 2 , . . ., ϕ n qy ˝(2.54) 1 read: with a field space given by sections of a vector bundle for ϕ 1 , . . ., ϕ n elements of the minimal model L ˝.This observation has been made many times, see e.g.[47,12] in the context of string field theory and [48,20,49] in the context of field theories.
Semi-classical equivalence.A suitable definition of equivalence between two classical field theories must start from the question of which properties equivalent quantum field theories have to share.An isomorphic solution space is clearly not enough, as e.g.theories of a single massless scalar field on Minkowski space R 1,n with canonical kinematic term and arbitrary polynomial potentials all have isomorphic solution spaces, parametrised e.g. by boundary data on some Cauchy surface.A good quantity to preserve is certainly the tree-level S-matrix containing the scattering amplitudes (2.54), as we think of these as determining all measurable quantities.We note that this notion of equivalence called semiclassical equivalence covers the standard operations of integrating fields in and out.
This form of equivalence is also mathematically preferable, as two field theories have S-matrices related by a similarity transformation if and only if their corresponding L 8algebras have isomorphic minimal models and are thus quasi-isomorphic.
Correspondences of L 8 -algebras.As mentioned above, semi-classically equivalent field theories have quasi-isomorphic L 8 -algebras, and as a consequence, isomorphic minimal models, which can be computed via homotopy transfer.In many situations, the two L 8 -algebras are directly linked by a homotopy transfer, e.g. when they are related by integrating out fields.In some situations, however, this is not the case, and physicists are already implicitly working with spans to describe these.In the following sections, we shall discuss three examples in detail: a simple example based on different blow ups of scalar field theory, (non-Abelian) T-duality in the case of the principal chiral model, and the Penrose-Ward transform.
An important remark regarding the application to field theory is the following.The correspondence L 8 -algebra we constructed in Theorem 2.4 using the minimal model is usually inconvenient from the field theoretic perspective, as it splits a field into its on-shell and off-shell components.Most of the time, we are interested in a correspondence L 8 -algebra with all elements true, unrestricted fields on a given space-time.This makes constructing a 'good' correspondence L 8 -algebra a bit more non-trivial.However, completing on-shell fields in a minimal model by trivial pairs to true, unrestricted fields on a given space-time is mostly straightforward 1 .In concrete computations with field theories, one therefore replaces the summand L ˝r´1s in L pcq with an enlarged copy, L˝r ´1s, which contains these 1 up to analytical difficulties, but these can be removed by restricting to functions bounded at infinity additional trivial pairs.Again, there are no fundamental obstructions to the existence of physically 'good' correspondence L 8 -algebras.
Quantum level.In this paper, we shall exclusively work at the tree level.Still, let us briefly comment on the extension to the quantum picture.In the BV formalism, the differential in the BV complex is deformed by a second order differential operator, and this deformation produces a differential graded algebra that is dual not to an L 8 -algebra, but to a loop L 8 -algebra, as defined in [11,22].Loop L 8 -algebras, however, are still homotopy algebras, and come with a homotopy transfer of their structure to a minimal model, cf.[13,21], which now governs the quantum scattering amplitudes.Two field theories are then quantum equivalent, if their loop L 8 -algebras have the same minimal model.
Note that all issues regarding regularisation and renormalisation have been ignored in this discussion.For a rigorous treatment of these, see [50,23,24].
At the quantum level, an additional benefit of lifting quasi-isomorphisms to spans of L 8algebras is that they make equivalences more evident.As a homotopy transfer essentially amounts to integrating in/out some fields, one can identify those integrations that are compatible with both homotopy algebra and loop homotopy algebra structures.That is, one can check whether the field redefinitions included in the homotopy transfer induce Jacobians when applied to the path integral measure.For more details on this as wells as the notion of equivalence of perturbative quantum field theories at quantum level, see also the discussion in [51].

Blowing up vertices in scalar field theory
Let us start with a simple example and consider scalar field theory with a sextic potential, i.e. the action for a single scalar field theory ϕ P Ω 0 pM d q on d-dimensional Minkowski space M d .Here, l is the d'Alembertian and λ P R. In the following, we shall blow up the interaction vertex by introducing auxiliary fields in two different ways: in one theory, as two quartic vertices, and in another as a cubic and a quintic vertex.In order to go from one theory to the other, one has to both integrate in and out auxiliary fields, leading naturally to a span of L 8 -algebras.

Homotopy algebraic formulation of the involved theories
Blow ups of interaction vertices.By introducing auxiliary fields χ 1 , χ 2 P Ω 0 pM d q, we can blow up the interaction vertex in (3.1) into two quartic vertices, S p1q :" Alternatively, we can introduce two auxiliary fields ψ 1 , ψ 2 P Ω 0 pM d q and blow up the interaction vertex in (3.1) into cubic and quintic vertices, Finally, there is the following blow-up using all the above auxiliary fields as well as ξ 1 , ξ 2 P Ω 0 pM d q, S pcq :" Note that this blow up is a 'common refinement' of the previous two in the following sense: if we integrate out the fields ψ 1 , ψ 2 and χ 1 , χ 2 in S pcq , we recover S p1q ; if we integrate out ξ 1 , ξ 2 and ψ 1 , ψ 2 , on the other hand, we recover S p2q .
Homotopy algebra for S pbq .The homotopy algebra L pbq corresponding to the action S pbq has the underlying cochain complex ChpL pbq q :" ¨ϕ Ω 0 pM d q looomooon ": .Note that we regard the integral as formal expressions; to be precise, we would have to restrict our field space to L 2 -functions, cf.e.g.[20] for a detailed discussion.
Homotopy algebra for S p1q .The homotopy algebra L p1q corresponding to the action S p1q has underlying cochain complex and the only non-trivial higher products are obtained by polarisation of for all pϕ, χ 1 , χ 2 q P L p2q 1 and pϕ `, χ 1 , χ 2 q P L p2q 2 , and where here and below the positions of the field components correspond to those in (3.6a).The metric structure is the evident one, for all pϕ, χ 1 , χ 2 q P L p2q 1 and pϕ `, χ 1 , χ 2 q P L p2q 2 .
Homotopy algebra for S p2q .The homotopy algebra L p2q corresponding to the action S p2q has underlying cochain complex and the only non-trivial higher products are obtained by polarisation of for all pϕ, ψ 1 , ψ 2 q P L p2q 1 and pϕ `, ψ 1 , ψ 2 q P L p2q 2 and where the positions of the component fields refer to diagram (3.7a).The metric structure is again evident, for all pϕ, ψ 1 , ψ 2 q P L p2q 1 and pϕ `, ψ 1 , ψ 2 q P L p2q 2 .
Homotopy algebra for S pcq .The homotopy algebra L pcq corresponding to the action S pcq has underlying cochain complex and the non-trivial higher products are the polarisations of The metric structure is, once more, the evident pairing of fields and anti-fields.

Span of L 8 -algebras
The four actions S pbq , S p1q , S p2q , and S pcq correspond to the four homotopy algebras L pbq , L p1q , L p2q , and L pcq which fit into where the arrows indicate quasi-isomorphisms.Moreover any downwards arrow can be formulated as a homotopy transfer, as we shall show in the following.In conclusion, the upper half of the diamond (3.9) forms a span of L 8 -algebras.
Homotopy transfer L pcq Ñ L p1q .This homotopy transfer starts from the special deformation retract with the following embedding and projection maps and contracting homotopy: so that and the side conditions (2.7) hold.
The homotopy transfer, by formulas (2.9), yields the new embedding with E 1 :" e and being the only non-trivial components.As a consequence, the induced higher products (2.9b) are only non-trivial when i " 3, and the single resulting higher product µ p1q 3 coincides with (3.6b).We conclude that there is a quasi-isomorphism between L pcq and L p1q that originates from a homotopy transfer.
Homotopy transfer L pcq Ñ L p2q .Let us now show the same for the quasi-isomorphism Here, we consider a homotopy transfer starting from the special deformation retract with the easily obtained maps and we have together with the side conditions (2.7).
Here, formulas (2.9) lead to the embedding with E 1 :" e and being the only non-trivial higher maps.The only non-trivial induced higher products (2.9b) resulting from (2.9) are then (3.7b).

(Non-Abelian) T-duality of the principal chiral model
We now turn to a physically more interesting pair of quasi-isomorphic theories: the principal chiral model and its (non-Abelian) T-dual.

Homotopy algebraic formulation of the involved theories
We start by listing the theories involved in the (non-Abelian) T-duality, together with their homotopy algebraic formulation in terms of cyclic L 8 -algebras.
Principal chiral model.We work on two-dimensional Minkowski space M 2 with metric η " diagp´1, 1q.We shall use the de Rham complex pΩ ‚ pM 2 q, dq together with the codifferential We then have where l is the d'Alembertian.We also note that The target space of the principal chiral model (PCM) is a Lie group G with Lie algebra g, and we assume that there is a non-degenerate, invariant, symmetric bilinear form x´, ´yg on g.The kinematical data is then given by a smooth map which we may parametrise as g " e ϕ for ϕ P Ω 0 pM 2 , gq.The pullback of the Maurer-Cartan form then reads as with ad ϕ p´q :" rϕ, ´s, and the action of the PCM is given by In rewriting this action, we have noted that only even powers of ad ϕ will contribute after inserting (4.5) and then substituted the identity ř 2n m"0 p2n`2q! .From the homotopy-algebraic perspective, the action (4.6) is given by a cyclic L 8algebra L p1q with the underlying cochain complex ChpL p1q q :" ¨ϕ Ω 0 pM 2 , gq loooomoooon non-trivial higher products that are obtained by polarising μp1q 2n`1 pϕ, . . ., ϕq :" ´d: ad 2n ϕ pdϕq (4.7b) for all n P N, and the metric structure xϕ, ϕ `yL p1q :" for all ϕ P L p1q 1 and ϕ `P L p1q 2 .We stress that the simple polarisation of the expressions (4.7b) is not sufficient to render them cyclic, which has to be done separately.To this end, it is useful to note that (4.9) The polarisation of these higher products are indeed cyclic with respect to (4.7c).
Gauged principal chiral model.Following [52,53], the first step to T-dualise the PCM is to gauge a normal Lie subgroup H of G that corresponds to the directions that we wish to T-dualise.Let h be the Lie algebra of H.The gauging is implemented by introducing an h-valued connection one-form ω P Ω 1 pM 2 , hq so that the current (4.5) generalises to j ω :" g ´1ωg `g´1 dg .(4.10) Evidently, with F ω :" dω `1 2 rω, ωs and F jω :" dj ω `1 2 rj ω , j ω s, we have F jω " g ´1F ω g implying equivalence of the flatness of j ω and ω.Furthermore, j ω is invariant under the local H-action for any smooth map h : M 2 Ñ H.To implement the flatness F ω " 0, we introduce a Lagrange multiplier Λ P Ω 0 pM 2 , hq subject to the local H-action This action corresponds to a cyclic L 8 -algebra L pcq with the underlying cochain complex  Note that the general higher products follow from polarisation of (4.15b) and cyclification via (4.15c) similar to (4.9).
T-dual principal chiral model.If we integrate out the gauge potential ω from the action (4.12), we obtain the action of the T-dual model [52,53].In particular, the equation of motion for ω is This is an algebraic equation for j ω which has the solution (4.17) Upon substituting this into (4.12),we obtain the T-dual action The corresponding homotopy algebra L p2q has the underlying cochain complex ChpL p2q q :" ¨Λ Ω 0 pM 2 , hq loooomoooon higher products defined by and the metric structure x Λ, Λ`y L p2q :" for all Λ P L p2q 1 and Λ`P L p2q 2 .Again, the general form of the higher products is obtained from the polarisation of (4.19b) followed by the cyclification with respect to (4.19c).Hence, similar to (4.9), we may equivalently take instead of (4.19b) whose polarisation is directly cyclic.

Span of L 8 -algebras
We now describe the homotopy transfers realising the quasi-isomorphisms that link the PCM to its T-dual model and produce a span of L 8 -algebras We start with the simpler transfer from L pcq to L p2q .
Homotopy transfer L pcq Ñ L p2q .Between the differential complexes underlying L pcq and L p2q , we have the following special deformation retract: pL pcq , µ pcq 1 q pL p2q , µ where the positions indicate the subspaces of the complexes in which the expressions take values, as displayed in (4.19) and (4.15a).In particular, we have (2.6b), and the side conditions (2.7) are satisfied as well.
We note that in the formulas (2.9), the arguments of the higher products are always applied to images of E n , which, in turn are images of either e or h.For degree for all n P N.These relations evidently hold for n " 1. Suppose now that they hold for 1, . . ., n ´1 with n ą 2.Then, where in the third step, we have used rα, ‹ k βs " p´1q k r‹ k α, βs for any two Lie-algebravalued differential one-forms α and β and in the last step, we have used the identity We assume that all fields are bounded at infinity, so that we can expand them in terms of plane waves x Þ Ñ e ip¨x with momentum p.Because the differential µ 1 in the complex (4.15a) preserves momenta, we can decompose Ω 1 pM 2 q as where we have introduced the subspaces Ω 1 e pM 2 q :" tone-forms with p 2 ‰ 0 and spanned by dx µ p µ e ip¨x u , Ω 1 c pM 2 q :" tone-forms with p 2 ‰ 0 and spanned by dx µ ε µν p ν e ip¨x u , Ω 1 ec pM 2 q :" tone-forms with p 2 " 0 and p ‰ 0 and spanned by dx µ p µ e ip¨x u , Ω 1 r pM 2 q :" tone-forms with p 2 " 0 and p ‰ 0 and spanned by dx µ δ µν p ν e ip¨x u , Ω 1 cm pM 2 q :" tone-forms with p " 0 and spanned by dx µ u , where ε µν is the Levi-Civita symbol and δ µν the Kronecker symbol, respectively.Elements of Ω 1 e pM 2 q are exact and elements of Ω 1 c pM 2 q are coexact.Furthermore, while elements of Ω 1 ec pM 2 q are both closed and coclosed since dx µ p µ " ˘dx µ ε µν p ν for p 0 " ˘p1 , elements of Ω 1 r pM 2 q are neither closed nor coclosed.We also have ‹ : Ω 1 e pM 2 q Ñ Ω 1 c pM 2 q and ‹ : Ω 1 c pM 2 q Ñ Ω 1 e pM 2 q , (4.31) and elements in Ω 1 ec pM 2 q and Ω 1 r pM 2 q with definite momentum p are either self-dual or anti-self-dual, depending on the sign in p 0 " ˘p1 .
It remains to show that the homotopy transfer indeed reproduces the higher products of L p1q .Considering formulas (2.9), we note the following.The embedding E 1 " e of L p1q into L pcq will map a field ϕ P L  Note that the ω `-component has no constant part, as the derivative of the functions that we are considering (i.e.bounded at infinity) either vanishes or is non-constant.So applying h to the result will only produce a field component Λ in L pcq 1 .In summary, the only arguments ever entering the higher products in the homotopy transfer will be the ϕ-and Λ-components of L pcq 1 .The only non-trivial higher products with these arguments, however, are the ones with all arguments being ϕ-components.The latter exclusively arise from the direct embedding via E 1 " e.The final projector p is only non-trivial on the component fields ϕ `and Λ `, and therefore the homotopy transfer is just a pullback of the higher product defined by which reproduces the higher products on L p1q .

Penrose-Ward transform
The purpose of this section is to briefly describe yet another example of spans of L 8algebras, arising in the context of the Penrose-Ward transform.
see [62,63] and also [64].Both the holomorphic Chern-Simons action and the Siegel action can be extended to evident BV actions, and the corresponding L 8 -algebras L Z 3|4 and L R 4|8 are quasi-isomorphic.Moreover, this quasi-isomorphism is a two-step homotopy transfer, cf.[62,63], see also [64].In a first step, we use the contracting homotopy with h 1 " B: CP 1 (5.8) the adjoint of the Dolbeault operator, restricted to π ´1 2 pxq -CP 1 for all x P R 4 to impose the space-time gauge.In a second step, we use a second homotopy transfer to integrate out all auxiliary fields, which leaves us with the space-time BV fields in L R 4|8 .These homotopy transfers are then concatenated as explained in (2.15).This quasi-isomorphism of L 8 -algebras has recently been used in the context of colour-kinematics duality [65] to derive kinematic Lie algebras from twistor spaces.
Span of L 8 -algebras with mini-twistors.As explained in detail in [66], the single fibration (5.6) is expanded into a double fibration again when considering its dimensional reduction to three space-time dimensions.Explicitly, R 4|8 is reduced to R 3|8 , but the twistor space Z 2|4 for the description of supersymmetric monopoles becomes a supersymmetric generalisation of the mini-twistor space introduced in [67], and Z 2|4 is the total space of the rank p1|4q-vector bundle Op2q ' C 0|4 b Op1qCP 1 .We end up with the double fibration R 3|8 ˆCP 1 (5.9) This, in turn, induces a span of L 8 -algebras L R 3|8 : supersymmetric monopole theory L R 3|8 ˆCP 1 : partially holomorphic Chern-Simons theory as defined in [66]   L Z 2|4 : holomorphic BF theory as defined in [66] (5.11) Evidently, these L 8 -algebras are quasi-isomorphic, and in the span of L 8 -algebras (5.10), the homotopy transfer p 2 is given by a real dimensional reduction of the homotopy transfer from L Z 3|4 to L R 4|8 , while the homotopy transfer p 1 amounts to a push-forward, as explained in [66].