Constraints in the BV formalism: six-dimensional supersymmetry and its twists
aa r X i v : . [ m a t h - ph ] S e p CONSTRAINTS IN THE BV FORMALISM:SIX-DIMENSIONAL SUPERSYMMETRY AND ITS TWISTS
INGMAR SABERI
Mathematisches Institut der Universit¨at HeidelbergIm Neuenheimer Feld 20569120 HeidelbergDeutschland
BRIAN R. WILLIAMS
School of MathematicsUniversity of EdinburghEdinburghUK
Abstract.
We formulate the abelian six-dimensional N = (2 ,
0) theory perturbatively, in a generalizationof the Batalin–Vilkovisky formalism. Using this description, we compute the holomorphic and non-minimaltwists at the perturbative level. This calculation hinges on the existence of an L ∞ action of the super-symmetry algebra on the abelian tensor multiplet, which we describe in detail. Our formulation appearsnaturally in the pure spinor superfield formalism, but understanding it requires developing a presymplecticgeneralization of the BV formalism, inspired by Dirac’s theory of constraints. The holomorphic twist consistsof symplectic-valued holomorphic bosons from the N = (1 ,
0) hypermultiplet, together with a degenerateholomorphic theory of holomorphic coclosed one-forms from the N = (1 ,
0) tensor multiplet, which canbe interpreted as representing the intermediate Jacobian. We check that our formulation and our resultsmatch with known ones under various dimensional reductions, as well as comparing the holomorphic twistto Kodaira–Spencer theory. Matching our formalism to five-dimensional Yang–Mills theory after reductionleads to some issues related to electric–magnetic duality; we offer some speculation on a nonperturbativeresolution.
E-mail addresses : [email protected], [email protected] . Date : September 8, 2020. ontents
1. Introduction 22. A presymplectic Batalin–Vilkovisky formalism 113. The abelian tensor multiplet 224. The minimal twists 355. The non-minimal twist 556. Comparison to Kodaira–Spencer gravity 637. Dimensional reduction 68References 791.
Introduction
There is a supersymmetric theory in six dimensions whose fields include a two-form with self-dual fieldstrength. Concrete and direct formulations of this theory, whose field content is referred to as the tensormultiplet, have remained elusive, despite an enormous amount of work and numerous applications, predic-tions, and consistency checks. One main difficulty is that the theory is believed not to admit a Lagrangiandescription, meaning that its equations of motion—even for the free theory—do not arise from a standardcovariant action functional via the usual methods of variational calculus.Part of the desire to better understand theories of tensor multiplets is due to their ubiquity in the contextof string theory and M -theory. The tensor multiplet with N = (2 ,
0) supersymmetry, valued in the Liealgebra u ( N ), famously appears as the worldvolume theory of N coincident M N >
1, although we believe thatsome of our structural insights should be of use in that setting as well.) On general grounds, the N = (2 , Theorem.
The abelian N = (2 , admits two inequivalent classes of twists described as follows.(1) The holomorphic twist exists on any complex three-fold X equipped with a square-root of the canonicalbundle K X . It is equivalent to a theory whose physical fields are a (1 , -form χ , , a (0 , -form χ , , and a pair of symplectic fermions ψ i , i = 1 , which transform as (0 , forms with values in X . These fields obey the equations ∂χ , + ∂χ , = 0 ,∂χ , = 0 ,∂ψ i = 0 . The gauge symmetries of this theory are parameterized by form fields α , and α , , together witha pair of symplectic fermion gauge fields ξ i , i = 1 , which are sections of K X . They act by theformulas χ , χ , + ∂α , + ∂α , ,χ , χ , + ∂α , ,ψ i ψ i + ∂ξ i . (2) The non-minimal twist exists on any manifold of the form M × Σ where M is a smooth four-manifoldand Σ is a Riemann surface. It is equivalent to a theory whose physical fields are a pair ( χ , , χ , ) ∈ Ω ( M ) ⊗ Ω (Σ) ⊕ Ω ( M ) ⊗ Ω , (Σ) . This pair obeys the equations of motion ∂χ , + d χ , = 0d χ , = 0 . Here ∂ is the ∂ -operator on Σ and d is the de Rham operator on M . The theory has gauge symmetriesby fields α , and α , , which act via χ , χ , +d α , and χ , χ , +d α , + ∂α , . The full statements of these results appear below in Theorems 4.2, 5.3, and 5.8. Making sense of thesetwists and proving the theorems rigorously requires a great deal of groundwork, which leads us to developsome general theoretical tools that we expect to be of use outside the context of six-dimensional supersym-metry.The main subtlety of the N = (2 ,
0) theory, and the twists above, is that they do not arise as the variationalequations of motion of a local action functional. Thus, our first goal is to give a precise mathematicalformulation of the perturbative theory of the free N = (2 ,
0) tensor multiplet. (Of course, a correspondingformulation of the N = (1 ,
0) tensor multiplet follows immediately from this.) Throughout the paper, wemake use of the Batalin–Vilkovisky (BV) formalism; see [1], [2] for a modern treatment of this setup, and [3],[4] for a more traditional outlook. Roughly, the data of a classical theory in the BV formalism is a gradedspace of fields E BV (given as the space of sections of some graded vector bundle on spacetime), together with symplectic form ω BV of cohomological degree ( −
1) on E BV and an action functional. The (degree-one)Hamiltonian vector field associated to the action functional defines a differential on E BV . Under appropriateconditions, this differential provides a free resolution to the sheaf of solutions to the equations of motion ofthe theory, modulo gauge equivalence.It is clear that this formalism does not extend to the tensor multiplet in a straightforward way. Theissue arises from the presence of the self-duality constraint on the field strength of the two-form, and isindependent of supersymmetry and of other details about this particular theory. In Lorentzian signature,self-dual constraints on real 2 k -form fields can be imposed in spacetime dimension 4 k +2, where k = 0 , , . . . . We will always work in Euclidean signature in this paper, and therefore also with complexified coefficients;for us, the self-dual constraint in six dimensions therefore takes the form(1) α ∈ Ω ( M ) , ⋆ d α = √− α. The Yang–Mills style action of a higher form gauge theory would be given by the L -norm k d α k L = R d α ∧ ⋆ d α . It is clear that the self-duality condition implies the norm vanishes identically, so an actionfunctional of Yang–Mills type is not feasible [5]. Writing a covariant Lagrangian of any standard form forthe tensor multiplet has been the subject of much effort, and is generally thought to be impossible (althoughvarious formulations have been proposed in the abelian case; see, for example, [6]–[9]). A standard BVformulation of the theory, along the lines of more familiar examples, is thus out of reach for this reasonalone.The formulation we use was motivated by the desire to understand the pure spinor superfield formalismfor N = (2 ,
0) supersymmetry; the relevant cohomology was first computed in [10], and was rediscovered andreinterpreted in [11]. Roughly speaking, this formalism takes as input an equivariant sheaf over the spaceof Maurer–Cartan elements, or nilpotence variety, of the supertranslation algebra, and produces a chaincomplex of locally free sheaves over the spacetime, together with a homotopy action of the correspondingsupersymmetry algebra. The resulting multiplet can be interpreted as the BRST or BV formulation ofthe corresponding free multiplet, according to whether the action of the supersymmetry algebra closes onshell or not; the differential, which is also an output of the formalism, corresponds in the latter case to theHamiltonian vector field mentioned above.In the case of N = (2 ,
0) supersymmetry, the action of the algebra is, as always, guaranteed on generalgrounds. The fields exhibit an obvious match to the content of the (2 ,
0) tensor multiplet, and the differentialincludes the correct linearized equations of motion. (In fact, the resulting multiplet contains no auxiliaryfields at all.) One thus expects to have obtained an on-shell formalism, but the interpretation of the resultingresolution as a BV theory is subtle for a new reason: there is no obvious or natural shifted symplectic pairing. In the literature, such constraints are sometimes called “chiral.” To avoid confusion, we will reserve this term for a differentconstraint that can be defined on complex geometries, and that will play a large role in what follows. n fact, developing a framework for studying the multiplets produced by pure-spinor techniques requires ageneralization of the standard formalism, which necessarily allows for degenerate pairings.In classical symplectic geometry, symplectic pairings that are not required to be nondegenerate are called presymplectic . In fact, presymplectic structures have played a role in physics before, in Dirac’s theoryof constrained systems. In this context, the origin is clear: while symplectic structures do not pull back,presymplectic structures (which are just closed two-forms) do. Any submanifold of a symplectic phase space,such as a constraint surface, thus naturally inherits a presymplectic structure.The simplest situation where the issues of self-duality constraints arise occurs for k = 0, in the context oftwo-dimensional conformal field theory. Here, the constraint is precisely the condition of holomorphy, andthe theory of a self-dual zero-form is just the well-known chiral boson. We take a brief intermezzo to remarkon this theory briefly, to offer the reader some familiar context for our more general considerations. The chiral boson.
In Lorentzian signature, the theory of the (periodic) chiral boson describes left-movingcircle-valued maps. Working perturbatively, as we do throughout this paper, the periodicity plays no role,so that the field is simply a left-moving real function; after switching to Euclidean signature (and corre-spondingly complexifying), we have a theory of maps ϕ that are simply holomorphic functions on a Riemannsurface.As discussed above, one role of the BV formalism is to provide a resolution of the sheaf of solutions to theequations of motion by smooth vector bundles. For the sheaf of holomorphic functions, such a resolution isstraightforward to write down: it is just given by the Dolbeault complex Ω , • (Σ).The chiral boson is not a theory in the usual sense of the word, perturbatively or otherwise, as it is notdescribed by an action functional: the equations of motion, namely that ϕ be holomorphic, do not arise asthe variational problem of a classical action functional. Relatedly, the free resolution Ω , • (Σ) is not a BVtheory, as it does not admit a nondegenerate pairing of an appropriate kind. Nevertheless, there is a way toformulate the chiral boson in a slightly modified version of the BV formalism, by interpreting holomorphy(which, in this setting, is the same as self-duality) as a constraint .To do this, we first consider a closely related theory, the (non-chiral) free boson, which does have a descrip-tion in the BV formalism. The free boson is a two-dimensional conformal field theory whose perturbativefields, in Euclidean signature, are just a smooth complex-valued function ϕ on Σ; the equations of motionimpose that ϕ is harmonic. In the BV formalism, one can model this free theory by the following two-termcochain complex(2) 0 1 E BV = Ω (Σ) ∂∂ −−→ Ω (Σ) . e can equip E with a degree ( −
1) antisymmetric non-degenerate pairing, which in this case is just givenby multiplication and integration. That is ω BV ( ϕ, ϕ + ) = Z ϕϕ + where ϕ ∈ Ω (Σ) and ϕ + ∈ Ω (Σ). This is the ( − i : Ω , • (Σ) → E BV which in degree zero is the identity map on smooth functions, and in degree one is defined by the holomorphicde Rham operator ∂ : Ω , (Σ) → Ω (Σ). We can pull back the degree ( −
1) symplectic form ω on E to atwo-form i ∗ ω on Ω , • (Σ), which is closed because i is a cochain map. Explicitly, this two-form on the spaceΩ , • (Σ) is ( i ∗ ω )( α, α ′ ) = R α∂α ′ .Since i is not a quasi-isomorphism, i ∗ ω is degenerate, and hence does not endow Ω , • (Σ) with a BVstructure. However, it is useful to think of i ∗ ω as a shifted presymplectic structure on the chiral boson,encoding “what remains” of the standard BV structure after the constraint of holomorphy has been imposed.In analogy with ordinary symplectic geometry, we will refer to the data of a pair ( E , ω ) where E is a gradedspace of fields, and ω is a closed two-form on E , as a presymplectic BV theory. We make this precise in Defi-nition 2.1, at least for the case of free theories. In the example of the chiral boson this pair is (Ω , • (Σ) , i ∗ ω BV ).The theory of the self-dual two-form in six-dimensions (more generally a self-dual 2 k -form in 4 k + 2dimensions) arises in an analogous fashion. There is an honest BV theory of a nondegenerate two-formon a Riemannian six-manifold, which endows the theory of the self-dual two-form with the structure of apresymplectic BV theory. Among other examples, we give a precise formulation of the self-dual two-form in § factorization algebras . Costello and Gwilliam have developed a mathematical approachto the study of observables in perturbative field theory, of which local operators are a special case. Thegeneral philosophy is that the observables of a perturbative (quantum) field theory have the structure of afactorization algebra on spacetime [1], [12]. Roughly, this factorization algebra of observables assigns to anopen set U of spacetime a cochain complex Obs( U ) of “observables with support contained in U .” Whentwo open sets U and V are disjoint and contained in some bigger open set W , the factorization algebrastructure defines a rule of how to “multiply” observables Obs( U ) ⊗ Obs( V ) → Obs( W ). For local operators,one should think of this as organizing the operator product expansion in a sufficiently coherent way. n the ordinary BV formalism, the factorization algebra of observables has a very important structure,namely a Poisson bracket of cohomological degree +1 induced from the shifted symplectic form ω BV . This isreminiscent of the Poisson structure on functions on an ordinary symplectic manifold, and is a key ingredientin quantization.In the case of a presymplectic manifold, the full algebra of functions does not carry such a bracket. Butthere is a subalgebra of functions, called the Hamiltonian functions, that does. This issue persists in thepresymplectic BV formalism, and some care must be taken to define a notion of observables that carries sucha shifted Poisson structure. We tentatively solve this problem, and for special classes of free presymplecticBV theories we provide an appropriate notion of “Hamiltonian observables.” The corresponding factorizationalgebra carries a shifted Poisson structure, which is a direct generalization of the work of Costello–Gwilliamthat works to include presymplectic BV theories. While the Hamiltonian observables provide a way ofunderstanding a large class of observables in presymplectic BV theories, we emphasize that a full theoryshould be expected to contain additional, nonperturbative observables: the Hamiltonian observables of thechiral boson, for example, agree with the U(1) current algebra, and therefore do not see observables (suchas vertex operators) that have to do with the bosonic zero mode.Using this formalism, we formulate the abelian tensor multiplet as a presymplectic BV theory, and go onto work out the full L ∞ module structure encoding the on-shell action of supersymmetry. Our formalismis distinguished from other formulations of the abelian tensor multiplet in that it extends supersymmetryoff-shell without using any auxiliary fields, in the homotopy-algebraic spirit of the BV formalism. Using this L ∞ module structure, we rigorously compute both twists; like the full theory, these are presymplectic BVtheories. In eliminating acyclic pairs to obtain more natural descriptions of the twisted theories, we are forcedto carefully consider what it means for a quasi-isomorphism to induce an equivalence of shifted presymplec-tic structures; understanding these equivalences is crucial for correctly describing both the presymplecticstructure on the holomorphic twist and the action of the residual supersymmetry there.Our results allow us to compare concretely to Kodaira–Spencer theory on Calabi–Yau threefolds, whichis expected to play a role in the proposed description of holomorphically twisted supergravity theories dueto Costello and Li [13]. It would be interesting to try and incorporate our results into the framework ofthe nonminimal twist of 11d supergravity, which we expect to agree with the proposals for “topologicalM-theory” considered in the literature [14]–[16]; branes in topological M-theory were considered in [17].However, we reserve more substantial comparisons for future work.We are also able to perform a number of consistency checks with known results on holomorphic twistsof theories arising by dimensional reduction of the tensor multiplet. At the level of the holomorphic twist,we show that the reduction to four-dimensions yields the expected supersymmetric Yang–Mills theories. The development of the theory of observables for more general presymplectic BV theories is part of ongoing work with EugeneRabinovich. urthermore, when we compactify along along a four-manifold we recover the ordinary chiral boson onRiemann surfaces.Finally, we discuss dimensional reduction to five-dimensional Yang–Mills theory at the level of the un-twisted theory. Issues related to electric–magnetic duality appear naturally and play a key role here; further-more, obtaining the correct result on the nose requires correctly accounting for nonperturbative phenomenathat are missed by our perturbative approach. Although we do not rigorously develop the presymplecticBV formalism at a nonperturbative level in this work, we speculate about a nonperturbative formulation forgauge group U(1), and argue that our proposal gives the correct dimensional reduction on the nose at thelevel of chain complexes of sheaves. Doing this requires a conjectural description of the theory of abelian p -form fields in terms of a direct sum of two Deligne cohomology groups, which can be interpreted as acomplete (nonperturbative) presymplectic BV theory in novel fashion. Previous work.
There has been an enormous amount of previous work in the physics literature on topicsrelated to M5 branes and N = (2 ,
0) superconformal theories in six dimensions, and any attempt to provideexhaustive references is doomed to fail. In light of this, our bibliography makes no pretense to be completeor even representative. The best we can offer is an extremely brief and cursory overview of some selectedpast literature, which may serve to orient the reader; for more complete background, the reader is referredto the references in the cited literature, and in particular to the reviews [18]–[20].Tensor multiplets in six dimensions were constructed in [21]. The earliest approaches to the M5-braneinvolved the study of relevant “black brane” solutions in eleven-dimensional supergravity theory [22]; perhapsthe first intimation that corresponding six-dimensional theories should exist was made by considering typeIIB superstring theory on K3 singularities in [23]. The abelian M5-brane theory was worked out, includingvarious proposals for Lagrangian formulations, in [8], [9], in [7], in [24], and in [6], following the generalframework for chiral fields in [25]. These formulations were later shown to be equivalent in [26]. Theconnection of the tensor multiplet to supergravity solutions on AdS × S was discussed in [27] with anemphasis on N = (2 ,
0) superconformal symmetry.As to twisting the theory, the non-minimal twist was studied in [28], [29], and a close relative earlier in [30].(The approach of the latter paper effectively made use of the twisting homomorphism appropriate to theunique topological twist in five-dimensional N = 2 supersymmetry; this is the dimensional reduction of thesix-dimensional non-minimal twist.) While these studies compute the nonminimal twist at a nonperturbativelevel, [28], [29] do so only after compactification to four dimensions along the Riemann surface in thespacetime Σ × M , and thus do not see the holomorphic dependence on Σ explicitly. Our results arethus in some sense orthogonal. The relevance of the full nonminimal twist for the AGT correspondencewas emphasized in [31]; it would be interesting to connect our results to the AGT [32] and 3d-3d [33]correspondences. he holomorphic twist has, as far as we know, not been considered explicitly before, although the super-symmetric index of the abelian theory was computed in [34]. We expect agreement between the characterof local operators in the holomorphic theory [35] and the index studied there, after correctly accounting fornonperturbative operators, but do not consider that question in the present work and hope to study it in thefuture. We note, however, that the P factorization algebra arising as the Hamiltonian observables of theholomorphically twisted (1,0) theory was studied in [36] as a boundary system for seven-dimensional abelianChern–Simons theory. (The relation between the six-dimensional self-dual theory and seven-dimensionalChern–Simons theory is the subject of earlier work by [37], among others.) We see both these resultsand our results here as progress towards an understanding of the holomorphically twisted version of theAdS /CFT correspondence.Recently, there has been new progress on the question of finding a formulation of the nonabelian theory;much of this progress makes use of higher algebraic or homotopy-algebraic structure. See, for example, [38],[39], and [40], [41]. It would be interesting either to study twisting some of these proposals, or to attemptto make further progress on these questions by searching for nonabelian or interacting generalizations of thetwisted theories studied here. These might be easier to find than their nontwisted counterparts and offernew insight into the nature of the interacting (2 ,
0) theory. We look forward to working on such questionsin the future, and hope that others are inspired to pursue similar lines of attack.For the physicist reader, we emphasize that we deal here with a formulation that is lacking, even at apurely classical level, in at least three respects. Firstly, we make no effort to formulate the theory non-perturbatively, even for gauge group U (1); in a sense, our discussion deals only with the gauge group R .(Some more speculative remarks about this, though, are given in § An outline of the paper.
We begin in § § N = (1 ,
0) and N = (2 ,
0) versions of the tensor multiplet in the presymplectic BV formalism. We reviewthe classification of possible twists, and then give an explicit description of the presymplectic BV theory s an L ∞ module for the supersymmetry algebra. We perform the calculation of the minimal twist of thetensor multiplet in §
4, and of the non-minimal twist in §
5. We touch back with string theory in §
6, wherewe relate our twisted theories to the conjectural twist of Type IIB supergravity due to Costello–Li. Finally,in §
7, we explore some consequences of our description of the twisted theories upon dimensional reduction.We perform some sanity checks with theories that are conjecturally obtained as the reduction of the theoryon the M5 brane, culminating in a computation of the dimensional reduction of the untwisted theory alonga circle. Some interesting issues related to electric-magnetic duality appear naturally; we discuss these, andend with some speculative remarks on nonperturbative generalizations of our results.
Conventions and notations. • If E → M is a graded vector bundle on a smooth manifold M , then we define the new vector bundle E ! = E ∗ ⊗ Dens M , where E ∗ is the linear dual and Dens M is the bundle of densities on M . Wedenote by E the space of smooth sections of E , and E ! the space of sections of E ! . The notation E c refers to the space of compactly supported sections of E . The notation ( E c ) E refers to the space of(compactly supported) distributional sections of E . • The sheaf of (smooth) p -forms on a smooth manifold M will be denoted Ω p ( M ) and Ω • ( M ) = L Ω p ( M )[ − p ] is the Z -graded sheaf of de Rham forms, with Ω p ( M ) in degree p . Often times when M is understood we will denote the space of p -forms by Ω p . More generally, our grading conventions arecohomological, and are chosen such that the cohomological degree of a chain complex of differentialforms is determined by the (total) form degree, but always taken to start with the lowest term ofthe complex in degree zero. Thus Ω p is a degree-zero object, Ω ≤ p is a chain complex with support indegrees zero to p , and Ω ≥ p ( R d ) begins with p -forms in degree zero and runs up to d -forms in degree d − p . • On a complex manifold X , we have the sheaves Ω i, hol ( X ) of holomorphic forms of type ( i, ∂ : Ω i, hol ( X ) → Ω i +1 , hol ( X ) is the holomorphic de Rham operator. The standard Dolbeaultresolution of holomorphic i -forms is (Ω i, • ( X ) , ∂ ) where Ω i, • ( X ) = ⊕ k Ω i,k ( X )[ − k ] is the complex ofDolbeault forms of type ( i, • ) with ( i, k ) in cohomological degree + k . Again, when X is understoodwe will denote forms of type ( i, j ) by Ω i,j . Acknowledgements.
We thank D. Butson, K. Costello, C. Elliott, D. Freed, O. Gwilliam, Si Li, N. Pa-quette, E. Rabinovich, S. Raghavendran, P. Safronov, P. Teichner, J. Walcher, P. Yoo for conversation andinspiration of all kinds. I.S. thanks the Fields Institute at the University of Toronto for hospitality, as wellas the Mathematical Sciences Research Institute in Berkeley, California and the Perimeter Institute for The-oretical Physics for generous offers of hospitality that did not take place due to Covid-19. The work of I.S. issupported in part by the Deutsche Forschungsgemeinschaft, within the framework of the cluster of excellence“STRUCTURES” at the University of Heidelberg. The work of B.R.W. is supported by the National Science oundation Award DMS-1645877 and by the National Science Foundation under Grant No. 1440140, whilethe author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, duringthe semester of Spring 2020.2. A presymplectic Batalin–Vilkovisky formalism
In the standard Batalin–Vilkovisky (BV) formalism [3], one is interested in studying the (derived) criticallocus of an action functional. On general grounds, derived critical loci are equipped with canonical ( − E are given as the space of sections of some graded vectorbundle E → M , where M is the spacetime. In this context, the ( − ω : E ∼ = E ! [ − E is of theform(3) E = T ∗ [ − F def = F ⊕ F ! [ − , where F is some graded vector bundle, which carries a natural ( − Q BV is constructed such that(4) H ( E , Q BV ) ∼ = Crit( S ) , i.e. so that the sheaf of chain complexes ( E , Q BV ) is a model of the derived critical locus.In general, we can think of the ( − ω as a two-form (with constant coefficients) onthe infinite-dimensional linear space E . Moreover, this two-form is of a very special nature: it arises locallyon spacetime. For a more detailed introduction to the BV formalism its description of perturbative classicalfield theory, see [1], [2].We will be interested in a generalization of the BV formalism, motivated by the classical theory ofpresymplectic geometry and its appearance in Dirac’s theory of constrained systems in quantum mechanics.In ordinary geometry, a presymplectic manifold is a smooth manifold M equipped with a closed two-form ω ∈ Ω ( M ), d ω = 0. Equivalently, ω can be viewed as a skew map of bundles T M → T ∗ M . This is ourstarting point for the presymplectic version of the BV formalism in the derived and infinite dimensionalsetting of field theory.2.1. Presymplectic BV formalism.
We begin by introducing the presymplectic version of the BV for-malism in terms of a two-form on the space of classical fields. This generalization shares many features withthe usual BV setup: the two-form of degree ( −
1) on arises “locally” on spacetime, in the sense that it isdefined by a differential operator acting on the fields. In this paper we are only concerned with free theories,so we immediately restrict our attention to this case. t is important for us that our complexes are bigraded by the abelian group Z × Z /
2. We will refer to theinteger grading as the cohomological or ghost degree , and the supplemental Z / parity or fermionnumber .Before stating the definition of a free presymplectic BV theory, we set up the following notion aboutthe skewness of a differential operator. Let E be a vector bundle on M and suppose D : E → E ! [ n ] is adifferential operator of degree n . The continuous linear dual of E is E ∨ = E ! c (see § D defines thefollowing composition D : E c ֒ → E D −→ E ! [ n ] ֒ → E ! [ n ] . The continuous linear dual of D is a linear map of the same form D ∨ : E c → E ! [ n ]. We say the originaloperator D is graded skew symmetric if D = ( − n +1 D ∨ . Definition 2.1.
A (perturbative) free presymplectic BV theory on a manifold M is a tuple ( E, Q BV , ω )where: • E is a finite-rank, Z × Z / M , equipped with a differential operator Q BV ∈ Diff( E , E )[1]of bidegree (1 , • a differential operator ω ∈ Diff (cid:0) E , E ! (cid:1) [ − − , Q BV satisfies ( Q BV ) = 0, and the resulting complex ( E , Q BV ) is elliptic;(2) the operator ω is graded skew symmetric with regard to the totalized Z / ω and Q BV are compatible: [ Q BV , ω ] = 0.We refer to the fields φ ∈ E of cohomological degree zero as the “physical fields”. For free theories, thelinearized equations of motion can be read off as Q BV φ = 0. As is usual in the BRST/BV formalism, gaugesymmetries are imposed by the fields of cohomological degree − ω determines a bilinear pairing of the form Z M ω : E c × E c → Dens M [ − R M −−→ C [ − E c with the structure of a ( − ω is induced from a bilinear map of vector bundles which is fiberwise non-degenerate. he notion of a free presymplectic BV theory is thus a weakening of the more familiar definition. Indeed,when ω is an order zero differential operator such that ω : E ∼ = −→ E ! [ −
1] is an isomorphism, the tuple(
E, Q BV , ω ) defines a free BV theory in the usual sense. Remark . There are two natural ways to generalize Definition 2.1 that we do not pursue here: − Non-constant coefficient presymplectic forms : More generally, one can ask that ω be given as a polydif-ferential operator of the form ω ∈ Y n ≥ PolyDiff( E ⊗ n ⊗ E , E ! )[ − . The right-hand side is what one should think of as the space of “local” two-forms on E . − “Interacting” presymplectic BV formalism : Here, we require that L = E [ −
1] be equipped with the struc-ture of a local L ∞ algebra. Thus, the space of fields E should be thought of as the formal modulispace given by the classifying space B L . In the situation above, the free theory corresponds to anabelian local L ∞ algebra, in which only the unary operation (differential) is nontrivial.There is a natural compatibility between these two more general structures that is required. Using thedescription of the fields as the formal moduli space B L , for some L ∞ algebra L , one can view ω as a two-form ω ∈ Ω (B L ) = C • ( L , ∧ L [1] ∗ ). There is an internal differential on the space of two-forms given by theChevalley–Eilenberg differential d CE corresponding to the L ∞ structure. There is also an external, de Rhamtype, differential of the form d dR : Ω (B L ) → Ω (B L ). In this setup we require d CE ω = 0 and d dR ω = 0.We could weaken this condition further by replacing strictly closed two-forms on B L by Ω ≥ (B L ) and askingthat ω be a cocycle here.Since we only consider free presymplectic BV theories in this paper, we will simply refer to them aspresymplectic BV theories.2.2. Examples of presymplectic BV theories.
We proceed to give some examples of presymplectic BVtheories, beginning with simple examples of degenerate pairings and proceeding to more ones more relevantto six-dimensional theories. The secondary goal of this section is to set up notation and terminology thatwill be used in the rest of the paper.
Example . Suppose (
V, w ) is a finite dimensional presymplectic vector space. That is, V is a finitedimensional vector space and w : V → V ∗ is a (degree zero) linear map which satisfies w ∗ = − w . Then, forany 1-manifold L , the elliptic complex ( E , Q BV ) = (Ω • ⊗ V, d dR )is a presymplectic BV theory on L with ω = id Ω • ⊗ w : Ω • ⊗ V → Ω • L ⊗ V ∗ = E ! [ − . imilarly, if Σ is a Riemann surface equipped with a spin structure K , then the elliptic complex( E , Q BV ) = (cid:16) Ω , • ⊗ K ⊗ V, ∂ (cid:17) is a presymplectic BV theory on Σ with ω = id Ω , • ⊗ K ⊗ w : Ω , • ⊗ K ⊗ V → Ω , • ⊗ K ⊗ V ∗ . Each theory in this example arose from an ordinary presymplectic vector space, which was also the sourceof the degeneracy of ω . The first example that is really intrinsic to field theory, and also relevant for thefurther discussion in this paper, is the following. Example . Let Σ be a Riemann surface and suppose (
W, h ) is a finite dimensional vector space equippedwith a symmetric bilinear form thought of as a linear map h : W → W ∗ . Then( E , Q BV ) = (cid:0) Ω , • ⊗ W, ∂ (cid:1) is a presymplectic BV theory with ω = ∂ ⊗ h : Ω , • ⊗ W → Ω , • ⊗ W ∗ = E ! [ − . We refer to this free presymplectic BV theory as the chiral boson with values in W , and will denote it by χ (0 , W ) (see the next example). In the case that W = C , we will simply denote this by χ (0). Remark . While we did not require (
W, h ) to be nondegenerate in the above example, the theory is agenuinely presymplectic BV theory even if h is nondegenerate. This corresponds to the standard notion ofthe chiral boson in the physics literature, and we will have no cause to consider degenerate pairings h inwhat follows. Example . Suppose X is a (2 k + 1)-dimensional complex manifold. Let Ω • , hol = (cid:0) Ω • , hol , ∂ (cid:1) be theholomorphic de Rham complex and let Ω ≥ k +1 , hol be the complex of forms of degree ≥ k + 1. By theholomorphic Poincar´e lemma, Ω ≥ k +1 , hol is a resolution of the sheaf of holomorphic closed ( k + 1)-forms.Further, Ω ≥ k +1 , hol [ − k −
1] is a subcomplex of Ω • , hol and there is a short exact sequence of sheaves of cochaincomplexes Ω ≥ k +1 , hol [ − k − → Ω • , hol → Ω ≤ k, hol which has a locally free resolution of the form(5) Ω ≥ k +1 , • [ − k − → Ω • , • → Ω ≤ k, • . In this sequence, all forms are smooth and the total differential is ∂ + ∂ in each term. We use this quotientcomplex Ω ≤ k, • to define another class of presymplectic BV theories. et ( W, h ) be as in the previous example. (Following Remark 2.5, it may as well be nondegenerate.) Theelliptic complex ( E , Q BV ) = (cid:16) Ω ≤ k, • X ⊗ W [2 k ] , d = ∂ + ∂ (cid:17) . is a presymplectic BV theory with ω = ∂ ⊗ h : Ω ≤ k, • X ⊗ W [2 k ] → Ω ≥ k +1 , • X ⊗ W ∗ [ k ] . We denote this presymplectic BV theory by χ (2 k, W ), which we will refer to as the chiral k -form withvalues in W . In the case W = C we will simply denote this by χ (2 k ). Example . Let M be a Riemannian (4 k + 2)-manifold, and ( W, h ) as above. The Hodge star operator ⋆ defines a decomposition(6) Ω k +1 ( M ) = Ω k +1+ ( M ) ⊕ Ω k +1 − ( M )on the middle de Rham forms, where ⋆ acts by ±√− k +1 ± ( M ).Consider the following exact sequence of sheaves of cochain complexes:(7) 0 → Ω ≥ k +1 − [ − k − → Ω • → Ω ≤ k +1+ → ≤ k +1+ = (cid:16) Ω −→ Ω [ − d −→ · · · d −→ Ω k [ − k ] d + −−→ Ω k +1+ [ − k − (cid:17) , d + = 12 (1 − √− ⋆ )d , and(9) Ω ≥ k +1 − = (cid:16) Ω k +1 − d −→ Ω k +2 [ − d −→ · · · d −→ Ω k +2 [ − k − (cid:17) Let(10) ( E , Q BV ) = (Ω ≤ k +1+ ⊗ W [2 k ] , d )and ω = d ⊗ h : Ω ≤ k +1+ ⊗ W [2 k ] → Ω ≥ k +1 − ⊗ W ∗ . This data defines a presymplectic BV theory χ + (2 k, W ) on any Riemannian (4 k + 2)-manifold, which wewill refer to as the self-dual k -form with values in W . Again, in the case W = C we will simply denotethis by χ + (2 k ). Remark . In general, the theories χ (2 k ) and χ + (2 k ) are defined on different classes of manifolds; theycan, however, be simultaneously defined when X is a complex manifold equipped with a K¨ahler metric. Evenin this case, they are distinct theories (although their dimensional reductions along C P both agree with the sual chiral boson; see § § N = (1 ,
0) tensor multiplet (which consistsof χ + (2) together with fermions and one scalar) becomes precisely χ (2) under a holomorphic twist.There is, however, one case where the two theories χ (2 k ) and χ + (2 k ) coincide. A choice of metric on aRiemann surface determines a conformal class, which then corresponds precisely to a complex structure. Assuch, both of the theories χ (0) and χ + (0) are always well-defined, and in fact agree; both are the theory ofthe chiral boson defined in Example 2.4.We now recall a couple of examples of nondegenerate theories, for later convenience and to fix notation,that fit the definition of a standard free BV theory [1, Definition 7.2.1.1]. Example . Let M be a Riemannian manifold of dimension d . Let ( W, h ) be a complex vector spaceequipped with a non-degenerate symmetric bilinear pairing h : W ∼ = W ∗ . The theory Φ(0 , W ) of the freeboson with values in W is the data(11) ( E , Q BV ) = (cid:16) Ω ( M ) ⊗ W d ⋆ d ⊗ id W −−−−−−→ Ω d ( M ) ⊗ W [ − (cid:17) , and ω = id Ω ⊗ h + id Ω d ⊗ h . Notice this is a BV theory, the ( −
1) presymplectic structure is non-degenerate.
Example . Let (
W, h ) be as in the previous example, p ≥ M is a Riemannianmanifold of dimension d ≥ p . The theory Φ( p, W ) of free p -form fields valued in W is defined [46] bythe data(12) ( E , Q BV ) = (cid:16) Ω ≤ p ⊗ W [ p ] d ⋆ d ⊗ id W −−−−−−→ Ω ≥ d − p ⊗ W [ p − (cid:17) , with ( − ω = id Ω ≤ p ⊗ h + id Ω ≥ d − p ⊗ h . Notice again this is an honest BV theory,the presymplectic structure is non-degenerate. If α ∈ E denotes a field, the classical action functional reads R h ( α, d ⋆ d α ).This example clearly generalizes the free scalar field theory, and also does not depend in any way onour special choice of dimension. We will simply write Φ( p ) for the case W = C when the spacetime M isunderstood. Example . Let M be as in the last example, and suppose in addition it carries a spin structure compatiblewith the Riemannian metric. Let ( R, w ) be a complex vector space equipped with an antisymmetric non-degenerate bilinear pairing. The theory Ψ − ( R ) of chiral fermions valued in R is the data(13) ( E , Q BV ) = Γ(Π S − ⊗ R ) /∂ ⊗ id R −−−−→ Γ(Π S + ⊗ R )[ − , with ( − ω = id S + ⊗ w + id S − ⊗ w .We depart from the world of Riemannian manifolds to exhibit theories natural to the world of complexgeometry that will play an essential role later on in the paper. xample . Suppose X is a complex manifold of complex dimension 3 which is equipped with a square-root of its canonical bundle K X . Let ( S, w ) be a Z / Abelian holomorphic Chern–Simons theory valued in S is the free BV theoryhCS( S ) whose complex of fields is Ω , • ( X, K X ⊗ S )[1]with ( − ω = id Ω , • ⊗ w . This theory is naturally Z × Z / R w ( α ∧ ∂α ). Notice that the fields in cohomological degree zero consist of α ∈ Ω , ( X, K X ⊗ S ),and the equation of motion is ∂α = 0. This theory thus describes deformations of complex structure of the Z / K X ⊗ S .We will be most interested in the case S = Π R where R is an ordinary (even) symplectic vector space,see Theorem 4.2.2.3. Presymplectic BV theories and constraints.
Perturbative presymplectic BV theories stand inthe same relationship to perturbative BV theories as presymplectic manifolds do to symplectic manifolds.Presymplectic structures obviously pull back along embeddings, whereas symplectic structures do not. Thereis thus always a preferred presymplectic structure on submanifolds of any (pre)symplectic manifold. In fact,this is the starting point for Dirac’s theory of constrained mechanical systems [47], [48].Each of the examples of presymplectic BV theories we have given so far can be similarly understood asconstrained systems relative to some (symplectic) BV theory.
Example . The chiral boson χ (0 , W ) on a Riemann surface Σ, fromExample 2.4, can be understood as a constrained system relative to the free scalar Φ(0 , W ), see Example2.9. At the level of the equations of motion this is obvious: the constrained system picks out the harmonicfunctions that are holomorphic.In the BV formalism, this constraint is realized by the following diagram of sheaves on Σ:(14) Ω , Ω , Ω , Ω , ∂ ¯ ∂ id ¯ ∂ ∂ It is evident that the diagram commutes, and that the vertical arrows define a cochain map upon tensoringwith W :(15) χ (0 , W ) → Φ(0 , W ) . Furthermore, a moment’s thought reveals that the ( − χ (0 , W ) arises bypulling back the ( − , W ). xample k -form and the free 2 k -form) . It is easy to form generalizations of the previousexample. Consider the following diagram of sheaves on a Riemannian (4 k + 2)-manifold:(16) Ω · · · Ω k Ω k +2 · · · Ω k +2 Ω · · · Ω k Ω k +1+d ∗ did id d + d Just as above, the vertical arrows of this commuting diagram define a cochain map(17) χ + (2 k, W ) → Φ(2 k, W ) , under which the natural ( − − k, W ).If X is a complex manifold of complex dimension 2 k +1, the presymplectic BV theory of the chiral 2 k -form χ (2 k ) is defined, see Example 2.6. As a higher dimensional generalization of Example 2.13, χ (2 k ) can alsobe understood as a constrained system relative to theory of the free 2 k -form Φ(2 k, W ), see Example 2.10. Itis an instructive exercise to construct the similar diagram that witnesses the presymplectic structure on thechiral 2 k -form χ (2 k, W ) by pullback from the ordinary (nondegenerate) BV structure on Φ(2 k, W ).2.4. The observables of a presymplectic BV theory.
The classical BV formalism, as formulated in[1], constructs a factorization algebra from a classical BV theory, which plays the role of functions on asymplectic manifold in the ordinary finite dimensional situation.In symplectic geometry, functions carry a Poisson bracket. In the classical BV formalism there is a shiftedversion of Poisson algebras that play a similar role. By definition, a P -algebra is a commutative dg algebratogether with a graded skew-symmetric bracket of cohomological degree +1 which acts as a graded derivationwith respect to the commutative product. Classically, the BV formalism outputs a P -factorization algebraof classical observables [1, § P -factorization algebra associated to a presymplectic BV theory,which agrees with the construction of [1] in the case that the presymplectic BV theory is nondegenerate.Unlike the usual situation, this algebra is not simply the functions on the space of fields, but consists ofcertain class of functions. We begin by recalling the situation in presymplectic mechanics.To any presymplectic manifold (
M, ω ) one can associate a Poisson algebra. This construction generalizesthe usual Poisson algebra of functions in the symplectic case, and goes as follows. Let Vect( M ) be the Liealgebra of vector fields on M , and define the space of Hamiltonian pairs (18) Ham(
M, ω ) ⊂ Vect( M ) ⊕ O ( M )to be the linear subspace of pairs ( X, f ) satisfying i X ω = d f . Correspondingly, we can define the spaceof Hamiltonian functions or Hamiltonian vector fields to be the image of Ham(
M, ω ) under the obvious forgetful) maps to O ( M ) or Vect( M ) respectively. We will denote these spaces by O ω ( M ) and Vect ω ( M ).Notice that O ω ( M ) is the quotient of Ham( M, ω ) by the Lie ideal ker( ω ) ⊂ Ham(
M, ω ).There is a bracket on Ham(
M, ω ), defined by[(
X, f ) , ( Y, g )] = ([
X, Y ] , i X i Y ( ω )) . On the right-hand side the bracket [ − , − ] is the usual Lie bracket of vector fields. Furthermore, there is acommutative product on Ham( M, ω ) defined by(
X, f ) · ( Y, g ) = ( gX + f Y, f g ) . Together, they endow Ham(
M, ω ) with the structure of a Poisson algebra. This Poisson bracket on Hamil-tonian pairs induces a Poisson algebra structure on the algebra of Hamiltonian functions O ω ( M ).In some situations, one can realize the Poisson algebra of Hamiltonian functions O ω ( M ) as functions ona particular symplectic manifold. Associated to the presymplectic form ω is the subbundle(19) ker( ω ) ⊆ T M of the tangent bundle. The closure condition on ω ensures that ker( ω ) is always involutive. If one furtherassumes that the leaf space M/ ker( ω ) is a smooth manifold, then ω automatically descends to a symplec-tic structure along the quotient map q : M → M/ ker( ω ). Pulling back along this map determines anisomorphism of Poisson algebras q ∗ : O ( M/ ker( ω )) ∼ = −→ O ω ( M ) . In particular, one can view the Poisson algebra of Hamiltonian functions as the ker( ω )-invariants of thealgebra of functions O ω ( M ) = O ( M ) ker( ω ) . Notice that this formula makes sense without any conditions onthe niceness of the quotient M/ ker( ω ).In our setting, the presymplectic data is given by a presymplectic BV theory. A natural problem is todefine and characterize a version of Hamiltonian functions in this setting.2.4.1. The factorization algebra of observables.
As we’ve already mentioned, given a (nondegenerate) BVtheory the work of [1] produces a factorization algebra of classical observables. If ( E , ω, Q BV ) is the spaceof fields of a free BV theory on a manifold M then this factorization algebra Obs E assigns to the openset U ⊂ M the cochain complex Obs E ( U ) = ( O sm ( E ( U )) , Q BV ). Here O sm ( E ( U )) refers to the “smooth”functionals on E ( U ), which by definition are O sm ( E ( U )) = Sym (cid:0) E ! c ( U ) (cid:1) . Notice E ! c ( U ) ֒ → E ( U ) ∨ , so O sm is a subspace of the space of all functionals on E ( U ). urthermore, since ω is an isomorphism, it induces a bilinear pairing ω − : E ! c × E ! c → C [1] . By the graded Leibniz rule, this then determines a bracket {− , −} : O sm ( E ( U )) × O sm ( E ( U )) → O sm ( E ( U ))[1]endowing Obs E with the structure of a P -factorization algebra, see [1, Lemma 5.3.0.1].In this section, we turn our attention to defining the observables of a presymplectic BV theory, modeled onthe notion of the algebra of Hamiltonian functions in the finite dimensional presymplectic setting. Supposethat ( E , ω, Q BV ) is a free presymplectic BV theory. The shifted presymplectic structure is defined by adifferential operator ω : E → E ! [ − . In order to implement the structures we recounted in the ordinary presymplectic setting, the first object wemust come to terms with is the solution sheaf of this differential operator ker( ω ) ⊂ E .In general ker( ω ) is not given as the smooth sections of a finite rank vector bundle, so it is outside of ourusual context of perturbative field theory. However, suppose we could find a semi-free resolution ( K • ω , D ) byfinite rank bundles ker( ω ) ≃ −→ ( K • ω , D )which fits in a commuting diagram ker( ω ) K • ω E ≃ π where the bottom left arrow is the natural inclusion, and π is a linear differential operator. In the moregeneral case, where ω is nonlinear, we would require that K • ω have the structure of a dg Lie algebra resolvingker( ω ) ⊂ Vect( E ).Given this data, the natural ansatz for the classical observables is the (derived) invariants of O ( E ) by K • ω .A model for this is the Lie algebra cohomology:C • ( K • ω , O ( E )) = C • ( K • ω ⊕ E [ − . In this free case that we are in, this cochain complex is isomorphic to functions on the dg vector space K • ω [1] ⊕ E where the differential is D + Q BV + π .As in the case of the ordinary BV formalism, in the free case we can use the smoothed version of functionson fields. efinition 2.15. Let ( E , ω, Q BV ) be a free presymplectic BV theory on M , and suppose ( K • , D ) is a semi-free resolution of ker( ω ) ⊂ E as above. The cochain complex of classical observables supported on theopen set U ⊂ M is Obs ω E ( U ) = O sm ( K • ω ( U ) ⊕ E ( U )[ − , D + Q BV + π )= (cid:18) Sym (cid:0) ( K • ω ) ! c ( U ) ⊕ E ! c ( U )[1] (cid:1) , D + Q BV + π (cid:19) . By [12, Theorem 6.0.1] the assignment U Obs ω E ( U ) defines a factorization algebra on M , which we willdenote by Obs ω E . Example . Consider the chiral boson presymplectic BV theory χ (0), see Example 2.4, on a Riemannsurface Σ. The kernel of ω = ∂ is the sheaf of constant functionsker( ω ) = C Σ ⊂ Ω , • (Σ) . By Poincar´e’s Lemma, the de Rham complex (cid:0) Ω • Σ , d dR = ∂ + ∂ (cid:1) is a semi-free resolution of C Σ . Thus, theclassical observables are given as the Lie algebra cohomology of the abelian dg Lie algebra (cid:16) Ω • Σ ⊕ Ω , • Σ [ − , d dR + ∂ + π (cid:17) where π : Ω • Σ → Ω , • Σ is the projection. This dg Lie algebra is quasi-isomorphic to the abelian dg Lie algebraΩ , • Σ [ − ωχ (0) ≃ O sm (Ω , • Σ ) = Sym (cid:16) Ω , • Σ ,c [1] (cid:17) . There are two special cases to point out.(1) Suppose the shifted presymplectic form ω is an order zero differential operator. Then, ker( ω ) isa subbundle of E , so there is no need to seek a resolution. Furthermore, in this case E / ker( ω ) isalso given as the sheaf of sections of a graded vector bundle E/ ker( ω ), and ω descends to a bundleisomorphism ω : E/ ker( ω ) ∼ = −→ ( E/ ker( ω )) ! [ − E / ker( ω ) , ω, Q BV ) defines a (nondegenerate) free BV theory. The factorizationalgebra of the classical observables of the pre BV theory Obs ω E agrees with the factorization algebraof the BV theory E / ker( ω )Obs E / ker( ω ) = ( O sm ( E / ker( ω ) , Q BV ) . In this case, the observables inherit a P -structure by [1, Lemma 5.3.0.1].
2) This next case may seem obtuse, but fits in with many of the examples we consider. Suppose thatthe two-term complex 0 1Cone( ω )[ −
1] :
E E ! [ − , ω defined by the presymplectic form ω , is itself a semi-free resolution of ker( ω ). (Though it is not quiteprecise, one can imagine this condition as requiring that ω have trivial cokernel.) In this case, it isimmediate to verify that the factorization algebra of observables isObs ω E = (cid:0) O sm ( E ! [ − , Q BV (cid:1) . We mention that in this case Obs ω E is also endowed with a P -structure defined directly by ω .We can summarize the discussion in the two points above as follows. Proposition 2.17.
If the presymplectic BV theory ( E , ω, Q BV ) satisfies (1) or (2) above then the classicalobservables Obs ω E form a P -factorization algebra.Remark . Generally speaking, the resolution of the solution sheaf ker( ω ) is given by the Spencer res-olution. We expect a definition of a P -factorization algebra of observables associated to any (non-linear)presymplectic BV theory, though we do not pursue that here.For any k , the self-dual 2 k -form χ (2 k, W ) and the chiral 2 k -form satisfy condition (2) and so give rise toa P -factorization algebra of Hamiltonian observables. We will study this factorization algebra in depth in §
6. 3.
The abelian tensor multiplet
We provide a definition of the (perturbative) abelian N = (2 ,
0) tensor multiplet in the presymplectic BVformalism, together with the N = (1 ,
0) tensor and hypermultiplets. As discussed in the previous section,in the BV formalism one must specify a ( − R -symmetry representation, and fermionstransforming in the positive spin representation of Spin(6). The degeneracy of the shifted symplectic structurearises from the presence of the self-duality constraint on the two-form in the multiplet, just as in the examplesin § e begin by defining the field content of each multiplet precisely and giving the presymplectic BV struc-ture. A source for the definition of the fields of the tensor multiplet in the BV formalism can be traced tothe description in terms of the six-dimensional nilpotence variety given in [49]. See Remark 3.2.The next step is to formulate the action of supersymmetry on the (1 ,
0) and (2 ,
0) tensor multiplets at thelevel of the BV formalism. Here, one makes use of the well-known linear transformations on physical fieldsthat are given in the physics literature. See, for example, [50] for the full superconformal transformations ofthe N = (2 ,
0) multiplet; we will review the linearized super-Poincar´e transformations below.However, these transformations do not define an action of p (2 , on the space of fields. In the physicsterminology, they close only on-shell (and after accounting for gauge equivalence). In the BV formalism,this is rectified by extending the action to an L ∞ action on the BV fields. (See, just for example, [51] for anapplication of this technique.) For the hypermultiplet, this was performed explicitly in [49]; the hypermulti-plet, however, is a symplectic BV theory in the standard sense. For the tensor multiplet, supersymmetry alsoonly exists on-shell; no strict Lie module structure can be given. We work out the required L ∞ correctionterms, which play a nontrivial role in our later calculation of the non-minimal twist.We will first recall the definitions of the relevant supersymmetry algebras; afterwards, we will constructthe multiplets as free perturbative presymplectic BV theories, and go on to give the L ∞ module structureon the N = (2 ,
0) tensor multiplet. Of course the N = (1 ,
0) transformations follow trivially from this byrestriction.3.1.
Supersymmetry algebras in six dimensions.
Let S ± ∼ = C denote the complex 4-dimensional spinrepresentations of Spin(6) and let V ∼ = C be the vector representation. There exist natural Spin(6)-invariantisomorphisms ∧ ( S ± ) ∼ = −→ V and a non-degenerate Spin(6)-invariant pairing( − , − ) : S + ⊗ S − → C . The latter identifies S + ∼ = ( S − ) ∗ as Spin(6)-representations. Under the exceptional isomorphism Spin(6) ∼ = SU (4), S ± are identified with the fundamental and antifundamental representation respectively.The odd part of the complexified six-dimensional N = ( n,
0) supersymmetry algebra is of the formΣ n = S + ⊗ R n , where R n is a (2 n )-dimensional complex symplectic vector space whose symplectic form we denote by ω R .There is thus a natural action of Sp( n ) on R n by the defining representation. Note that we can identify thedual Σ ∗ n = S − ⊗ R n as representations of Spin(6) × Sp( n ). he full N = ( n,
0) supertranslation algebra in six dimensions is the super Lie algebra t ( n, = V ⊕ ΠΣ n with bracket(20) [ − , − ] = ∧ ⊗ ω R : ∧ (ΠΣ n ) → V. This algebra admits an action of Spin(6) × Sp( n ), where the first factor is the group of (Euclidean) Lorentzsymmetries and the second is called the R -symmetry group G R = Sp( n ). Extending the Lie algebra ofSpin(6) × Sp( n ) by this module produces the full N = ( n,
0) super-Poincar´e algebra, denoted p ( n, . Remark . We can view p ( n, as a graded Lie algebra by assigning degree zero to so (6) ⊕ sp ( n ), degreeone to Σ n , and degree two to V . In physics, this consistent Z -grading plays the role of the conformal weight.Both this grading and the R -symmetry action become inner in the superconformal algebra , which is thesimple super Lie algebra(21) c ( n, = osp (8 | n ) . The abelian N = (2 ,
0) multiplet in fact carries a module structure for osp (8 | n = 1 or 2. In the latter case, an accidental isomorphismidentifies Sp(2) with Spin(5), which further identifies R with the unique complex spin representation ofSpin(5).3.1.1. Elements of square zero.
With an eye towards twisting, we recall the classification of square-zeroelements in p ( n, for n = 1 and 2, following [11], [53]. As above, we are interested in odd supercharges(22) Q ∈ ΠΣ n = Π S + ⊗ R n , which satisfy the condition [ Q, Q ] = 0. Such supercharges define twists of a supersymmetric theory.We will find it useful to refer to supercharges by their rank with respect to the tensor product decompo-sition (22) (meaning the rank of the corresponding linear map R n → ( S + ) ∗ ). It is immediate from the formof the supertranslation algebra that elements of rank one square to zero for any n .When n = 1, it is also easy to see that any square-zero element must be of rank one, so that the spaceof such elements is isomorphic to the determinantal variety of rank-one matrices in M × ( C ). This can in urn be thought of as the image of the Segre embedding(23) P × P ֒ → P . For n = 2, there are two distinct classes of such supercharges: those of rank one, which we will also referto as minimal or holomorphic, and a certain class of rank-two elements, also called non-minimal or partiallytopological. A closer characterization of the two types of square-zero supercharges is the following: Minimal (or holomorphic):
A supercharge of this type is automatically square-zero. Moreover, such asupercharge has three invariant directions, and so the resulting twist is a holomorphic theory definedon complex three-folds. Similarly to the n = 1 case, the space of such elements is isomorphic to thedeterminantal variety of rank-one matrices in M × ( C ), which is the image of the Segre embedding(24) P × P ֒ → P . We remark that in the case n = 2, the supercharge Q of rank one defines a N = (1 ,
0) subalgebra p (1 , ∼ = p Q (1 , ⊂ p (2 , . Non-minimal (or partially topological):
Suppose Q ∈ ΠΣ is a rank-two supercharge (there is no suchsupercharge when n = 1). It can be written in the form(25) Q = ξ ⊗ r + ξ ⊗ r . Since ∧ S + ∼ = V , such an element must satisfy a single quadratic condition(26) w ( r , r ) = 0in order to be of square zero. Such a supercharge has five invariant directions, and the resultingtwist can be defined on the product of a smooth four-manifold with a Riemann surface. The space ofall such supercharges is a subvariety of the determinantal variety of rank-two matrices in M × ( C ),cut out by this single additional quadratic equation. Just as for the determinantal variety itself, itssingular locus is precisely the space of rank-one (holomorphic) supercharges.We will compute the holomorphic twist below in § §
5. There, we will also recallsome further details about nilpotent elements in t (2 , , showing how the non-minimal twist can be obtainedas a deformation of a fixed minimal twist. Remark . In fact, a study of the space of Maurer–Cartan elements in p (2 , was also a major motivationfor the formulation of the supersymmetry multiplets that we use throughout this paper. In physics, the purespinor superfield formalism [54] has been used as a tool to construct multiplets for some time. The relevantcohomology, corresponding to the field content of T (2 , , was first computed in [10]. n [11], the pure spinor superfield formalism was reinterpreted as a construction that produces a super-multiplet (in the form of a cochain complex of vector bundles) from the data of an equivariant sheaf over thenilpotence variety. It was further observed that, when the nilpotence variety is Calabi–Yau, Serre dualitygives rise to the structure of a shifted symplectic pairing on the resulting multiplet, so that the full data ofa BV theory is produced. More generally, when the canonical bundle is not trivial, the multiplet resultingfrom the canonical bundle admits a pairing with the multiplet associated to the structure sheaf.As mentioned before, applying this formalism to the structure sheaf of the nilpotence variety for p (2 , —thegeometry of which was reviewed above—produces a cochain complex with a homotopy action of p (2 , thatcorresponds precisely to the formulation we use in this paper and explore in detail in the following section.For this space, however, the canonical bundle is not trivial; the multiplet associated to the canonical bundleis, roughly speaking, T !(2 , , which can be identified with the space of linear Hamiltonian observables of T (2 , .It would be extremely interesting to give a geometric description of the origin of the presymplectic pairingon T (2 , , but we do not pursue this here; our use of this pairing, as described above, is motivated byinterpreting self-duality as a constraint and pulling back the pairing from the standard structure on thenondegenerate two-form.3.2. Supersymmetry multiplets.
The two theories we are most interested in are the abelian (1 ,
0) and(2 ,
0) tensor multiplets. We define these here at the level of (perturbative, free) presymplectic BV theories,and then go on to discuss the N = (1 ,
0) hypermultiplet, which will also play a role in what follows.First, we define the (1 ,
0) theory. Recall that R denotes the defining representation of Sp(1). Definition 3.3.
The six-dimensional abelian N = (1 , tensor multiplet is the presymplectic BV theory T (1 , defined by the direct sum of presymplectic BV theories:(27) T (1 , = χ + (2) ⊕ Ψ − ( R ) ⊕ Φ(0 , C ) , defined on a Riemannian spin manifold M . This theory has a symmetry by the group G R = Sp(1) whichacts on R by the defining representation and trivially on the summands χ + (2), Φ(0 , C ).This theory admits an action by the supertranslation algebra t (1 , , which will be constructed explicitlybelow in § • a two-form β ∈ Ω ( M ), satisfying the linear constraint d + ( β ) = 0 ∈ Ω ( M ); • a spinor ψ ∈ Ω ( M, S − ⊗ R ), satisfying the linear equation of motion ( /∂ ⊗ id R ) ψ = 0 ∈ Ω ( M, S + ⊗ R ); • a scalar ϕ ∈ Ω ( M ), satisfying the linear equation of motion d ⋆ d ϕ = 0 ∈ Ω ( M ).Next, we define the (2 ,
0) theory. Recall, R denotes the defining representation of Sp(2). Let W be thevector representation of Spin(5) ∼ = Sp(2). efinition 3.4. The six-dimensional abelian N = (2 , multiplet is the presymplectic BV theory T (2 , defined by the direct sum of presymplectic BV theories:(28) T (2 , = χ + (2) ⊕ Ψ − ( R ) ⊕ Φ(0 , W ) . defined on a Riemannian spin manifold. This theory has a symmetry by the group G R = Sp(2) which acts on R by the defining representation and W by the vector representation upon the identification Sp(2) ∼ = Spin(5).Note, G R = Sp(2) acts trivially on the summand χ + (2).This theory admits an action by the supertranslation algebra t (2 , , which will be constructed explicitlybelow in § • a two-form β ∈ Ω ( M ), satisfying the linear constraint d + ( β ) = 0 ∈ Ω ( M ); • a spinor ψ ∈ Ω ( M, S − ⊗ R ), satisfying the linear equation of motion ( /∂ ⊗ id R ) ψ = 0 ∈ Ω ( M, S + ⊗ R ); • a scalar ϕ ∈ Ω ( M, W ), satisfying the linear equation of motion (d ⋆ d ⊗ id W ) ϕ = 0 ∈ Ω ( M, W ).Lastly, we discuss the six-dimensional N = (1 ,
0) hypermultiplet.
Definition 3.5.
Let R be a finite-dimensional symplectic vector space over C , as above. The N = (1 , hypermultiplet valued in R is the following free (nondegenerate) BV theory in six dimensions:(29) T hyp(1 , ( R ) = Φ(0 , R ⊗ R ) ⊕ Ψ − ( R )The theory admits an action of the flavor symmetry group Sp( R ). (Note that R ⊗ R obtains a symmetricpairing from the tensor product of the symplectic pairings on R and R .)Exhibiting each of these theories as an L ∞ -module for the relevant supersymmetry algebra is the subjectof the next subsection.3.3. The module structure.
The main goal of this section is to define an action of the (2 ,
0) supersymmetryalgebra p (2 , on the tensor multiplet T (2 , . The action of the (1 ,
0) supersymmetry algebra on the constituentmultiplets T (1 , and T hyp(1 , ( R ′ ) will then be obtained trivially by restriction, which we will spell out at theend of this section.This action is only defined up to homotopy, which means we will give a description of T (2 , as an L ∞ - module over p (2 , . This amounts to giving a Lorentz- and R -symmetry invariant L ∞ action of thesupertranslation algebra t (2 , .Associated to the cochain complex T (2 , is the dg Lie algebra of endomorphisms End( T (2 , ). Sitting insideof this dg Lie algebra is a sub dg Lie algebra consisting of linear differential operators Diff( T (2 , , T (2 , ).The differential is given by the commutator with the classical BV differential Q BV . For us, an L ∞ -actionwill mean a homotopy coherent map, or L ∞ map, of dg Lie algebras ρ : p (2 , Diff( T (2 , , T (2 , ). uch an L ∞ map is encoded by the data of a sequence of polydifferential operators { ρ ( j ) } j ≥ of the form(30) X j ≥ ρ ( j ) : M j Sym j (cid:0) t (2 , [1] (cid:1) ⊗ T (2 , → T (2 , [1] , satisfying a list of compatibilities. For instance, the failure for ρ (1) : t (2 , ⊗ T (2 , → T (2 , to define a Liealgebra action is by the homotopy ρ (2) :(31) ρ (1) ( x ) ρ (1) ( y ) − ρ (1) ( y ) ρ (1) ( x ) − ρ (1) ([ x, y ]) = [ Q BV , ρ (2) ( x, y )] . In the case at hand, ρ (1) will be given by the known supersymmetry transformations from the physicsliterature, extended to the remaining complex by the requirement that it preserve the shifted presymplecticstructure. While ρ (1) does not define a representation of t (2 , , we can find ρ ( j ) , j ≥ L ∞ module structure. In fact, we will see that ρ ( j ) = 0 for j ≥
3, so we will only need to work out the quadraticterm ρ (2) . Theorem 3.6.
There are linear maps { ρ (1) , ρ (2) } that define an L ∞ -action of t (2 , on T (2 , . Furthermore,both ρ (1) and ρ (2) strictly preserve the ( − -shifted presymplectic structure. We split the proof of this result into two steps. First, we will construct the linear component ρ (1) andverify that it preserves the BV differential and shifted presymplectic form. Then we will define the quadratichomotopy ρ (2) and show that together with the linear term defines an L ∞ -module structure on T (2 , .3.3.1. The physical transformations.
We define the linear component ρ (1) of the action of supersymmetry on T (2 , . The map ρ (1) consists standard supersymmetry transformations on the physical fields (in cohomolog-ical degree zero), together with certain transformations on the antifields which guarantee that ρ (1) preservethe shifted presymplectic structure on T (2 , .The linear term ρ (1) is a sum of four components:(32) ρ V : V ⊗ T (2 , → T (2 , ρ Ψ : Σ ⊗ Ψ − ( R ) → χ + (2) ⊕ Φ(0 , W ) ρ Φ : Σ ⊗ Φ(0 , W ) → Ψ − ( R ) ρ χ : Σ ⊗ χ + (2) → Ψ − ( R )We will define each of these component maps in turn.The first transformation is simply the action by (complexified) translations on the fields. An translationinvariant vector field X ∈ V ⊂ Vect( R ) acts via the Lie derivative L X α , where α is any BV field. That is, ρ V ( X ⊗ α ) = L X α . e now turn to describe the supersymmetry transformations. We will first describe the action on thephysical fields, that is, the fields in cohomological degree zero. We will deduce the action on the antifieldsin the next subsection.The transformation of the physical fermion field (the component (Π S − ⊗ R ) of the BV complex Ψ − ( R )in degree zero) is given by ρ Ψ , which is defined as follows. Consider the isomorphism(33) (Π S + ⊗ R ) ⊗ (Π S − ⊗ R ) ∼ = (cid:0) C ⊕ ∧ V (cid:1) ⊗ (cid:0) C ⊕ W ⊕ Sym ( R ) (cid:1) of Spin(6) × Sp(2) representations. It is clear by inspection that there are equivariant projection mapsonto the irreducible representations ∧ V ⊗ C and C ⊗ W . These projections allow us to define ρ Ψ as thecomposition of the following sequence of maps:(34) Ω ⊗ W ΠΣ ⊗ Γ(Π S − ⊗ R ) ( S + ⊗ R ) ⊗ ( S − ⊗ R ) χ + (2) ⊕ Φ(0 , W ) . Ω ⊂ = ρ Ψ , ρ Ψ , ⊂ Of course, this map is canonically decomposed as the sum of two maps (along the direct sum in the target),which we will later refer to as ρ Ψ , and ρ Ψ , respectively.The transformation of the physical scalar field (the component C ∞ ( R ; W ) of the BV complex Φ(0 , W )in degree zero) is defined as follows. We observe that there is a map of Spin(6) × Sp(2) representations ofthe form(35) (Π S + ⊗ R ) ⊗ ( V ⊗ W ) → S − ⊗ R , which can be thought of (using the accidental isomorphism B ∼ = C ) as the tensor product of the six- andfive-dimensional Clifford multiplication maps. ρ (1)Φ can then be defined as the composition of the maps inthe diagram(36) ΠΣ ⊗ (Ω ⊗ W ) (Π S + ⊗ R ) ⊗ (Ω ⊗ W )Γ(Π S − ⊗ R ) Ψ − ( R ) . d ⊂ The vertical map is induced by (35).On the degree zero component Ω ( R ) of the presymplectic BV complex χ + (2), the map ρ χ is defined asfollows. Recall that there is a projection map of Spin(6) representations π : S + ⊗ ∧ − ( V ) → S − btained via the isomorphism ∧ − ( V ) ⊗ S + ∼ = S − ⊕ [012]. This isomorphism is most easily seen using theaccidental isomorphism with SU(4), where it can be derived using the standard rules for Young tableauxand takes the form(37) ⊗ ∼ = ⊕ . The map ρ χ is then defined on physical fields by the following sequence of maps:(38) ΠΣ ⊗ Ω (Π S + ⊗ R ) ⊗ Ω − Γ(Π S − ⊗ R ) Ψ − ( R ) . d − ⊂ Supersymmetry transformations on the anti-fields.
In the standard BV approach, there is a prescribedway to extend the linear action of any Lie algebra on the physical fields to an action on the BV complex ina way that preserves the shifted symplectic structure. The idea is that the action of a physical symmetryalgebra g is usually defined by a map(39) ρ : g → Vect( F )that implements the physical symmetry transformations on the physical (BRST) fields, just as in the previoussection. Of course there are strong conditions on ρ coming from, for example, the requirement of locality. Inthe BV formalism, there is additionally the requirement that the action of g on the BV fields must preservethe shifted symplectic structure. There is an immediate way to extend the vector fields (39) to symplectic vector fields on the space E = T ∗ [ − F of BV fields: one can take the transformation laws of the antifieldsto be determined by the condition of preserving the shifted symplectic form. (In fact, such vector fieldsare always Hamiltonian in the standard case.) The induced transformations of the antifields are sometimesknown as the anti-maps of the original transformations, and we will denote them with the superscript ρ + .For the anti-map component of ρ Φ , no complexity appears: we can simply define it as the composition(40) Ω ⊗ W [ −
1] Φ(0 , W ) . ΠΣ ⊗ Γ(Π S + [ − ⊗ R ) ( S + ⊗ R ) ⊗ Γ( S − [ − ⊗ R ) ⊂ /∂ The anti-map component of ρ Ψ , is similarly straightforward, and can be expressed with the diagram(41) ΠΣ ⊗ Φ(0 , W ) (Π S + ⊗ R ) ⊗ (Ω ⊗ W )[ −
1] Γ(Π S + [ − ⊗ R ) Ψ − ( R ) . ∼ = ⊂ The other two maps is determined by the nature of the shifted presymplectic pairing ω χ + on χ + (2). Assuch, the number of derivatives appearing is, at first glance, somewhat surprising. The anti-map to ρ Ψ , The notation refers to the Dynkin labels of type D . akes the form(42) Γ(Π S + [ − ⊗ R ) Ψ − ( R ) . ΠΣ ⊗ Ω [ −
1] (Π S + ⊗ R ) ⊗ Ω [ − ⊂ d Finally, the anti-map component of ρ χ takes the form(43) ΠΣ ⊗ Γ( S + [ − ⊗ R ) (Π S + ⊗ R ) ⊗ Γ(Π S + [ − ⊗ R ) Ω [ − χ + (2) . = ⊂ We have thus constructed the linear component of supersymmetry. It is straightforward to check that ρ (1) commutes with the classical BV differential and preserves the ( − The L ∞ terms. We turn to the proof of the remaining part of Theorem 3.6. We will show that ρ (1) sits as the linear component of an L ∞ -action of t (2 , on the (2 ,
0) theory. In fact, we will only need tointroduce a quadratic action term ρ (2) : t (2 , ⊗ t (2 , ⊗ T (2 , → T (2 , [ − ρ (2) χ : t (2 , ⊗ t (2 , ⊗ χ + (2) → χ + (2)[ − ρ (2)Ψ : t (2 , ⊗ t (2 , ⊗ Ψ − ( R ) → Ψ − ( R )[ − ρ (2)Φ : t (2 , ⊗ t (2 , ⊗ Φ(0 , W ) → χ + (2)[ − . First, ρ (2) χ = P ρ (2) χ,j is defined by the sum over form type of the linear maps ρ (2) χ,j : (Σ ⊗ Σ ) ⊗ Ω j [ · , · ] ⊗ −−−−→ V ⊗ Ω j i ( · ) −−→ Ω j − where [ · , · ] is the Lie bracket defining the (2 ,
0) algebra and i X denotes contraction with the vector field X .The next map, ρ (2)Ψ , acts on a fermion anti-field and produces a fermion field. To define it, we introducethe following notation. Recall that ∧ ( S + ) ∼ = V as Spin(6)-representations and ∧ R ∼ = C ⊕ W as Sp(2)-representations. Thus, there is the following composition of Spin(6) × Sp(2)-representations ⋆ : Σ ⊗ Σ → ( ∧ S + ) ⊗ ( ∧ R ) → V ⊗ W. So, given Q , Q ∈ Σ the image of Q ⊗ Q along this map is an element in V ⊗ W that we will denote by Q ⋆ Q . Now, we define ρ (2)Ψ as the sum ρ (2)Ψ , + ρ (2)Ψ , where ρ (2)Ψ , is the composition ρ (2)Ψ , : (Σ ⊗ Σ ) ⊗ Γ( S + ⊗ R ) ⋆ ⊗ −−→ ( V ⊗ W ) ⊗ Γ( S + ⊗ R ) → Γ( S − ⊗ R ) ⊗ C Ω ⊗ C Ω ⊗ C Ω ⊗ C S − ⊗ R S + ⊗ R Ω ⊗ W Ω ⊗ W i [ Q ,Q d i [ Q ,Q d+d i [ Q ,Q ρ χ i [ Q ,Q d + +d i [ Q ,Q ρ Ψ , d + i [ Q ,Q ( Q ⋆Q +[ Q ,Q ]) /∂ ρ Ψ , ρ Ψ , ( Q ⋆Q +[ Q ,Q ]) /∂ρ χ ρ Φ ρ Φ ( Q ⋆Q )d ρ Ψ , Figure 1.
The failure of ρ (1) to be a Lie map.where the second arrow is the map of Spin(6) × Sp(2)-representations in (35). Next, ρ (2)Ψ , is defined by thecomposition ρ (2)Ψ , : (Σ ⊗ Σ ) ⊗ Γ( S + ⊗ R ) [ · , · ] ⊗ −−−−→ V ⊗ Γ( S + ⊗ R ) → Γ( S − ⊗ R )where the last map is Clifford multiplication.Finally, the map ρ (2)Φ acts on a scalar field and produces a ghost one-form in χ + (2). Using the map ⋆ above, ρ (2)Φ is described by the composition(Σ ⊗ Σ ) ⊗ (cid:0) Ω ⊗ W (cid:1) ⋆ −→ Ω ⊗ ( W ⊗ W ) → Ω where the last map utilizes the symmetric form on W .To finish the proof of Theorem 3.6 we must show that ρ (1) and ρ (2) satisfy (31) for all x, y ∈ t (2 , .It will be convenient to define the following linear map.(45) µ : t (2 , ⊗ t (2 , ⊗ T (2 , → T (2 , ,x ⊗ y ⊗ f ρ (1) ([ x, y ] , f ) − ρ (1) ( x, ρ (1) ( y, f )) ± ρ (1) ( y, ρ (1) ( x, f ))This map µ represents the failure of ρ (1) to define a strict Lie algebra action. In terms of µ , (31) simplyreads(46) [ Q BV , ρ (2) ( x, y )] = µ ( x, y ) . We have represented µ via the orange arrows in Figure 1. In this figure, the dashed and dotted arrows denotethe action of Q and Q through the linear term ρ (1) .It is sufficient to consider the case when x = Q , y = Q ∈ Σ . We observe that the first term in µ simplyproduces the Lie derivative of any field in the direction [ Q , Q ]. Since µ is an even degree-zero map, we can onsider each degree and parity separately, beginning with the ghosts: here, it is easy to see that(47) µ ( Q , Q ) | Ω = L [ Q ,Q ] : Ω [2] → Ω [2] ,µ ( Q , Q ) | Ω = L [ Q ,Q ] : Ω [1] → Ω [1] , since the supersymmetry variations make no contribution. We next work out the action of µ on the two-formfield, which is given by(48) µ ( Q , Q ) | Ω = L [ Q ,Q ] − ρ Ψ ( Q ) ◦ ρ χ ( Q ) − ρ Ψ ( Q ) ◦ ρ χ ( Q )which is a map of the form Ω → Ω ⊕ (Ω ⊗ W ) ⊂ Φ(0 , W ). The map must be symmetric in the twofactors of Σ ; since Ω is neutral under Sp(2) R -symmetry, the only possible contractions of ( R ) ⊗ land inthe trivial representation or in W , and both are antisymmetric. So the pairing on (Π S + ) ⊗ must also beantisymmetric, showing that(49) µ ( Q , Q ) | Ω = L [ Q ,Q ] − i [ Q ,Q ] d − = d i [ Q ,Q ] + i [ Q ,Q ] d + . In degree one, there is also a unique equivariant map that can contribute: it is not difficult to show that(50) µ ( Q , Q ) | Ω = L [ Q ,Q ] − π + i [ Q ,Q ] d . Since [ Q , Q ] is a constant vector field, the Lie derivative preserves the self-duality condition; from this, itfollows via Cartan’s formula that the anti-self-dual part of i [ Q ,Q ] d is equal to d − i [ Q ,Q ] , so that(51) µ ( Q , Q ) | Ω = d + i [ Q ,Q ] . Similar arguments apply for the component of µ acting on the scalar field. One can check that therestriction of µ to the scalar field is of the form(52) µ ( Q , Q ) | Ω ⊗ W : Ω ⊗ W → Ω ⊂ χ + (2) . The diagonal term in µ restricted to Ω ⊗ W is seen to vanish upon applying Cartan’s magic formula. Thesame argument shows that µ , Φ also vanishes.The component (52) comes from a contraction of the supersymmetry generators with the de Rham dif-ferential acting on the scalar. There is precisely one such map, which takes the form(53) µ ( Q , Q ) | Ω ⊗ W = d ◦ ( Q ⋆ Q , · ) W where ( · , · ) W is the symmetric form on W . inally, the component of µ acting on Ψ − ( R ) maps a fermion to itself and a fermion anti-field to itself.For the fermion field, the restriction of µ is given as a sum of two terms µ ( Q , Q ) | Γ( S + ⊗ R ) = µ Ψ , ( Q , Q ) + µ Ψ , ( Q , Q )where µ Ψ , is given by the composition(54) µ Ψ , : Γ( S − ⊗ R ) Q ⋆Q −−−−→ Γ( S + ⊗ R ) /∂ −→ Γ( S − ⊗ R )and µ Ψ , is given by the composition(55) µ Ψ , : Γ( S − ⊗ R ) [ Q ,Q ] −−−−−→ Γ( S + ⊗ R ) /∂ −→ Γ( S − ⊗ R ) . The action of µ ( Q , Q ) on the anti-fermion fields is completely analogous.We proceed to verify (46). For the restriction of µ ( Q , Q ) to χ + (2) the equation follows from repeateduse of Cartan’s formula.Next, the restriction of µ ( Q , Q ) to the scalar is given by (52). The restriction of the left-hand side of(46) to the scalar is Q BV ◦ ρ (2)Φ ( Q , Q ) = d Ω → Ω ◦ ( Q ⋆ Q , · ) W as desired.Finally, the restriction of µ ( Q , Q ) to the fermion is given by the sum of (54) and (55). The left-handside of (46) also splits into two pieces. Note that[ Q BV , ρ (2)Ψ , ( Q , Q )] = /∂ ◦ ( Q ⋆ Q · ( · ))which is precisely µ Ψ , ( Q , Q,
2) acting on Ψ − ( R ).The other term is [ Q BV , ρ (2)Ψ , ( Q , Q )] = /∂ ◦ ( Q ⋆ Q · ( · ))which is precisely µ Ψ , ( Q , Q,
2) acting on Ψ − ( R ).We conclude by noting the following result: Proposition 3.7.
With respect to a fixed N = (1 , subalgebra of p (2 , , the abelian tensor multiplet decom-poses as (56) T (2 , ∼ = T (1 , ⊕ T hyp (1 , ( R ′ ) . Proof.
At the level of field content, the statements reduce to simple representation-theoretic facts: underthe subgroup Sp(1) × Sp(1 ′ ) ⊆ Sp(2), the vector and spinor representations decompose as(57) W ∼ = ( R ⊗ R ′ ) ⊕ C , R ∼ = R ⊕ R ′ espectively. (Here Sp(1) denotes the R -symmetry of p (1 , , and Sp(1) ′ its commutant inside of the (2 , R -symmetry.)The L ∞ module structure for p (2 , obviously restricts to an L ∞ module structure for p (1 , , and it istrivial to see that the resulting module structure extends the physical N = (1 ,
0) transformations. (At thelevel of physical transformations, the proposition is standard.) (cid:3) The minimal twists
In this section we will compute the holomorphic twist of the abelian N = (1 ,
0) and (2 ,
0) tensor multiplets,using the formulation and supersymmetry action developed in the preceding sections. We will begin byplacing the theory on a K¨ahler manifold and decomposing the fields with respect to the K¨ahler structure;at the level of representation theory, this corresponds to recalling the branching rules from SO(6) to U(3)(more precisely, at the level of the double covers MU(3) ֒ → Spin(6)), followed by a regrading.We will then deform the differential by a compatible holomorphic supercharge. (As is usual in twistcalculations, choices of holomorphic supercharge are in one-to-one correspondence with choices of complexstructure on R .) Since the L ∞ module structure worked out in the previous section preserves the presym-plectic structure, we are guaranteed that the twisted theory T Q (1 , is a well-defined presymplectic BV theoryafter deforming the differential.One subtlety appears when we attempt to simplify the resulting theory by discarding acyclic portions ofthe BV complex. There is a natural quasi-isomorphism of chain complexes of the form(58) Φ : T Q (1 , → χ (2) , whose kernel consists of an acyclic subcomplex of T Q (1 , . However, Φ does not respect the presymplecticstructure on T Q (1 , in a naive fashion!Given a general quasi-isomorphism(59) Φ : T → T ′ of cochain complexes underlying some presymplectic BV theories, the appropriate notion of compatibility isto ask that the two shifted presymplectic forms are equivalent in the larger theory; in other words, that(60) Φ ∗ ω ′ − ω = [ Q BV , h ] . Here h is a degree-( −
2) element in the space of symplectic structures, witnessing a homotopy between thetwo ( − Q BV -exact. emark . Indeed, suppose ker(Φ) is a nondegenerate BV theory whose differential is acyclic. For instance,suppose the complex of fields is of the form(61) T ∗ [ −
1] ( V ⊕ V [ − ∼ = ( V ⊕ V [ − ⊕ ( V ∨ [ − ⊕ V ∨ ) , where V is some chain complex of vector bundles, and the acyclic differential is the shift morphism betweenthe two copies of V and its anti-map between the two copies of V ∨ . Now, the symplectic form pairs V with V ∨ [ −
1] and V [ −
1] with V ∨ ; there is an obvious nullhomotopy given by the degree-( −
2) pairing thatpairs V [ −
1] with V ∨ [ − T Q (1 , , so that the homotopy equivalenceplays an essential role in determining the appropriate presymplectic BV structure on the holomorphic theory.With this in mind, we demonstrate an equivalence between T Q (1 , and χ (2), not just as chain complexes,but as presymplectic BV theories. Of course, an identical phenomenon occurs in the holomorphic twist ofthe (2 ,
0) multiplet, which can be thought of as one (1 ,
0) tensor multiplet and one (1 ,
0) hypermultiplet.Together, these rigorous twist computations are our main result in this section, which we state precisely asfollows:
Theorem 4.2.
Let Q be a rank one supercharge in either of the supersymmetry algebras p (1 , or p (2 , . Therespective twists of the abelian (1 , and (2 , tensor multiplets on C are as follows. • (1,0) The holomorphic twist T Q (1 , is equivalent to the Z -graded presymplectic BV theory of the chiral -form: T Q (1 , ≃ χ (2) . • (2,0) The holomorphic twist T Q (2 , is equivalent to the Z × Z / -graded presymplectic BV theory defined bythe chiral -form plus abelian holomorphic Chern–Simons theory with values in the odd symplecticvector space Π R ′ : (62) T Q (2 , ≃ χ (2) ⊕ hCS(Π R ′ ) . Moreover, this equivalence is
Sp(1) ′ -equivariant. The remainder of the section is devoted to a detailed proof. We start off with some reminders about thegeneral yoga of twisting.4.1.
Supersymmetric twisting.
In this section we briefly recall the procedure of twisting a supersymmetricfield theory. For a more complete formulation see [49], [55], though we modify the construction very slightly(see [49, Remark 2.19]). As we’ve already mentioned, the key piece of data is that of a square-zero supercharge Q . Roughly, the twisted theory is given by deforming the classical BV operator Q BV by Q . n the cited references, the twisting procedure is performed starting with the data of a supersymmetrictheory in the BV formalism. This means that one starts with the data of a classical theory in the BVformalism together with an ( L ∞ ) action by the super Lie algebra of supertranslations. In our context, we haveexhibited an L ∞ action of the supersymmetry algebra on a presymplectic BV theory, which acts compatiblywith the ( − Z × Z / Q is ofbidegree (0 , , Q BV Q BV + Q , one could choose to remember just the totalized Z / regrade the theory so that Q, Q BV have the same homogenousdegree. In addition to the action by supertranslations, a classical supersymmetric theory on R d carries anaction by the Lorentz group Spin( d ). It also often carries an action by the R -symmetry group G R , whichis the set of automorphisms of ΠΣ n preserving the pairing. For us, d = 6 and G R = Sp( n ) for N = ( n, R -symmetry to define a consistent graded structure, as well as toensure that the twisted theory is well-defined not just on affine space, but on all manifolds with appropriateholonomy group. To do this, we use two additional pieces of data, which we now describe in turn. Definition 4.3.
Given a square-zero supercharge Q , a regrading homomorphism is a homomorphism α :U(1) → G R such that the weight of Q under α is +1.Suppose E = ( E , Q BV ) is the cochain complex of fields of the classical theory, and for ϕ ∈ E , denote by | ϕ | = ( p, q mod 2) ∈ Z × Z / α , we define a new Z × Z / E α = ( E α , Q BV ) which agrees with ( E , Q BV ) as a totalized Z / | ϕ | α = | ϕ | + ( α ( ϕ ) , α ( ϕ ) mod 2) ∈ Z × Z / α ( ϕ ) denotes the weight of the field ϕ ∈ E under α . Note that Q BV and Q are both of bidegree (1 , E α . Our convention is that E α denotes the cochain complex offields that are regraded, but equipped with the original BV differential Q BV . The shifted (pre) symplecticstructure remains unchanged. here is one last step before performing the deformation of the classical differential by the supercharge Q in the regraded theory. In general, the symmetry group Spin( n ) × G R will no longer act on the deformedtheory since Q is generally not invariant under this group action. Definition 4.4.
Let Q be a square-zero supercharge, and suppose ι : G → Spin( n ) is a group homomorphism.A twisting homomorphism (relative to ι ) is a homomorphism φ : G → G R such that Q is preserved underthe product ι × φ : G → Spin( n ) × G R .Given such a φ , we can restrict the regraded theory to a representation for the group G , which we willdenote by φ ∗ e E α . We will refer to as the G -regraded theory.Given a square-zero supercharge Q , a regrading homomorphism α , and twisting homomorphism φ we canfinally define a twist of a supersymmetric theory E . It is the Z × Z / E Q = ( φ ∗ E α , Q BV + Q ) . Holomorphic decomposition.
Throughout the rest of this section we fix the data of a rank-onesupercharge Q ∈ Σ (which is automatically square-zero in p (1 , ), and characterize the resulting twist of the(1 ,
0) tensor multiplet T (1 , . As discussed in § Q defines a theory with three invariant directions,so we will refer to the twist as holomorphic. In addition to Q , to perform the twist we must prescribe acompatible pair of a twisting homomorphism φ and regrading homomorphism α .Geometrically, the supercharge Q defines a complex structure L = C ⊂ V = C equipped with the choiceof a holomorphic half-density on L .Under the subgroup MU(3) ⊂ Spin(6), the spin representations decompose as(63) S + = det( L ) ⊕ L ⊗ det( L ) − , S − = det( L ) − ⊕ L ∗ ⊗ det( L ) . ‘ In particular, the odd part Σ = S + ⊗ R of the super Lie algebra p (1 , decomposes under MU(3) asdet( L ) ⊗ R ⊕ L ⊗ det( L ) − ⊗ R . The holomorphic supercharge Q lies in the first factor.There exists a unique embedding U(1) ⊂ G R = Sp(1) under which Q has weight +1. The twistinghomomorphism is defined by the composition φ : MU(3) det −−−→ U(1) ֒ → Sp(1) . Under this twisting homomorphism, the defining representation R of Sp(1) splits as(64) R = det( L ) − ⊕ det( L ) . dditionally, we fix the regrading homomorphism to agree with the natural inclusion above: α : U(1) ֒ → Sp(1) . As outlined in § φ and α allow us to consider the G = MU(3)-regraded theory.We observe that the odd part Σ of p (1 , decomposes under these twisting data as(65) − − L ⊗ det( L ) − C · Q L L )Here, the horizontal grading is by the ghost Z -degree determined by α and the vertical grading is by spinU(1) ⊂ MU(3). Note that Q lives in a scalar summand of ghost degree +1.The decomposition of the (1 ,
0) tensor multiplet with respect to the twisting data is described in thefollowing proposition.
Proposition 4.5.
The
MU(3) -regraded (1 , tensor multiplet φ ∗ T α (1 , decomposes as φ ∗ T α (1 , = χ + (2) ⊕ Ψ α − ( R ) ⊕ Φ(0 , C ) . The result is depicted in Figure 2.
Notice that the MU(3)-action descends to a U(3)-action, so without confusion we will refer φ ∗ T α (1 , as theU(3)-regraded theory. Proof of Proposition 4.5.
The components χ + (2) and Φ(0 , C ) of T (1 , are acted on trivially by the R -symmetry group G R = Sp(1), so we only need to focus on how Ψ − ( R ) is regraded. According to Equations(63) and (64), the physical fields decompose under the twisting homomorphism φ by:(66) Π (cid:0) Ω ⊗ S − ⊗ R (cid:1) = Π(Ω , ⊕ Ω , ) ⊕ Π(Ω , ⊕ Ω , ) . Similarly, the antifields decompose as(67) Π (cid:0) Ω ⊗ S + ⊗ R (cid:1) [ −
1] = Π(Ω , ⊕ Ω , )[ − ⊕ Π(Ω , ⊕ Ω , )[ − α : U(1) ֒ → Sp(1) = G R . At thelevel of the decomposed fields in Equation (66), this U(1) acts by weight − , ⊕ Ω , ,and by weight +1 on the second summand Ω , ⊕ Ω , . Thus, we see that the regraded fields of Equation(66) become (Ω , ⊕ Ω , )[1] ⊕ (Ω , ⊕ Ω , )[ − . ∗ T α (1 , − − , Ω , Ω , ⊥ χ + (2) : Ω Ω , Ω , Ω , ω Ω , Ω , Ψ α − ( R ) : Ω , Ω , Ω , Ω , Ω , Ω , Ω , Ω , Φ(0 , C ) : Ω , Ω , ∂ ω ∂ ω ∂ ω ∂ ω ∂ ∗ ∂ ∗ △ Figure 2.
The regraded N = (1 ,
0) tensor multiplet. The unlabeled arrows denote theobvious ∂ or ∂ operators.Similarly, the regraded anti-fields of Equation (67) become(Ω , ⊕ Ω , )[ − ⊕ (Ω , ⊕ Ω , ) . It remains to identify the linear BV operator Q BV in the regraded theory. This follows from the well-knowndecomposition of the Dirac operator, on a K¨ahler manifold:(68) (cid:18) S − ⊗ R S + ⊗ R /∂ (cid:19) ∼ = Ω , ⊗ K − ⊗ R Ω , ⊗ K − ⊗ R Ω , ⊗ K − ⊗ R Ω , ⊗ K − ⊗ R . ∂∂∂ ∗ The components χ + (2) , Φ(0 , C ) remain unaffected by both the twisting homomorphism φ and regradinghomomorphism α . However, it is necessary in what follows to decompose these cochain complexes as U(3)-representations, using information about the decomposition of the de Rham forms on a K¨ahler manifold.We recall that multiplication by the K¨ahler form determines a cochain map of degree (1 , , = Ω , ⊥ ⊕ Ω , ω , (1 , φ ∗ T α (1 , χ + (2) β ∈ Ω β ∈ Ω = Ω , ⊕ Ω , ⊕ Ω , Ψ − ( R ) ψ − ∈ Π( S − ⊗ R ) ψ − ∈ (Ω ⊕ Ω , )[1] ⊕ (Ω , ⊕ Ω , )[ − , C ) φ ∈ Ω ⊗ C φ ∈ Ω ⊗ C Table 1.
The physical fields in the regraded (1 ,
0) theory.with the latter summand being the image of Ω , under the K¨ahler form. (Such a splitting is of coursedetermined on compactly supported forms by the choice of K¨ahler metric.) We correspondingly decomposethe ∂ and ∂ operators with respect to this splitting; we will sometimes use the subscript ∂ ω to indicate aprojection onto nonprimitive forms, and the superscript ∂ ω for projection onto primitive forms. Verifyingthe isomorphism(70) Ω ∼ = Ω , ⊕ Ω , ω ⊕ Ω , ⊥ is then a straightforward representation-theoretic exercise. (cid:3) In Table 1 we have summarized what happens to the physical fields (cohomological degree zero in theoriginal theory) of the (1 ,
0) tensor multiplet in the regraded theory.4.3.
Proof of (1,0) part of Theorem 4.2.
The proof proceeds in two steps. In the first, we use theholomorphic decomposition discussed above, and deform the theory by the holomorphic supercharge toobtain a description of the twist T Q (1 , . In the second, we give an explicit projection map which defines aquasi-isomorphism onto χ (2), and check that it defines an equivalence of presymplectic BV theories.4.3.1. Calculation of T Q (1 , . Throughout this section, we refer to Figure 3, which uses the decomposition ofthe fields we found in the previous section and shows the additional differentials generated by the holomorphicsupercharge. The black text denotes the fields in the component χ + (2) of the tensor multiplet. The red textdenotes the fields in the Ψ α − ( R ) component, as in Proposition 4.5. Finally, the green text denotes the fieldsin the Φ(0 , C ) component. Each of the solid lines denotes the linear BV differential in the original, untwistedtheory, see Figure 2. We will use superscripts to label the components of each field by their form degree.We have labeled the differential generated by the supercharge Q by the dotted and dashed arrows, whichwe now proceed to justify. The dotted arrows denote order zero differential operators, and thedashed arrows are given by the labeled differential operator. Throughout, we extensively refer tothe notation in § χ + (2). Inthe notation of § ρ Ψ , . In the holomorphic decomposition of the fields this map − − , − , Ω , − , Ω , ⊥ Ω , , Ω , Ω , ⊥ Ω , ω Ω , Ω , Ω , , Ω , Ω , ω , Ω , , Ω , ∂ ω ∂ ∂ ∂ ω ω∂ ω ∂ ω∂ Figure 3.
The holomorphic twist of the N = (1 ,
0) tensor multiplet. The horizontal gradingis the cohomological grading. The vertical grading is the weight with respect to U(1) ⊂ U(3).is the following projection ρ Ψ , ( Q ⊗ − ) : Ω ⊕ Ω , ⊕ Ω , ⊕ Ω , ։ ω Ω ⊕ Ω , ⊂ Ω , ⊕ Ω , ⊂ χ + (2)which reads ρ Ψ , ( Q ⊗ ψ − ) = ωψ , − + ψ , − . This term accounts for the dotted arrows in Figure 3 labeled 1and 2 . On the anti-fields, the map ρ Ψ , ( Q ⊗ − ) is given by the composition π ◦ d where π + is the projection π + : ( S + ⊗ R ) ⊗ Ω ։ S + ⊗ R . hen restricted to the holomorphic supercharge Q ∈ S + ⊗ R this projection defines a linear map π ′ ( Q ⊗ − )which reads, in holomorphic coordinates: π ( Q ⊗ − ) : Ω , ⊕ Ω , ⊕ Ω , ։ Ω , ⊕ Ω , → Ω , ⊕ Ω , where the last map uses the K¨ahler form as a linear map ω : Ω , → Ω , . Thus, acting on the anti-fields, ρ Ψ , ( Q ⊗ − ) reads Ω , /ω Ω , Ω , Ω , ρ Ψ , ( Q ⊗ − ) : Ω = ω Ω , ω Ω , Ω , ⊂ Ψ α − ( R )[1]Ω , Ω , ∂∂ = ∂∂ ω∂ This accounts for the dashed arrows Ω , /ω Ω , Ω , ∂ , ω Ω , Ω , ω∂ , and Ω , /ω Ω , Ω , ω∂ .We turn to the part of the supersymmetry which transforms fermions into the scalar Φ(0 , C ). In thenotation of § ρ Ψ , which is defined using the projection of a tensor product ofspin representations onto a trivial summand. When applied to the holomorphic supercharge Q ∈ S + ⊗ R ,the resulting linear map is given by the projection ρ Ψ , ( Q ⊗ − ) : Ω ⊕ Ω , ⊕ Ω , ⊕ Ω , ։ Ω . In the decomposition of the fields above, this reads ρ Ψ , ( Q ⊗ ψ − ) = ψ , − ∈ Ω , ⊂ Φ(0 , C ). This termaccounts for the dotted arrow in Figure 3 labeled 3 . Similarly, on the anti-fields we have ρ Ψ , ( Q ⊗ φ + ) = φ + ∈ Ω , ⊂ Ψ α − ( R )[2]. which one could write as δ Q ψ + − = φ + . This term accounts for thedotted arrow labeled 4 .Next, consider the supersymmetry which transforms Φ(0 , C ) into the fermion. In the notation of § ρ Φ . Applied to the supercharge Q , this is the composition ρ Φ ( Q ⊗ − ) : Ω −→ Ω , ⊕ Ω , Q −→ ω Ω , ⊂ Ψ α − ( R )[1]which reads ρ Φ ( Q ⊗ φ ) = ω∂φ and accounts for the dashed arrow Ω ω Ω , ω∂ . On antifields this isthe dashed arrow Ω , Ω , ω ∂ .Next, consider the supersymmetry which transforms χ + (2) into Ψ α − ( R ). In the notation of § ρ χ , which when acting on the physical fields is the composition π ◦ d − where π is theprojection π : ( S + ⊗ R ) ⊗ Ω − → S − ⊗ R . In standard physics notation, one would write this as δ Q φ = ψ , − . ( β ,j ) = β ,j ∈ Ω ,j , j = 0 , , ψ , − ) = ψ , − ∈ Ω , ;Φ( β , ) = β , ∈ Ω , ;Φ( β , + φ ) = β , ⊥ + ( β , ω − ωφ ) ∈ Ω , ;Φ([ β , ] ω + ωψ , − ) = β , + ωψ , − ∈ Ω , ;Φ( ψ +1 , − ) = ψ +1 , − ∈ Ω , . Table 2.
A component description of the projection map ΦWhen restricted to the holomorphic supercharge Q ∈ S + ⊗ R this projection defines a linear map π ( Q ⊗ − ) :Ω − → S − ⊗ R which reads, in holomorphic coordinates: π ( Q ⊗ − ) : Ω , ⊕ ω Ω , ⊕ Ω , /ω Ω , ։ Ω , ⊕ ω Ω , and is given by the obvious projection. Thus, acting on the physical fields, the map ρ χ ( Q ⊗ − ) is thecomposition Ω , Ω , Ω , ρ χ ( Q ⊗ − ) : Ω = Ω , ω Ω , Ω , ⊂ Ψ α − ( R )[1]Ω , Ω , /ω Ω , ∂∂ ω = ∂ ω ∂ ω = ∂ ω This accounts for the dashed arrows Ω , Ω , ∂ , Ω , ω Ω , ∂ ω , and Ω , ω Ω , ∂ ω . Theanti map for this component of supersymmetry acts on Ω , ⊕ Ω , ⊂ Ψ α − ( R ) and is defined by the projection: ρ χ ( Q ⊗ − ) : Ω , ⊕ Ω , ⊕ Ω , ⊕ Ω , ։ ω Ω , ⊕ Ω , ⊂ χ + (2)[1] . This accounts for the dotted arrows labeled 5 and 6 . All arrows have been accounted for, and we havethus verified that the twisted theory T Q (1 , is described by Figure 3.4.3.2. Verification of the equivalence between T Q (1 , and χ (2) . We now move to the second step of the proof.To begin, we note that there is a projection Φ from the total complex T Q (1 , in Figure 3 to the cochaincomplex χ (2) = Ω ≤ , • [2]:(71) Φ : T Q (1 , → χ (2) . On components, Φ is defined by the formulas in Table 2; it sends all other fields to zero. Here, β , ⊥ and β , ω denote the components of the (1 , , = Ω , ⊥ ⊕ ω Ω . Notice thatΦ( e Q BV ψ − ) = Φ( ∂ψ − + ωψ − + ψ − ) = 0 + ( ωψ − − ωψ − ) = 0 hich is the only nontrivial check that Φ is a cochain map. Notice that Φ is a map of underlying gradedvector bundles, so its kernel is well-defined. Since all the dotted arrows are isomorphisms, the kernel of thismap is acyclic, and so Φ defines a quasi-isomorphism of sheaves of cochain complexes.Since the supercharge Q preserves the presymplectic structure ω T on T (1 , , we know that T Q (1 , hasthe induced structure of a presymplectic BV theory. As discussed above, there is also a natural shiftedpresymplectic structure on χ (2), defined by the formula ω χ = R C α∂α ′ . To check that Φ defines an equivalenceof presymplectic BV theories, we will need to check its compatibility with these pairings.We note that the quasi-isomorphism Φ does not preserve the shifted presymplectic structures in any strictsense. However, there does exist a two-form on the space of fields h : T Q (1 , ,c × T Q (1 , ,c → C of degree − ω T − Φ ∗ ω χ = ( Q BV + Q ) h, where Q BV + Q denotes the internal differential on the cochain complex of two-forms in field space, withrespect to the total differential on T Q (1 , . In writing elements of the space of two-forms, we will alwayssuppress the integration symbol over C , which should be understood implicitly. We also suppress thesubscripts indicating the chirality of the (untwisted) fermions. Proposition 4.6.
Consider the two-form on the space of fields (73) h = β , ψ , + β , ( ψ + ) , + β , ω · ω − ψ , + φ , ( ψ + ) , . Then h defines a homotopy between ω T and the pullback Φ ∗ ω χ . That is, (72) is satisfied.Proof. The proof is a straightforward computation. The pairing on χ (2) is given by(74) ω χ = χ∂χ = χ , ∂χ , + ( χ , ⊥ + χ , ω ) ∂ ( χ , ⊥ + χ , ω ) , containing a total of five terms. Applying the pullback, we obtain(75) Φ ∗ ω χ = ( ψ + ) , ∂β , + (cid:16) β , ⊥ + ω − ψ , (cid:17) ∂ (cid:16) β , ⊥ + β , ⊥ + ωφ , (cid:17) . The pairing on T Q (1 , is given by(76) ω T = φ , φ , + ψ i, ψ − i, + β , ∂β , + β , ω (cid:16) ∂β , + ∂β , ⊥ + ∂β , ω (cid:17) + β , ⊥ (cid:16) ∂β , ⊥ + ∂β , ω + ∂β , (cid:17) . In writing the term ψ i, ψ − i, , we have suppressed the antifield symbols; of course, this means the nonde-generate pairing on the fermi fields, and would more properly be written ψ ev , ( ψ + ) odd , + ( ψ + ) odd , ψ ev , .Note also that we make no claim that all of the terms we write are nonvanishing (for example, many willidentically vanish on a compact K¨ahler manifold); the point is that our claim holds formally even withoutusing these facts. hen taking the difference of the pairings, the sixth and seventh terms of ω T cancel with correspondingterms, and the result is(77) ω T − π ∗ ω χ = φ , φ , + ψ i, ψ − i, + β , ∂β , + β , ω (cid:16) ∂β , + ∂β , ⊥ + ∂β , ω (cid:17) + β , ⊥ ∂β , − ( ψ + ) , ∂β , − β , ⊥ ω∂φ , − ω − ψ , ∂ (cid:16) β , ⊥ + β , ⊥ + ωφ , (cid:17) . This is obviously nonzero as a two-form on T Q (1 , , but we will show that it is the BV variation of thehomotopy (73). Note that the last term of the homotopy crucially contains only the scalar field, and not β , ω .To compute the BV variation Q BV h , we will need to consider all differentials in T Q (1 , entering terms thatappear in the homotopy. As usual, it is helpful to refer to Figure 3.As a first step, note that the homotopy h pairs the fermions at the upper right of Figure (3) with anisomorphic subcomplex of T Q (1 , . All of the “internal” arrows in each of these Z-shaped subdiagrams canthus be ignored; the terms they generate in the variation will occur twice, once from each side of the pairing,and will cancel after an integration by parts.It is also clear that the arrows that do not contain differential operators—the dotted arrows 2 through 6in the figure—generate precisely the terms of the pairing which do not contain differential operators, on thescalar and between ψ = and ψ + . This accounts for the first two terms in (77).It thus remains to consider only terms involving differential operators in both ω T − π ∗ ω χ and Qh , where wemay ignore the “internal” differentials in computing the latter. We proceed term by term in the homotopy.The first term is(78) Q ( β , ψ , ) = β , ∂β , , which cancels with the third term of (77). The second term is(79) Q ( β , ( ψ + ) , ) = ∂β , ( ψ + ) , + β , ∂β , ⊥ . These two terms cancel with the seventh and eighth terms of (77) after an integration by parts.The third term in the homotopy generates the largest number of terms: we have(80) Q ( β , ω · ω − ψ , ) = β , ω (cid:16) ∂β , + ∂β , ⊥ + ∂β , ω + ω∂φ , (cid:17) + (cid:16) ∂β , ⊥ + ∂β , ω + ω∂φ , (cid:17) ω − ψ , . The last three terms in this variation cancel with the last three terms in (77), and the first three terms cancelwith the fourth, fifth, and sixth terms of (77). The fourth term in the variation is left over.It remains to calculate the variation of the fourth and last term of the homotopy, which is(81) Q ( φ , ( ψ + ) , ) = φ , (cid:16) ∂β , ⊥ + ∂β , ω (cid:17) . he first of these terms cancels the ninth and final term of (77), and the last term cancels the leftover piecefrom the variation of the third term of the homotopy (after another integration by parts). The proposition,and thus this portion of the main theorem, is proved. (cid:3) Holomorphic decomposition for the (2,0) theory.
In this section we finish the second part ofTheorem 4.2 concerning the holomorphic twist of the (2 ,
0) tensor multiplet. Again, we fix the data of arank one supercharge Q , this time viewed as an odd element of the super Lie algebra p (2 , .Recall that the R -symmetry group of (2 ,
0) supersymmetry is G R = Sp(2). As in the (1 ,
0) case, thesupercharge Q defines a complex structure L = C ⊂ V = C equipped with the choice of a holomorphichalf-density on L . The twist carries a symmetry by the subgroup group MU(3) ⊂ Spin(6) whose action isdefined by the twisting homomorphism φ : MU(3) det −−−→ U(1) i × −−→ Sp(1) × Sp(1) ′ ⊂ Sp(2) = G R . Here, i : U(1) ֒ → Sp(1) denotes the embedding for which Q has weight +1. Also we use primes as inSp(1) × Sp(1) ′ ⊂ Sp(2) to differentiate between the two abstractly isomorphic groups.Under the twisting homomorphism φ the defining representation R of Sp(2) decomposes as(82) R = det( L ) − ⊕ det( L ) ⊕ R ′ where MU(3) acts trivially on R ′ . The vector representation W of Sp(2) = Spin(5) decomposes under φ as W = C ⊕ (cid:16) det( L ) − ⊕ det( L ) (cid:17) ⊗ R ′ . The regrading datum is specified by the homomorphism α : U(1) ֒ → Sp(1) i × −−→ Sp(1) × Sp(1) ′ ⊂ Sp(2) = G R . Note that this factors through the twisting homomorphism we used in the (1 ,
0) case along the embeddingSp(1) ֒ → Sp(2).In addition to MU(3), the twist enjoys a global symmetry by the group Sp(1) ′ . Moreover, these actionscommute for the trivial reason that MU(3) acts trivially on Sp(1) ′ . Using Equation (63), we observe that,after applying the twisting homomorphism φ , the odd part Σ of the super Lie algebra p (2 , transformsunder MU(3) × Sp(1) ′ ⊂ Spin(6) × Sp(2) as: − L )5 / / L ) ⊗ Π R ′ L / C · Q − / L ⊗ det( L ) − ⊗ Π R ′ − − / − L ⊗ det( L ) − In this table, the vertical grading organizes spin number, and the horizontal grading is by ghost Z -degree.The terms involving R ′ are all odd with respect to the new Z / Q lies in the red summand. Its only nonzero bracket occurs with thesupercharges in L represented in green above, using the degree-zero pairing on the R -symmetry space.As remarked above, this bracket witnesses a nullhomotopy of the translations in L with respect to theholomorphic supercharge.In Proposition 3.7, we described the Sp(1) × Sp(1) ′ decomposition of the (2 ,
0) tensor multiplet as a sumof the (1 ,
0) tensor multiplet plus the (1 ,
0) hypermultiplet valued in the symplectic representation R ′ : T (2 , = T (1 , ⊕ T hyp(1 , ( R ′ )Analogously, accounting for the twisting data φ, α just introduced we have the following description of theregraded (2 ,
0) tensor multiplet.
Proposition 4.7.
The
MU(3) -regraded (2 , tensor multiplet φ ∗ T α (2 , decomposes as φ ∗ T α (2 , = φ ∗ T α (1 , ⊕ Π φ ∗ T α hyp ( R ′ ) where φ ∗ T α (1 , is the regraded (1 , tensor multiplet as in Proposition 4.5 and φ ∗ T α hyp ( R ′ ) is the free BVtheory of the regraded hypermultiplet whose complex of fields is displayed in Figure 4. In Figure 4, the operator ∂ ∗ denotes the adjoint of ∂ corresponding to the standard K¨ahler form on C .Under the regrading T hyp ( R ′ ) = Φ(0 , R ′ ) ⊕ Ψ − ( R ′ ) Π φ ∗ T α hyp ( R ′ ), we will denote the decomposition of ∗ T α hyp ( R ′ ) − , ( K ⊗ R ′ ) Ω , ( K ⊗ R ′ )Ω , ( K ⊗ R ′ ) Ω ( K ⊗ R ′ )Ω ( K ⊗ R ′ ) Ω ( K ⊗ R ′ ) Ω , ( K ⊗ R ′ ) Ω , ( K ⊗ R ′ ) ∂ ∗ ∂∂ ∗ △ △ Figure 4.
The subcomplex φ ∗ T α hyp ( R ′ ) of the MU(3)-regraded (2 ,
0) tensor multiplet, seeProposition 4.7. The top complex is the result of regrading the fermions in the (1 , , , R ′ ) ∋ ν = ν , + ν , ∈ Ω ( K ⊗ R ′ )[1] ⊕ Ω , ( K ⊗ R ′ )[ − − ( R ′ ) ∋ λ = λ , + λ , ∈ Ω , ( K ⊗ R ′ ) ⊕ Ω , ( K ⊗ R ′ )for the fermions. A similar decomposition holds for the anti-fields which will be denoted ν + , , etc.. Proof.
The N = (2 ,
0) multiplet splits as a sum of three complexes T (2 , = χ + (2) ⊕ Ψ − ( R ) ⊕ Φ(0 , W ) . As in the case of the N = (1 ,
0) multiplet, the component χ + (2) is not charged under the R -symmetry group G R = Sp(2).The physical fields of Ψ − ( R ) decompose under the twisting homomorphism φ as:(86) Π (cid:0) Ω ⊗ S − ⊗ R (cid:1) = (cid:18) Π(Ω , ⊕ Ω , ) ⊕ Π(Ω , ⊕ Ω , ) (cid:19) ⊕ Π (cid:18) Ω ⊗ S − ⊗ R ′ (cid:19) . The first component in parentheses contributes to the regraded N = (1 ,
0) tensor as in Proposition 4.5. Thesecond component Ω ⊗ S − ⊗ R ′ = Ω ( K − ⊗ R ′ ) ⊕ Ω , ( K ⊗ R ′ )contributes to the regraded hypermultiplet Π e T hyp ( R ′ ). There is a similar decomposition for the anti-fieldsin Ψ − ( R ). − , /
22 Ω , Ω , / , (Π R ′ ) Ω , (Π R ′ ) Ω , (Π R ′ )1 Ω , Ω , Ω , (Π R ′ )0 Ω Ω , − Ω , (Π R ′ ) − , − / , (Π R ′ ) Ω , (Π R ′ ) Ω , (Π R ′ ) ∂∂ Figure 5.
The holomorphically twisted N = (2 ,
0) theory T Q (2 , . The horizontal grading isthe cohomological Z -grading. Note that the green and red text sits in odd Z / ⊂ MU(3).Next, the physical fields of the scalar theory Φ(0 , W ) decompose as(87) Ω ⊗ W = Ω ⊕ Ω ⊗ (cid:16) K − ⊕ K (cid:17) ⊗ R ′ The first summand, the single copy of smooth functions Ω , contributes to the regraded (1 ,
0) tensor multiplet.The second summand contributes to Π e T hyp ( R ′ ). There is a similar decomposition for the anti-fields inΦ(0 , W ).By Proposition 4.5, upon regrading, we see that the components χ + (2), the first summand of 86, and thefirst summand of (87), combine to give the regraded (1 ,
0) tensor multiplet.Of the remaining terms, the only component which is acted upon nontrivially by Sp(2) is the secondsummand in (87) (and the corresponding antifields). Under α , we see that the factor proportional to K has weight − K has weight +1. It remains to check that the BV differential decomposesas stated, but this is nearly identical to the proof of Proposition 4.5. (cid:3) .5. Proof of (2,0) part of Theorem 4.2.
We now complete the proof of Theorem 4.2, which involvesdeforming the regraded theory described in Proposition 4.7 by the holomorphic supercharge Q . Throughoutthis section we refer to the description of the twisted theory in Figure 5.According to Proposition 4.7, the Q -twisted theory splits as a sum of two complexes(88) T Q (1 , ⊕ T hyp ( R ′ ) Q where T Q (1 , is the Q -twist of the (1 ,
0) tensor multiplet and T hyp ( R ′ ) Q is the theory obtained by deformingthe MU(3)-regraded hypermultiplet Π φ ∗ T α hyp ( R ′ ) by Q .In Figure 5, the black solid arrows represent the twist of the N = (1 ,
0) tensor multiplet, as we computedin § T Q (1 , in (88). The red text refers to the MU(3)-regradedhypermultiplet Π φ ∗ T α hyp ( R ′ ). The red solid arrows represent the underlying classical BV differential of theregraded hypermultiplet. Note that we use the shorthand notation Ω ± ,ℓ ( R ′ ) to mean the Dolbeault formsof type (0 , ℓ ) valued in the holomorphic vector bundle K ± ⊗ R ′ . We have labeled the differentials generatedby the holomorphic supercharge Q acting on the hypermultiplet by the dotted and dashed arrows. As inthe N = (1 ,
0) case, the dotted arrows denote isomorphisms, and the dashed arrowsare given by the labeled differential operator, which we now proceed to characterize. Again, we refer to thenotation in § § ρ Ψ , . In the holomorphic decomposition, see Equation (85), ofthe fields we read off ρ Ψ , ( Q ⊗ λ ) = λ , ∈ Ω , ⊂ Φ(0 , R ′ )[1] , This term accounts for the dotted arrow in Figure 5 labeled 1 . Similarly, on the anti-fieldswe have ρ Ψ , (cid:0) Q ⊗ ν + (cid:1) = ν , ∈ Ω , ⊂ Ψ − ( R ′ ) , see the notation of Equation (84). This term accounts for the dotted arrow labeled 2 .Next, we look at the component of supersymmetry which transforms a scalar into a fermion. In thenotation of § ρ Φ . In the holomorphic decomposition of fields we have ρ Φ ( Q ⊗ ν ) = ∂ν , ∈ Ω , ⊂ Ψ − ( R ′ ) . Similarly, on the anti-fields we have ρ Φ ( Q ⊗ λ + ) = ∂λ , ∈ Ω , ⊂ Φ(0 , R ′ ) . These maps account for each of the dashed arrows in Figure 5. ext, we will describe an equivalence of presymplectic BV theoriesΦ : T Q (2 , → χ (2) ⊕ hCS(Π R ′ )On the (1 ,
0) tensor multiplet summand of the (2 ,
0) theory, the map Φ is defined to be the map (71) thatwe used in the twist of the (1 ,
0) multiplet.On the (1 ,
0) hypermultiplet summand, the map is defined as follows.(89) Φ( ν , ) = ν , ∈ Ω , Φ( λ , ) = λ , ∈ Ω , Φ( λ , + ν , ) = λ , − ∂ ∗ ν + , ∈ Ω , Φ( ν , ) = ν , ∈ Ω , . The map Φ annihilates the remaining fields of the (1 ,
0) hypermultiplet.On the hypermultiplet, we note that this map is not the obvious projection map of graded vector spaces,but is “corrected” to account for the differentials mapping out of the acyclic subcomplex at the upper leftof the hypermultiplet in Figure 5. The correction in this case is analogous to standard twist calculations,and follows the general rubric presented in Proposition 1.23 of [49]. By this result, and the theorem for the(1 ,
0) tensor multiplet, it follows that Φ is a quasi-isomorphism.We have already shown how the map on the (1 ,
0) tensor multiplet is compatible with the degree ( − −
1) presymplectic structures.4.5.1.
An alternative description.
There is an alternative to the twisting data ( φ, α ) in the case of the (2 , × Sp(1) ′ , even after twisting. This alternative twist breaks this global Sp(1) ′ symmetry completely, but further descendsthe MU(3)-action to an action by U(3).The reason this twist enjoys a smaller symmetry group is because it depends on the choice of a polarizationof the 2-dimensional symplectic vector space R ′ . Such a polarization determines an embedding i ′ : U(1) ֒ → Sp(1) ′ which we now fix.Define the new twisting homomorphism by the composition e φ : MU(3) det −−−→ U(1) diag −−−→
U(1) × U(1) i × i ′ −−→ Sp(1) × Sp(1) ′ ⊂ Sp(2) . As in the previous section, i : U(1) → Sp(1) denotes the homomorphism for which Q has weight +1.Additionally, we have the regrading homomorphism e α : U(1) diag −−−→ U(1) × U(1) i × i ′ −−→ Sp(1) × Sp(1) ′ ⊂ Sp(2) . − , Ω , Ω , , , , Ω , Ω , Ω , Ω , Ω , − , − , − , Ω , Ω , ∂∂∂∂ Figure 6.
The description of the subcomplex A ⊂ e T Q (2 , using the alternative twisting data.To simplify the notation in the next section, we will denote by e T (2 , the MU(3)-regraded (2 ,
0) theoryusing this twisting data. The Q -twisted theory will be denoted by e T Q (2 , .With this choice of a regrading homomorphism, the twisted theory e T Q (2 , descends to a U(3)-equivarianttheory and is concentrated in even Z / Z -graded theory. Aside from this, theonly part of the calculation that changes is the subcomplex defined by the green and red text of Figure 5,which we will henceforth denote by A ⊂ e T Q (2 , .For example, in the original description of the twist the scalar field lives in ΠΩ , ( R ′ )[1]. According tothis new twisting data this becomesΠΩ , ( R ′ )[1] ⊕ ΠΩ , ( R ′ )[ − (cid:0) Ω , ⊕ Ω , [2] (cid:1) ⊕ (cid:0) Ω , [ − ⊕ Ω , [1] (cid:1) . Similarly, using the original twisting data, the fermion field lives in ΠΩ , ⊕ ΠΩ , . According to this newtwisting data this becomesΠΩ , ⊕ ΠΩ , (cid:0) Ω , [ − ⊕ Ω , [1] (cid:1) ⊕ (cid:0) Ω , [ − ⊕ Ω , [1] (cid:1) . In total, using this alternative twisting data, the green and red subcomplex of the diagram in Figure5, which we denote A , is displayed in Figure 4.5.1. As before the solid arrows denote the differentials inthe original untwisted theory. The dotted arrows denote isomorphisms and the dashed arrowsarrows are given by the labeled differential operators induced by the action by Q . The greentext labels the components arising from the scalar part of the untwisted theory, the red text labels thecomponents arising from the fermion. e recognize that the complex of Figure 4.5.1 admits a cochain map to the βγ system on C . Sincethe dotted arrows are isomorphisms, this cochain map is a quasi-isomorphism. The following propositionfollows from tracing through the presymplectic BV structures, which is completely similar to the previouscalculations. Proposition 4.8.
There is an equivalence of presymplectic BV theories
Φ : e T Q (2 , ≃ −→ χ (2) ⊕ βγ ( C ) . Moreover, this equivalence is
U(3) -equivariant.
The map Φ is defined nearly identically to the quasi-isomorphism defined in the previous section for thetwisting data ( φ, α ). The only difference is that one must decompose (and twist) the formula for Φ actingon the hypermultiplet as in Equation (89).4.6.
The twisted factorization algebras. In § Q , each of the twistedpresymplectic BV theories T Q (1 , , T Q (2 , and e T Q (2 , satisfy Condition (2) in § P -factorization algebra of Hamiltonian observables.The twist of the (1 ,
0) theory T Q (1 , is defined on any complex three-fold X . We denote the correspondingfactorization algebra of observables on X by Obs (1 , , with the supercharge Q understood. We can describethis P -factorization algebra explicitly as follows. Recall T Q (1 , ≃ χ (2) which, as a cochain complex, isΩ ≤ , • [2] equipped with the differential ∂ + ∂ . Keeping track of shifts, one has χ (2) ! = Ω ≥ , • [2], againequipped with the differential ∂ + ∂ . Thus, the factorization algebra is described byObs (1 , = (cid:0) O sm (Ω ≥ , • [1]) , ∂ + ∂ (cid:1) where O sm denotes the “smooth” functionals as defined in § U ⊂ X , thefactorization algebra assigns the cochain complexObs (1 , ( U ) = (cid:18) Sym (cid:0) Ω ≤ , • c ( U )[3] (cid:1) , ∂ + ∂ (cid:19) . With this description in hand, the P -structure is also easy to interpret. Given two linear observables O , O ∈ Ω ≤ , • c ( U )[3], the P -bracket is(90) { O , O ′ } = Z U O ∂ O ′ . The bracket extends to non-linear observables by the graded Leibniz rule. In [36] this P -factorization algebrahas appeared as the factorization algebra of boundary observables of abelian 7-dimensional Chern–Simonstheory. For more discussion on the relationship to 7-dimensional Chern–Simons theory and topologicalM-theory we refer to § e will not explicitly need to mention the factorization algebra associated to the twist of the (2 ,
0) theory T Q (2 , . However, we will study the factorization algebra associated to its alternative twist e T Q (2 , , which wewill denote by Obs (2 , . Again, this theory exists on any complex three-fold X . Similarly to the (1 ,
0) case,we obtain the following explicit description of this factorization algebra. To an open set U ⊂ X , it assignsthe cochain complexObs (2 , ( U ) = (cid:18) Sym (cid:0) Ω ≤ , • c ( U )[3] ⊕ Ω , • c ( U )[3] ⊕ Ω , • c ( U )[1] (cid:1) , ∂ + ∂ (cid:19) . The P -bracket on linear observables is again straightforward. The first linear factor is the same as in the(1 ,
0) case. The second two linear factors are the linear observables of the βγ system on C . For linearobservables in Ω ≤ , • ( U )[3] it is given by the same formula as in (90). The only other nonzero bracketbetween linear observables occurs between elements O ∈ Ω , • c ( U )[3] and O ′ ∈ Ω , • c ( U )[1] where it is given by { O , O ′ } = Z U OO ′ . The non-minimal twist
We have classified in § ,
0) supersymmetry algebra. Wefound that they were characterized by the rank of the supercharge, which for a non-trivial square-zeroelement could be either one or two. The minimal, rank one, case was studied in the last section. We nowturn to the further, non-minimal, twist of the (2 ,
0) theory.Upon applying a twisting homomorphism more natural to the non-minimal twisting supercharge, thenon-minimal twist exists on manifolds of the form M × Σ where M is a smooth four-manifold and Σ isa Riemann surface. Since the non-minimal supercharge leaves five directions invariant, this theory dependstopologically on M and holomorphically on Σ.Our main result is the following; see Theorems 5.3 and 5.8 for more careful statements. Theorem 5.1.
The non-minimal twist of the abelian (2 , tensor multiplet on R × C is equivalent, as apresymplectic BV theory, to the theory whose complex of fields is Ω • ( R ) b ⊗ Ω , • ( C )[2] . The ( − -shifted presymplectic structure is (91) ( α, α ′ ) Z α∂ C α ′ . Here, ∂ C denotes the holomorphic de Rham operator on C . The non-minimal deformation.
Before computing the twist, it is instructive to get a handle on theexplicit data involved in choosing a non-minimal twisting supercharge. As a Spin(6) × Sp(2)-module, theodd part of the supertranslation algebra p (2 , is Σ ∼ = Π S + ⊗ R . It is thus easy to compute the stabilizer f a chosen rank-one supercharge, which is the product of the respective stabilizers of fixed vectors in S + and R separately. This is the subgroup MU(3) × Sp(1) ′ × U(1) ⊂ Spin(6) × Sp(2). As representations ofthe stabilizer, S + and R decompose as(92) S + = det( L ) ⊕ L ⊗ det( L ) − , R = C − ⊕ ( R ′ ) ⊕ C +1 . Here, the superscripts C ± denote the charges under U(1).We can thus consider the following diagram representing the decomposition of Σ as a MU(3) × U(1) ⊂ Spin(6) × Sp(2) representation:(93) det( L ) ⊗ C − det( L ) ⊗ ( R ′ ) det( L ) ⊗ C +1 L ⊗ det( L ) − ⊗ C − L ⊗ det( L ) − ⊗ ( R ′ ) L ⊗ det( L ) − ⊗ C +1 The holomorphic supercharge is indicated in red (note that we have not yet applied any twisting homomor-phism). Its only nonzero bracket occurs with the supercharges in L ⊗ det( L ) − ⊗ C +1 , represented in greenabove, using the degree-zero pairing on the R -symmetry space. As remarked above, this bracket witnesses anullhomotopy of the translations in L with respect to the holomorphic supercharge. The other bracket mapof interest to us pairs the supercharges represented in blue with themselves, via the map(94) ( L ⊗ det( L ) − ⊗ ( R ′ ) ) ⊗ → ∧ L ⊗ det( L ) − ⊗ ∧ R ′ ∼ = L ∨ . Remark . This equivariant decomposition makes clear the structure of the tangent space to the nilpo-tence variety at a holomorphic supercharge. The dimension of the normal bundle is 3, represented by thecomponent colored green above; all other supercharges anticommute with Q , and therefore define first-orderdeformations, which are tangent vectors to the nilpotence variety. The dimension of the tangent space ata holomorphic supercharge is thus 12, although the projective variety is in fact only 10-dimensional. Thefibers of the tangent bundle are “too large” because the holomorphic locus is in fact the singular locus ofthe variety. In fact, as remarked above, the singular locus (or space of holomorphic supercharges) is a copyof P × P , consisting of four-by-four matrices of rank one; its tangent space is spanned by the black entriesin the diagram (83). The red entry is Q itself, representing the tangent direction along the affine cone of theprojective variety.The deformations represented by the blue elements are of interest here; they generate the non-minimaltwist (and therefore represent deforming away from the holomorphic locus of the nilpotence variety, intothe locus of nonminimal supercharges). However, not all such infinitesimal deformations give rise to finitedeformations of Q ; geometrically, this corresponds to the fact that the nilpotence variety is singular, andnot all vectors in the algebraic tangent space correspond to paths in the variety. Since the nilpotenceconditions are quadratic, though, this can be checked at order two: for a deforming supercharge Q ′ ∈ ⊗ det( L ) − ⊗ ( R ′ ) , we just need the condition that[ Q ′ , Q ′ ] = 0inside p (2 , . Examining the bracket map discussed above shows immediately that the deforming superchargeswith zero self-bracket are precisely the rank-one elements:(95) Q ′ = α ⊗ w : α ∈ L ⊗ det( L ) − , w ∈ ( R ′ ) . The data of Q ′ has an especially nice interpretation through the lens of the holomorphic twist. Werecall the alternative twisting homomorphism e φ of the (2 ,
0) theory from § ′ symmetry by fixing a polarization of R ′ . Further, upon twisting by e φ therelevant component of the spinor representation decomposes under MU(3) as(96) L ⊗ det( L ) − ⊗ ( R ′ ) L ⊗ det( L ) − ⊕ L. Without loss of generality we can assume Q ′ lies in the first factor. Thus, from the perspective of theholomorphic twist, the datum of a further nonminimal twist therefore consists precisely of a polarization ofthe symplectic vector space R ′ , together with a nonzero translation invariant section of ∧ T , C , where T , C is the holomorphic tangent bundle. We choose holomorphic coordinates ( w , w , z ) on C and identify,without loss of generality, the supercharge Q ′ with the translation invariant bivector Q ′ = ∂ w ∧ ∂ w . The non-minimal twisting supercharge is of the form Q := Q + Q ′ where Q is the minimal supercharge lying in the red component of (93) and Q ′ is a rank one superchargelying in the blue component of (93).Most of the remainder of this section is devoted to the proof of the following description of the non-minimaltwist. Theorem 5.3.
Using the twisting data ( e φ, e α ) , the Q -twist of the (2 , tensor multiplet T Q (2 , is equivalentto the free presymplectic BV theory whose complex of fields is (97) T = (cid:18) Ω ≤ , • ( C ) / (d z )[2] Ω , • ( C ) (cid:19) . Π ◦ ∂ where Π is the translation invariant bivector ∂ w ∧ ∂ w . − , Ω , Ω , Ω , Ω , Ω , Ω , Ω , Ω , Ω , Ω , Ω , Ω , Ω , Ω , Ω h Π , ·i ∂ C h Π , ·i ∂ C h Π , ·i ∂ C h Π , ·i ∂ C Figure 7.
The solid arrows represent the holomorphic twist of the N = (2 ,
0) theory. Thedashed and dotted arrows represent the action by the supercharge Q ′ which deforms theminimal twist to the non-minimal twist.In the statement of the theorem, the ( − T is( α, α ′ ) Z α∂ C z α ′ . Here, ∂ C denote the holomorphic de Rham operator on C . Remark . In the description of the non-minimal twist in (97), we have used the Calabi–Yau form d w d w d z on C and the fact that h Π , d w d w d z i = d z . On a general, not necessarily Calabi–Yau, three-fold X equipped with a bivector Π ∈ PV ,hol ( X ) we can write this description more invariantly asΩ ≤ , • ( X ) / (Im(Π)) [2] Ω , • ( X ) Π ◦ ∂ Here, Im(Π) ⊂ T ∗ X is the image of the bundle map Π : ∧ T ∗ X → T ∗ X .In (96), we could have alternatively chosen the deformation to a non-minimal supercharge Q ′ to be thedata of a holomorphic one-form. On C , these lead to equivalent non-minimal twists. Globally, however, the ata of a non-vanishing holomorphic one-form η ∈ Ω ,hol ( X ) leads to the following description of the twist:Ω ≤ , • ( X ) / ( η ) [2] Ω , • ( X ) . η ∧ ∂ Here, Ω ≤ , • ( X ) / ( η ) denotes the quotient of Ω ≤ , • ( X ) by the subspace η Ω , • ( X ).We pose the question as to what extent these global descriptions of the non-minimal twist of the (2 , Symmetries of the holomorphic twist.
The first step in the proof of Theorem 5.3 is to exhibit theaction of the non-minimal deformation Q ′ on the description of the minimal, holomorphic, twist of the (2 , βγ system.At the level of the twisted theory we break the symmetry by the (2 ,
0) super Poincar´e algebra p (2 , to its Q -cohomology. Upon regrading and applying the twisting data of § Q -cohomology of p (2 , , thisgives us an action of a Z -graded Lie algebra p Q (2 , on the holomorphic twist of the (2 ,
0) theory.Let g ⊂ p (2 , be the Z / Q , we obtain a subalgebra g Q ⊂ p Q (2 , .If L is a complex three-dimensional vector space spanned by the symbols ∂ z , ∂ w , ∂ w , then as a Z -gradedvector space we have g Q = L ∗ [1] ⊕ L ⊕ ∧ L [ − . whose elements we will denote by ( η, v, π ). This space has a Z -graded Lie algebra structure whose bracketis defined by the U( L )-invariant pairing of L ∗ with ∧ L as in [ η, π ] = h η, π i ∈ L .In the notation of §
4, the minimal Q -twist of the (2 ,
0) theory, using the twisting data ( e φ, e α ), was denoted e T Q (2 , . In Proposition 4.8 we have shown that e T Q (2 , is equivalent to the presymplectic BV theory χ (2) ⊕ βγ ( C )through an explicit quasi-isomorphism Φ : e T Q (2 , → χ (2) ⊕ βγ ( C ) . As an element of p (2 , that commutes with Q , the deformation Q ′ acts on e T Q (2 , . Naively, one couldtransfer the action of Q ′ along Φ. However, since Φ does not strictly preserve the presymplectic form, whatresults is an action of Q ′ on χ (2) ⊕ βγ ( C ) that does not preserve the shifted presymplectic pairing.By pulling back along Φ, the ( − ω χ + ω βγ on χ (2) ⊕ βγ ( C ) defines a ( − e T Q (2 , . Since we know the two shifted presymplectic structures are equivalent,we know abstractly that given any symmetry of the (2 ,
0) theory that is compatible with the holomorphicsupercharge and the original presymplectic structure ω T , that we can find a homotopy equivalent symmetrythat preserves Φ ∗ ( ω χ + ω βγ ).For symmetries arising from the sub Lie algebra g ⊂ p (2 , , we have the following explicit result. emma 5.5. Let X ∈ g ⊂ p (2 , and denote by ρ Q ( X ) the associated endomorphism of e T Q (2 , of Z -degree | X | . There exists a degree- ( | X | − endomorphism H X of e T Q (2 , such that (98) Φ ( ρ Q ( X ) + [ Q BV + Q, H X ]) strictly preserves the ( − -presymplectic form ω χ + ω βγ . We denote by e ρ Q ( X ) the endomorphism Φ( ρ Q ( X ) + [ Q BV + Q, H X ]) of χ (2) ⊕ βγ ( C ). Proof. If X is a holomorphic translation we can simply take e ρ Q ( X ) = ρ Q ( X ) and H X = 0.Suppose that X ∈ p (2 , becomes an element π ∈ ∧ L ⊂ g Q under the twisting homomorphism e φ . Forclarity, we will denote the holomorphic decomposition of the bosons and fermions in the (1 ,
0) hypermultipletby ν • , • and λ • , • respectively. We denote by β • , • , φ • , • , ψ • , •− elements of the holomorphically decomposed (1 , H π ( β , ) = h π, β , i ∈ Ω , ν . One can compute that the resulting endomorphism Φ ( ρ Q ( π ) + [ Q BV + Q, H π ]) is given by the dotted anddashed arrows of Figure 7. This is readily seen to preserve the shifted presymplectic structure.For an element X ∈ g which becomes an element η ∈ L ∗ ⊂ g Q under the twisting homomorphism e φ thedefinition of the homotopy is similar. (cid:3) To comment briefly on this calculation, we refer again to the dotted and dashed arrows in Figure 7. Whilemost of these originate in standard supersymmetry transformations of the untwisted theory, three are subtle:the leftmost dotted arrow, which carries a physical scalar field to a one-form ghost of the two-form field; theleftmost dashed arrow, which carries a one-form ghost to a physical scalar field via a differential operator;and the dashed arrow third from left, a component of which carries a physical fermi field to a fermi antifield,but is not part of the original BV differential.These three mysterious arrows have three different origins. The first is generated by the L ∞ closure term ρ (2)Φ . The others, though, are of (untwisted) BV degree +1, and so cannot originate from any subtleties of themodule structure. The third term is, in fact, generated by the projection map Φ in (89), used in computingthe twist of the (1 ,
0) hypermultiplet; the second term is generated by the homotopy H π discussed above,which replaces the fermi field ψ , —whose antifield is not eliminated after the holomorphic twist—by its“new” antifield ∂β , .The endomorphisms ρ Q ( X ) and ρ Q ( X )+[ Q BV + Q, H X ] are homotopy equivalent. Thus, as a consequenceof this proposition, we obtain an equivalent action of g Q on e T Q (2 , which is presymplectic upon applying thequasi-isomorphism Φ. This action is described explicitly in Proposition 5.6 below, which we take a briefmoment to foreground. s a graded vector space, χ (2) ⊕ βγ ( C ) decomposes as( c , A , β, γ ) ∈ Ω , • [2] ⊕ Ω , • [1] ⊕ Ω , • [2] ⊕ Ω , • . The first two components comprise the theory χ (2) and the second two comprise the βγ system.Recall, there is the internal ∂ differential and also the differential c ∂ c = A where ∂ is the holomorphicde Rham operator on C . As in the case of the full supersymmetry algebra, the action of g Q on e T Q (2 , through e ρ Q is an action only up to homotopy. We decompose e ρ Q = e ρ (1) Q + e ρ (2) Q where e ρ (1) Q is linear and e ρ (2) Q is quadratic in g Q . Unpacking the action of supersymmetry given in § g Q at the level of the holomorphic twist. Proposition 5.6.
The action e ρ Q = e ρ (1) Q + e ρ (2) Q of g Q on χ (2) ⊕ βγ ( C ) is given by:(1) the linear term e ρ (1) Q is defined on holomorphic translations v ∈ L by e ρ (1) Q ( v ) α = L v α , where α is anyfield. On the remaining part of the algebra, e ρ (1) Q is e ρ (1) Q ( η ) A = η ∧ ∂ A ∈ Ω , • β , e ρ (1) Q ( π ) A = h π, ∂ A i ∈ Ω , • γ e ρ (1) Q ( η ) γ = η ∧ γ ∈ Ω , • A , e ρ (1) Q ( π ) β = h π, β i ∈ Ω , • A . Whenever it appears, the symbol h· , ·i refers to the obvious U( L ) -invariant pairing.(2) The quadratic term e ρ (2) is given by e ρ (2) Q ( η ⊗ π ) A = ι h η,π i A ∈ Ω , • c . Conceptually, as remarked above, the key step in proving this proposition is to observe that the homotopy H π described in Lemma 5.5 generates the transformation e ρ (1) ( π ) A = h π, ∂ A i via (98). Remark . It is instructive to verify directly that the action described in the above proposition is an L ∞ -action. To see this, the key relation to observe is η · ( π · A ) − π · ( η · A ) = ι h η,π i ∂ A . for any η, π, A where the · denotes the linear action e ρ (1) Q .We can now give a proof of the main result of this section. Proof of Theorem 5.3.
The shifted presymplectic action of X ∈ g Q at the level of the holomorphic twist χ (2) ⊕ βγ ( C ) is given by e ρ Q ( X ).The non-minimal deformation Q ′ determines a nontrivial element in ∧ L [ − ⊂ g Q that we identify with ∂ w ∧ ∂ w . Schematically, the action e ρ Q ( Q ′ ) is given by the dotted and dashed arrows in Figure 7. Let e Q BV denote the solid arrows in this figure, which describes the linear BV differential of the holomorphic twist. e observe that each of the dotted arrows is of the form h Π , −i : Ω , • ( C ) → Ω , • ( C ) . If we decompose Ω , • ( C ) as Ω , • ( C w ) b ⊗ Ω , • ( C z ) ⊕ Ω , • ( C ) b ⊗ Ω , • ( C z ) we see that this map is an isomorphismonto the second component.So, we see that there is a projection from the total complex (cid:16)e T Q (2 , , e Q BV + Q ′ (cid:17) to T whose kernel isacyclic. (cid:3) A refined twisting homomorphism.
We proceed to describe the twisting data that is somewhatmore natural to the non-minimal twist.Consider the twisting homomorphism is defined by the composition φ top : U(2) × U(1) → U(3) × U(3) det × det −−−−−−−−→ U(1) × U(1) ( i, ( i ′ ) − ) −−−−−−→ Sp(1) × Sp(1) ′ ֒ → Sp(2) . The first map is the block diagonal embedding of (
A, x ) ∈ U(2) × U(1) into U(3) via A x in the firstfactor and via A x − into the second factor. Also, i : U(1) → Sp(1) is the unique homomorphism forwhich Q has weight +1 and i ′ : U(1) → Sp(1) is defined by the polarization determined by Q ′ .Additionally, we have the regrading homomorphism α top : U(1) diag −−−→ U(1) × U(1) i × i ′ −−→ Sp(1) × Sp(1) ′ ⊂ Sp(2) . Note that this is identical to the regrading homomorphism e α of § φ top , α top constitute twisting data for the the nonminimal twistingsupercharge Q = Q + Q ′ .The affect of φ top is to “twist” the graded vector space underlying T of Theorem 5.3 to(99) φ ∗ T = (cid:0) Ω , • ( C )[2] ⊕ Ω , • ( C )[1] ⊕ Ω , • ( C ) (cid:1) ⊗ Ω , • ( C z ) . The differential then becomes ∂ C + ∂ C + ∂ C z .Moreover, φ top satisfies the following property. First, note that there is a natural embedding j : U(2) ֒ → SO(4). Unwinding the definition of φ top above, one finds that it can be factored asU(2) × U(1) Sp(2)SO(4) × U(1) j × φ top φ SO here φ SO is the following composition of maps:SO(4) × U(1) ∼ = −→ SU(2) × SU(2) × U(1) pr , −−−→ Spin(3) × Spin(2) → Spin(5) ∼ = −→ Sp(2) . The pair ( φ SO , α ) constitutes yet another set of twisting data for the non-minimal supercharge Q = Q + Q ′ .(To see that the embedding along a short root of sp (2) is the correct one, recall that Q is defined by thewedge pairing of a Lagrangian subspace in R with a two-dimensional subspace in S + ; the stabilizer of aLagrangian subspace in sp ( n ) is gl ( n ), embedded along the short roots via the diagram inclusion A n − ֒ → C n .The twisting homomorphism φ SO allows one to identify (99) with Ω • ( R ) ⊗ Ω , • ( C z )[2]. As a consequence,we have the following. Proposition 5.8.
Using the twisting data ( φ SO , α SO ) , the Q -twist of the (2 , tensor multiplet T Q (2 , isequivalent to the free presymplectic BV theory whose BV complex of fields is Ω • ( R ) b ⊗ Ω , • ( C )[2] and whose ( − -shifted presymplectic structure is given in (91). The equivalence is SO(4) × U(1) -equivariant.
This description of the nonminimal twist makes sense on a manifold of the form M × Σ where M is asmooth four-manifold and Σ is a Riemann surface.6. Comparison to Kodaira–Spencer gravity
In this section we document a relationship between the twist of the tensor multiplet and a holomorphictheory defined on Calabi–Yau manifolds that has roots in string theory and theories of supergravity. Thistheory, which we will refer to as Kodaira–Spencer theory, is gravitational in the sense that it describes vari-ations of the Calabi–Yau structure; it was first introduced in [56] as the closed string field theory describingthe B -twisted topological string on three-folds. Work of Costello–Li [57]–[59] has began to systematicallyexhibit the relationship of Kodaira–Spencer theory on more general manifolds to twists of other classes ofstring theories and theories of supergravity.The main result of this section (Proposition 6.1) can be stated heuristically as follows: up to topologicaldegrees of freedom, the theory of the field strengths of the holomorphic twist of the abelian (2 , tensormultiplet on a Calabi–Yau three-fold is equivalent to the free limit of (minimal) Kodaira–Spencer theory .There is a similar statement for the (1 ,
0) tensor multiplet and a Type I Kodaira–Spencer theory. Thisresolves a simple form of a conjecture in Costello–Li in [57]. This can also be seen as an enhancement of aresult of Mari˜no, Minasian, Moore, and Strominger [60], where it is shown that the equations of motion ofthe M5 brane theory on a Calabi–Yau three-fold include the Kodaira–Spencer equations of motion.We consider Kodaira–Spencer theory on a Calabi–Yau three-fold X , and we denote by Ω the nowherevanishing holomorphic volume form. Denote by PV i,j ( X ) = Γ( X, ∧ i T X ⊗∧ j T ) the j th term in the Dolbeault esolution of polyvector fields of type i . The fields of Kodaira–Spencer theory are T KS def = PV • , • ( X )[[ t ]][2] . Here t denotes a formal parameter of degree +2. The gradings are such that the degree of the component t k PV i,j is i + j + 2 k −
2. The complex of fields carries the differential Q KS = ∂ + t∂ Ω where ∂ Ω fits into the diagram PV i,j ( X ) PV i − ,j ( X )Ω − i,j ( X ) Ω − i, • ( X ) . ∂ Ω Ω ∼ = Ω ∼ = ∂ for i ≥
1. Note that ∂ Ω is an operator of degree − T KS , so that ∂ + t∂ Ω is an operator of homogenousdegree +1. The fields of Kodaira–Spencer theory are not the sections of a finite rank vector bundle, but wewill pick out certain subspaces of fields which are the sections of a finite rank bundle.Kodaira–Spencer theory fits into the BV formalism as a (degenerate) Poisson BV theory [57]. For a precisedefinition of a Poisson BV theory see [61]. The degree +1 Poisson bivector Π KS on T KS which endows T KS with a Poisson BV structure is defined byΠ KS = ( ∂ ⊗ δ ∆ ∈ T KS ( X ) b ⊗ T KS ( X ) . Here, δ ∆ is the Dirac delta-function on the diagonal in X × X .Any Poisson BV theory defines a P -factorization algebra of observables [61]. For the free limit of Kodaira–Spencer theory this P -factorization algebra is completely explicit. To an open set U ⊂ X one assigns thecochain complex: Obs KS ( U ) = (cid:18) O sm ( T KS ( U )) , Q KS (cid:19) = (cid:18) Sym (cid:0) T !KS ,c ( U ) (cid:1) , Q KS (cid:19) . The BV bracket is defined via contraction with Π KS . We denote the resulting P -factorization algebra forKodaira–Spencer theory by Obs KS .There are variations of the theory obtained by looking at certain subcomplexes of T KS and by restrictingthe P -bivector. They are called: minimal Kodaira–Spencer theory, denoted by e T KS ; Type I
Kodaira–Spencer theory, denoted T Type I ; and minimal Type I
Kodaira–Spencer theory, denoted e T TypeI . They fit into he following diagram of embeddings of complexes of fields: e T KS e T Type I T KS T Type I
The corresponding P factorization algebras of classical observables will be denoted g Obs KS , Obs Type I , and g Obs
Type I (whose definitions we recall below).The goal of this section is relate Kodaira–Spencer theory to the twists of the (1 ,
0) and (2 ,
0) superconfor-mal theories using factorization algebras. Recall that in § presymplectic BV theories, suchas the chiral 2 k -form χ (2 k ), admit a P -factorization algebra consisting of the “Hamitlonian” observables.We have provided a detailed description of the factorization algebras associated to the holomorphic twistsof the (1 ,
0) and (2 ,
0) theories in § Proposition 6.1.
Let X be a Calabi–Yau three-fold and Q be a holomorphic supercharge. The followingstatements are true regarding the holomorphic twists T Q (2 , and T Q (1 , of the N = (2 , and N = (1 , tensormultiplets, respectively:(1) There is a sequence of morphisms of complexes of fields: (100) βγ ( C ) ⊕ Ω ≥ , • [1] T Q (2 , e T KS . fF which induces a morphism of P -factorization algebras on X : g Obs KS → Obs (2 , whose fiber is a locally constant factorization algebra.(2) There is a sequence of morphisms of complexes of fields: (101) Ω ≥ , • [1] T Q (1 , e T Type I . f ∼ = F which induces a quasi-isomorphism of P -factorization algebras on X : g Obs
Type I ≃ −→ Obs (1 , hese result may be summarized as follows. For the (1 ,
0) theory, one finds that the factorization algebraof Hamiltonian observables of the presymplectic BV theory T Q (1 , is equivalent to the free limit of theobservables of Type I Kodaira–Spencer theory. For the (2 ,
0) theory, the observables of the presymplecticBV theory T Q (2 , differ from the free limit of the observables of minimal Kodaira–Spencer theory by a locallyconstant factorization algebra. This locally constant part has been appeared in [62] in what they refer to asKodaira–Spencer theory with potentials .The connection between Kodaira–Spencer theory and the tensor multiplet is through the field strength.Indeed, the map labeled F in the above statement is a holomorphic version of the field strength of the chiraltwo-form. In the sections below, it is defined as the obvious extension of the following map of Dolbeaultcomplexes ∂ : Ω ≤ , • [2] → Ω ≥ , • [1]given by the holomorphic de Rham operator. By our results in the previous sections, given a two-formelement χ ∈ Ω ≤ , • ( X ), the component of the three-form field strength ∂χ ∈ Ω ≥ , • is the only piece thatsurvives the holomorphic twist.Finally, we remark that the P -factorization algebra g Obs
Type I ≃ Obs (1 , has appeared as the factorizationalgebra of boundary observables of 7-dimensional abelian Chern–Simons theory. Likewise, minimal Kodaira–Spencer theory e T KS also appears as a boundary condition of 7-dimensional abelian Chern–Simons theory.6.1. Minimal theory.
Many of the fields in the complex T KS are invisible to the shifted Poisson structurewe have just introduced. There is a piece of T KS that “sees” the Poisson bracket, called the minimaltheory. The fields of the minimal theory form the subcomplex of fields of full Kodaira–Spencer theory e T BCOV ⊂ PV • , • ( X )[[ t ]][2] defined by e T KS def = M i + k ≤ t k PV i, • [ − i − k + 2] . The shifted Poisson tensor Π KS restricts to one on this subcomplex, thus defining another Poisson BV theorywhose fields are e T KS .6.1.1. Proof of part (1) of Proposition 6.1.
This is a direct calculation. Observe that the minimal fieldsdecompose into six graded summands: e T KS = PV , • [2] ⊕ PV , • [1] ⊕ t PV , • ⊕ PV , • ⊕ t PV , • [ − ⊕ t PV , • [ − . nd the differential takes the form:(102) − − , • PV , • t PV , • PV , • t PV , • t PV , • t∂ Ω t∂ Ω t∂ Ω Using the Calabi–Yau form Ω we can identify each line above with some complex of differential forms. Forthe first line, we have PV , • Ω ∼ = Ω , • . The second line is isomorphic to the cochain complex Ω ≥ , • [1], where ∂ Ω is identified with the holomorphic de Rham operator. This is the standard resolution of closed two-formsup to a shift. Similarly, the third line is isomorphic to Ω ≥ , • . This is the standard resolution for closedone-forms.In total, the cochain complex of minimal Kodaira–Spencer theory T KS is isomorphic toΩ , • [2] ⊕ Ω ≥ , • [1] ⊕ Ω ≥ , • . We define the morphism f in the first diagram (100) of Proposition 6.1. Recall, the cochain complex offields of the βγ system with values in C is Ω , • ⊕ Ω , • [2] . On the components Ω , • [2] and Ω ≥ , • [1], we take f to be the identity morphism. On the component Ω , • we take f to be the holomorphic de Rham operator ∂ : Ω , • → Ω ≥ , • . Using the description of the holomorphic twist in § N = (2 ,
0) theory with T Q (2 , ∼ = χ (2) ⊕ βγ ( C ). The morphism F is defined to be the identity on the βγ ( C )component. On χ (2) = Ω ≤ , • [2], F is defined by the holomorphic de Rham operator ∂ : Ω ≤ , • [2] → Ω ≥ , • [1] . To finish the proof, we introduce an intermediate factorization algebra that we think of as the observablesassociated to the Poisson BV theory βγ ( C ) ⊕ Ω ≥ , • [1]. Let F be the factorization algebra which assigns to U ⊂ X the cochain complex F ( U ) = (cid:18) Sym (cid:0) βγ ! c ( U ) ⊕ Ω ≤ , • c ( U )[3] (cid:1) , ∂ βγ + ∂ + ∂ (cid:19) . he maps f, g induce maps of factorization algebrasObs KS f ∗ −→ F F ∗ −−→ Obs (2 , Following the description of Obs (2 , given in § F ∗ is a quasi–isomorphism. Theresult follows from the fact that the kernel of f is the sheaf of constant functions C .6.2. Type I theory.
Type I Kodaira–Spencer theory has underlying complex of fields T Type I = M i + k = odd t k PV i, • [ − i − k − . This describes the conjectural spacetime string field theory of the Type I topological string, see [59].The complex of fields of minimal Type I Kodaira–Spencer theory e T Type I is the intersection of the fieldsof the minimal theory with the Type I theory. The only polyvector fields that appear are of arity zero andone, so that: e T Type I = PV , • [1] ⊕ t PV , • As before, the differential is the internal ∂ operator plus the operator t∂ Ω which maps the first componentto the second. Notice that e T Type I ⊂ e T KS as the middle line in diagram (102).The proof of part (2) of Proposition 6.1 is more direct than the last section. We have already explainedhow to identify e T Type I with the resolution of closed two-forms Ω ≥ , • [1]. This is the isomorphism f in diagram(101).The morphism F in diagram (101) is the holomorphic de Rham operator ∂ . The same argument as in thelast section shows that F ◦ f defines the desired quasi-isomorphism( F ◦ f ) ∗ : g Obs
Type I ≃ −→ Obs (1 , . Dimensional reduction
In this final section, we place the N = (2 ,
0) theory on various geometries, including both naive dimensionalreduction and compactification on product manifolds. We begin with the twisted theories, showing that theholomorphic twist reduces to the twist of five-dimensional supersymmetric Yang–Mills theory up to a copyof the constant sheaf. We then go on to give a description of the holomorphic twist after compactificationalong a complex surface, as well as the two-dimensional theory obtained by placing the nonminimal twiston a smooth four-manifold. Finally, we consider dimensional reduction to five dimensions at the level ofthe untwisted theory, and show that this produces untwisted five-dimensional Yang–Mills as expected, up tothe same copy of the constant sheaf. The calculation leads us into some considerations related to electric–magnetic duality, through which we argue that the presence of the constant sheaf is reasonable and in factexpected from a physics perspective. In the final portion, we offer some more speculative thoughts on howthe zero modes can be correctly handled by passing to a nonperturbative description of the theory. .1. Compactifications of the twisted theories.
Reduction to twisted four-dimensional Yang–Mills.
In this section, let E be an elliptic curve and Y acomplex surface. We consider the holomorphic twists of the (1 ,
0) and (2 ,
0) theories on the complex three-fold Y × E . Recall, in § ,
0) theory Obs (1 , and of the holomorphic twist (using the alternative twisting homomorphism of § ,
0) theory Obs (2 , . We look at the dimensional reduction of these factorization algebras along theelliptic curve E , meaning we consider the pushforward along the projection map Y × E → Y .Upon reduction along E , we find a relationship of the factorization algebras Obs (1 , and Obs (2 , to thefactorization algebras associated to the holomorphic twists of pure 4d N = 2 and N = 4 Yang–Mills theoryfor the abelian one-dimensional Lie algebra.Following [49], [55], we recall the description of the holomorphic twist of supersymmetric Yang–Mills infour-dimensions. Each of these holomorphic twists exists on any complex surface Y .For the case of 4 d N = 2, the holomorphic twist is described by the underlying complex of fields(103) Ω , • ( Y )[ ε ][1] ⊕ Ω , • ( Y )[ ε ][1]where ε is a formal parameter of degree +1. This theory is free and is equipped with the linear BRSToperator given by the ∂ -operator. The degree ( −
1) pairing on the space of fields is given by the integrationpairing along Y together with the Berezin integral in the odd ε direction. That is, given fields A + εA ′ and B + εB ′ where A, A ′ ∈ Ω , • ( Y ), B, B ′ ∈ Ω , • ( Y ), the pairing is( A + εA ′ , B + εB ′ ) Z Y ( AB ′ + A ′ B ) . Since the pure supersymmetric Yang–Mills theory for an abelian Lie algebra is a free theory, we consider the“smooth” version O sm of the classical observables, just as in § d N =2 . Via the degree ( −
1) pairing this factorization algebra isequipped with a P -structure.The description of the holomorphic twist of 4 d N = 4 supersymmetric Yang–Mills theory for abelian Liealgebra is similar. The underlying complex of fields is(104) Ω , • ( Y )[ ε, δ ][1] ⊕ Ω , • ( Y )[ ε, δ ][2]The degree ( −
1) pairing is given by the integration pairing along Y together with the Berezin integral inthe odd ε, δ directions. Proposition 7.1.
Let π : Y × E → Y be the projection. • There is a morphism of P -factorization algebras on Yπ ∗ Obs (1 , → Obs d N =269 hose cofiber is a locally constant factorization algebra with trivial P -structure. • There is a morphism of P -factorization algebras on Yπ ∗ Obs (2 , → Obs d N =4 whose cofiber is a locally constant factorization algebra with trivial P -structure.Proof. We consider the (1 ,
0) case first. Following the description in § (1 , is given by the “smooth” functionals on the sheaf of cochain complexes Ω ≥ , • [1] on Y × E . Since E is formal,there is a quasi-isomorphism C [ ε ] ≃ −→ Ω , • ( E ). Here, ε is a chosen generator for the sheaf of sections of theanti-holomorphic canonical bundle on E . Thus, there is a quasi-isomorphism of sheaves on Y :Ω , • ( Y )[ ε ] ⊕ d z Ω ≥ , • ( Y )[ ε ] ≃ −→ π ∗ Ω ≥ , • . Here, d z denotes the holomorphic volume form on the elliptic curve.The sheaf of cochain complexes Ω ≥ , • ( Y ) is a resolution for the sheaf of closed one-forms on the complexsurface Y . The ∂ -operator determines a map of cochain complexes ∂ : Ω , • ( Y ) → Ω ≥ , • ( Y ) whose kernel isthe sheaf of constant functions.Putting this together, we find that there is a map of sheaves of cochain complexes on Y :Ω , • ( Y )[ ε ][1] ⊕ Ω , • ( Y )[ ε ][1] ∂ −→ π ∗ Ω ≥ , • [1] . We recognize the left-hand side as the complex of fields underlying the holomorphic twist of 4 d N = 2. Apply-ing the functor of taking the “smooth” functionals O sm ( − ) we obtain the first statement of the proposition.It is immediate to verify that this map intertwines the P -structures.The second statement is not much more difficult. Recall, the complex of fields of the holomorphic twistof the (2 ,
0) theory is obtained by adjoining the βγ system on the three-fold Y × E to the holomorphic twistof the (1 ,
0) theory. As a sheaf on Y × E , the complex of fields of the βγ system isΩ , • ( Y × E ) ⊕ Ω , • ( Y × E )[2] . Pushing forward along π the complex becomesΩ , • ( Y )[ ε ] ⊕ d z Ω , • ( Y )[ ε ][2] . Notice that this is a symplectic BV theory with the wedge and integrate pairing. The statement then followsfrom the observation that there is an isomorphism of symplectic BV theories (cid:18) Ω , • ( Y )[ ε ][1] ⊕ Ω , • ( Y )[ ε ][1] (cid:19) ⊕ (cid:18) Ω , • ( Y )[ ε ] ⊕ d z Ω , • ( Y )[ ε ][2] (cid:19) with the holomorphic twist of 4 d N = 4 as in (104). (cid:3) .1.2. Reduction to 2d CFT.
Consider the higher dimensional βγ system βγ ( X, C ) on a three-fold X withvalues in the trivial bundle. The space of fields is Ω , • ( X ) ⊕ Ω , • ( X )[2]. If Y is a compact K¨ahler surface, Σa Riemann surface, and π : Y × Σ → Σ the projection, then there is an equivalence of free BV theories on Σ:(105) π ∗ βγ ( X ; C ) = βγ (Σ; H • ( Y, O Y )) . This is the βγ system on Σ with values in the (graded) vector space H • ( Y, O Y ).Let χ (2) be the theory of the chiral two-form on Y × Σ with values in C . The following result follows froma direct calculation of the sheaf-theoretic pushforward of the complex χ (2) along the map π : Y × Σ → Σ.In the result, we use the fact that on a compact surface Y there is a symmetric bilinear form on the gradedvector space H • ( Y, Ω Y )[1] provided by Serre duality. Lemma 7.2.
Let Y be a compact K¨ahler surface and Σ a Riemann surface. There is an equivalence ofpresymplectic BV theories on Σ(106) π ∗ χ (2)( Y × Σ) ≃ (Ω • (Σ) ⊗ H • ( Y, O Y )) [2] ⊕ χ (cid:0) , H • ( Y, Ω Y )[1] (cid:1) (Σ) . On the right-hand side, the ( − -shifted presymplectic form is trivial on the first summand and the secondsummand is the chiral boson on Σ with values in the graded vector space H • ( Y, Ω Y )[1] . Combining this lemma with (105) we obtain the following description of the reduction of the holomorphictwist of the (2 ,
0) theory along π : Y × Σ → Σ. Proposition 7.3.
Suppose Y is a compact K¨ahler surface. The compactification of the holomorphic twistof the abelian (2 , theory along Y is equivalent to the direct sum of the following three presymplectic BVtheories on Σ : (107) βγ ( H • ( Y, O Y )) ⊕ χ (cid:0) , H • ( Y, Ω Y )[1] (cid:1) ⊕ (Ω • (Σ) ⊗ H • ( Y, O Y )) [2] . The first summand in (107) is the βγ chiral CFT with values in H • ( Y, O Y ) and the second summand isthe chiral boson with values in H • ( Y, Ω Y ). The last summand is quasi-isomorphic to the constant sheaf indegree −
2, and so has only topological degrees of freedom. Moreover, the last summand is equipped withthe trivial shifted presymplectic structure.Next, consider the non-minimal twist of the (2 ,
0) theory which we will place on M × Σ, where M is aclosed four-manifold. The presymplectic BV complex is of the form Ω • ( M ) ⊗ Ω , • (Σ)[2]. Similarly as inthe holomorphic twist, the compactification along M produces the theory of the chiral boson. This time,however, it is valued in the graded vector space H • ( M, C )[2], the de Rham cohomology of M shifted by two.Note that the integration pairing endows this graded space with a graded symmetric bilinear form. roposition 7.4. Let M be a closed four-manifold. The compactification of the non-minimal twist of theabelian (2 , theory along M is equivalent, as a presymplectic BV theory, to the chiral boson χ (0 , H • ( M, C )[2]) on Σ . Untwisted dimensional reduction.
We close this paper by giving some results on dimensional re-duction at the level of the untwisted theory. It is expected that dimensional reduction along a circle shouldproduce five-dimensional N = 2 super Yang–Mills theory, but with an inverse dependence of the 5d couplingconstant on the compactification radius, compared to the dependence expected from a typical Kaluza–Kleinreduction. As in the results for the twisted theory above, we will be able to show that our formulation reducescorrectly, up to a copy of the constant sheaf. We check that the presymplectic structure agrees with thestandard BV pairing after dimensional reduction. Accounting correctly for this copy of the constant sheafshould require passing to a nonperturbative description of the theory; we offer some speculative commentson this, and plan to return to a more rigorous analysis in future work.In five-dimensional N = 2 supersymmetry, the R -symmetry group is Sp(2) ∼ = SO(5), just as in sixdimensions. The chiral spinor repreduces to the (unique) five-dimensional spinor representation; dimensionalreduction of the fermions and the scalars in T (2 , is thus trivial. The only difficulty is thus to check that ourdescription of χ + (2) reduces to the (nondegenerate) BV theory of a one-form gauge field in five dimensions.We check this in the theorem below; the full statement for supersymmetric theories follows immediately(Corollary 7.6). Proposition 7.5.
Let χ + (2) red denote the dimensional reduction of χ + (2) to five dimensions along S .There is a map of theories (108) Ξ : χ + (2) red → Φ(1) , where Φ(1) is the standard nondegenerate theory of perturbative abelian one-form fields on R . The kernel ofthis map is a copy of the constant sheaf in BV degree − . Furthermore, the shifted presymplectic structure Ξ ∗ ω Φ agrees with the shifted presymplectic structure inherited from χ + (2) .Proof. We begin by placing the theory on R × S . For all of the fields other than the self-dual antifield, thedecomposition is straightforward, using the fact that(109) Ω i ( R ) ∼ = (cid:0) Ω i ( R ) ⊗ Ω ( S ) (cid:1) ⊕ (cid:0) Ω i − ( R ) ⊗ Ω ( S ) (cid:1) . The self-dual forms Ω , however, sit diagonally with respect to this decomposition. If p : R × S → R denotes the obvious projection and α ∈ Ω ( R ) is a fixed two-form, then(110) π + p ∗ α = 12 (cid:0) p ∗ α − √− dt ∧ p ∗ ( ⋆α ) (cid:1) . ote that ⋆ here denotes the five -dimensional Hodge star operator; ⋆α is thus an element of Ω ( R ). θ is acoordinate on S .In light of this decomposition, we can rewrite χ + (2) on R × S in the following fashion:(111)Ω ( R ) ⊗ Ω ( S )[2] Ω ( R ) ⊗ Ω ( S )[1] Ω ( R ) ⊗ Ω ( S )Ω ( R ) ⊗ Ω ( S )[1] Ω ( R ) ⊗ Ω ( S ) Ω ( R ) ⊗ Ω ( S )[ − . d R d S d R d S d S +( ⋆ d) R d R d R (We have made use of the fact that the spaces Ω ( R ) ⊗ Ω ( S ) and Ω ( R ) ⊗ Ω ( S ) are isomorphic via thesix-dimensional Hodge star.)To dimensionally reduce along the circle, we pass to the cohomology of d S . This results in the followingcochain complex χ + (2) red of vector bundles on R :(112) Ω ( R )[2] Ω ( R )[1] Ω ( R )Ω ( R )[1] Ω ( R ) Ω ( R )[ − . d d ⋆ dd d Of course, the shifted presymplectic structure on χ + (2) red is inherited from the structure on χ + (2). It is easyto verify that, as a skew map from χ + (2) red to ( χ + (2) red ) ! [ − ω red is just the pair of first-orderdifferential operators represented by dashed arrows in the below diagram:(113) Ω ( R )[2] Ω ( R )[1] Ω ( R )Ω ( R )[1] Ω ( R ) Ω ( R )[ − ( R )[ −
1] Ω ( R )[ −
2] Ω ( R )[ − ( R ) Ω ( R )[ −
1] Ω ( R )[ − . ⋆ d dd d ⋆ There is a single check that ω red defines a cochain map, which is trivial.Now, there is a strict isomorphism of cochain complexes of vector bundles, defined to be the identitymap in all nonpositive degrees and the Hodge star map in degree 1. This isomorphism allows us to replace χ + (2) red by the cochain complex of vector bundles in the diagram below, which we will call U for brevity:(114) Ω ( R )[2] Ω ( R )[1] Ω ( R ) Ω ( R )[ − ( R )[1] Ω ( R ) . d d dd ⋆ d73 he shifted presymplectic structure on U is analogous to (in fact, just isomorphic to) that depicted in (113).We are dealing with the complex(115) U = (Ω ≤ ( R )[1] ⊕ Ω ≤ ( R )[2] , ⋆ d) , and it is immediate that(116) U ! [ − ∼ = (Ω ≥ ( R )[ − ⊕ Ω ≥ ( R ) , d ⋆ ) . The map ω U consists of the two obvious de Rham operators that define maps of degree zero from U to U ! [ − , C ) from §
2, which takes theform(117) Φ(1) = (Ω ≤ ( R )[1] ⊕ Ω ≥ ( R )[ − , d ⋆ d) . There is then a map of cochain complexes defined by the vertical arrows in the following diagram:(118) Ξ : U → Φ(1) , Ω ≤ ( R )[1] Ω ≤ ( R )[2]Ω ≤ ( R )[1] Ω ≥ ( R )[ − . ⋆ did dd ⋆ d We now form the cone of this map. Using [49, Proposition 1.23], we can eliminate the acyclic piece, obtainingthe description(119) Cone(Ξ) ∼ = (cid:16) Ω ≤ ( R )[3] Ω ≥ ( R )[ − d (cid:17) . This complex has cohomology only in the left-hand term; after totalizing, we obtain Ω • ( R )[3], thus findingthat the kernel is a copy of the constant sheaf in degree −
2. It is entirely straightforward to check thatΞ ∗ ω Φ = ω U . Since U ∼ = χ + (2) red , we have completed the proof. (cid:3) As remarked above, the full statement, pertaining to dimensional reduction of the full N = (2 ,
0) abeliantensor multiplet, follows immediately from Proposition 7.5:
Corollary 7.6.
Let T red (2 , denote the dimensional reduction of T (2 , to five dimensions along S , and letYM N =2 denote the N = 2 supersymmetric (perturbative) abelian Yang–Mills multiplet on R . There is amap of theories (120) ˆΞ : T red (2 , → YM N =2 , efined by extending Ξ by identity morphisms. The kernel of this map is a copy of the constant sheaf inBV degree − . Furthermore, the shifted presymplectic structure Ξ ∗ ω YM agrees with the shifted presymplecticstructure inherited from T (2 , . Electric–magnetic duality and the physical interpretation of the proof of Proposition 7.5.
For the physi-cist reader, the language of the proof of Proposition 7.5 may be unfamiliar, but the manipulations should atleast have a familiar flavor. We briefly recall the typical description of electric–magnetic duality that is folkwisdom among physicists: A theory of p -form gauge fields in dimension d has a gauge potential A ∈ Ω p ( R d ),whose field strength is a gauge-invariant ( p + 1)-form given (in the abelian case) just by F = dA . F satisfiesan equation of motion d ⋆F = 0, but also a “Bianchi identity” d F = 0, which is (at least in contractibleopen sets) equivalent to the existence of the potential A . These equations could be just as well phrased interms of the “dual” field strength, the ( d − p − G = ⋆F , with the roles of the equations of motionand the Bianchi identity reversed. In light of the Bianchi identity, G can be written as the field strengthof a potential B ∈ Ω d − p − ( R d ). One can sum this up by saying that an equivalence is expected betweenthe theories of p -forms and ( d − p − F that are the physical electric and magnetic fields. (In Maxwell theory in fourdimensions, both the electric and magnetic gauge fields are one-forms.)One might therefore expect an equivalence between the theories we have called Φ( p ) and Φ( d − p − p ) = Ω ≤ p ( R d )[ p ] Ω ≥ ( d − p ) ( R d )[ − , d ⋆ d thinking of it as the electric description. There is another cochain complex F ( p ) of vector bundles on R d ,defined by(122) F ( p ) = Ω ≥ ( p +1) ( R d ) Ω ≥ ( d − p ) ( R d )[ − , d ⋆ hich can be thought of as a BV or on-shell version of the field strength, subjected to its equation of motion.( F ( p ) freely resolves the sheaf of solutions to the equations d F = d ⋆F = 0.) There is a curvature mapcurv : Φ( p ) → F ( p ), defined by the vertical arrows in the diagram(123) Ω ≤ p ( R d )[ p ] Ω ≥ ( d − p ) ( R d )[ − ≥ ( p +1) ( R d ) Ω ≥ ( d − p ) ( R d )[ − . d d ⋆ d idd ⋆ It extends the usual curvature map on fields by the identity on antifields. The cone of curv is a shift of the deRham complex, as in the proof above, and so there is a kernel, consisting of the constant sheaf representingzero modes of the zero-form ghost in BV degree − p .Now, applying the Hodge star in degree zero defines an isomorphism of F ( p ) with F ( d − p − p ) F ( p ) F ( d − p −
2) Φ( d − p − , curv ∼ = curv encapsulating a BV description of the argument above. If electric–magnetic duality were to hold at aperturbative level, all of these maps would be quasi–isomorphisms; the curvature map, however, is not,and the duality fails at the level of the constant sheaves described above. It is interesting to note that, inthe description we are giving, the antifields to the electric degrees of freedom in some sense play the roleof the magnetic degrees of freedom. Furthermore, we remark that F ( p ) does not admit a natural shiftedpresymplectic structure; it does, however, admit a shifted Poisson tensor.In the proof of Proposition 7.5, a very analogous set of arguments play a role. However, the object U thatappears there is not the curvature; in fact, it maps into both Φ(1) and Φ(2) on R in symmetric fashion,defining a roof of maps between them, rather than receiving maps from each. To phrase the situation ingeneral language, we would define(125) U ( p ) = Ω ≤ p ( R d )[ p ] Ω ≤ ( d − p − ( R d )[ d − p − . ⋆ d It is immediate to see that U ( p ) and U ( d − p −
2) are isomorphic via the Hodge star in BV degree +1, andthat the map Ξ can be generalized to a map Ξ( p ) : U ( p ) → Φ( p ). U ( p ), moreover, does admit a naturalshifted presymplectic structure, as in (113).The proof of Proposition 7.5 relies on electric–magnetic duality, in the sense that χ + (2) red = U (2)( R )is, at first glance, most naturally interpreted as a theory of a two-form. The additional copy of a constantsheaf in the theorem is also, in some sense, dual to the issue that appeared in our attempt to perturbativelyformalize the standard argument. We can sum up all of these considerations, in somewhat greater generality, ith the following diagram:(126) U ( p ) U ( d − p − p ) Φ( d − p − F ( p ) F ( d − p − . ∼ =Ξ Ξcurv EM dual curv ∼ = The kernel of each vertical map in (126) is an appropriately shifted copy of the constant sheaf. The failureof these vertical maps to define quasi-isomorphisms reflects the nonperturbative nature of the duality; weoffer some speculation on the correct fix for this in the next section.7.2.2.
Speculative remarks on global structure.
There is no doubt that the reader will have been disappointedby all of the “errors” in the above results, having to do with zero modes (or, for mathematicians, undesirablecopies of constant sheaves). Part of the reason for the discursiveness of the above remarks on electric–magnetic duality is to emphasize that we see these as representing familiar phenomena from the physicsperspective: Any on-the-nose equivalence of perturbative theories cannot possibly be a correct representationof electric–magnetic duality. The fact that electric–magnetic duality plays a role in passing from the N =(2 ,
0) multiplet to supersymmetric Yang–Mills theory in five dimensions is also not unreasonable; in fact, thisis the key reason that the dependence on the coupling constant is inverted from the standard Kaluza–Kleinexpectation, as remarked above. For interacting theories, electric–magnetic duality requires an inversion ofthe coupling constant. (The coupling constant that scales “correctly” with the compactification radius is notthe Yang–Mills coupling constant, but the coupling constant of its magnetically dual theory of two-forms.)In fact, it is tempting to speculate that insisting on the correct dimensional reduction at the nonpertur-bative level will shine a light on the nonperturbative BV formulation of electric–magnetic duality. Recallthat the correct nonperturbative generalization of the BRST complex of an abelian gauge field—which isperturbatively just Ω ≤ p ( R d )[ p ]—is the smooth Deligne cohomology group(127) Z α ( p ) ∞ D = Z [ p + 1] Ω ≤ p [ p ] . (2 πi ) p α (Here α denotes a choice of real number, which plays the role of the coupling constant or radius of thegauge group; our notation here differs from the standard notation for Deligne cohomology by indicating α explicitly.) We should thus expect that it is possible to formulate a BV, or possibly presymplectic BV,description of abelian Yang–Mills theory, using Deligne cohomology groups to represent both the electricand magnetic gauge fields. In light of the considerations above, and by directly generalizing (125), one wouldattempt to write down a complex of the form(128) Z α U ( p ) = Z α ( p ) ∞ D Z /α ( d − p − ∞ D [ − . ⋆ d77 he inverse choices of coupling constants are necessitated by the requirement that the complex have nontrivialcohomology in degree zero. In particular, Deligne cohomology represents the curvature of a connection in aU(1) (or GL(1)) bundle, and so admits a curvature map whose image is an integral class (for α = 1). Wecan rewrite the complex above in the form(129) Z α ( p ) ∞ D Ω d − p − [ − . Z /α ( p ) ∞ D ⋆ dd By passing to the cohomology of the internal differentials of the Deligne complexes, we see that the curvaturesof the electric and magnetic degrees of freedom must be related by the Hodge star. Choosing a volume formso that the Hodge star preserves integral classes makes the requirement on the coupling constants immediate,at least up to discrete choices corresponding to finite coverings of U(1) by U(1).Describing things in this way makes our considerations seem almost trivial; of course abelian Yang–Millstheory consists of an electric gauge field A with curvature F , and a magnetic gauge field B with curvature G , subject to the constraint that F = ⋆G . We emphasize that the novelty in this way of thinking consists ofthe fact that this pair is interpreted as a complete (presymplectic) BV theory , where the pairing is definedby differential operators as done for ω U above. In this formulation, the equations of motion (and thereforethe antifields) for F have been replaced by the Bianchi identities (and therefore the gauge invariances)for G . This is the sense in which the magnetic gauge fields and the electric antifields are one and the same.Electric–magnetic duality then just amounts to the trivial or manifest statement that(130) Z α U ( p ) ∼ = Z /α U ( d − p − . It would be interesting to make contact with other BV approaches to electric–magnetic duality, such as [46].Identical considerations suggest a nonperturbative definition of the theory of self-dual (2 k )-forms; thereader will probably have guessed that the complex we have in mind is(131) Z α χ + (2 k ) = Z α (2 k ) ∞ D Ω k +1+ [ − d + = Z [2 k + 1] χ + (2 k ) . (2 πi ) k α We note that, for k = 0, this theory describes periodic (circle-valued) chiral bosons; in general, it describes aconnection on an abelian gerbe, subject to the constraint that the curvature (which now defines an integralclass) must be self-dual.Now, placing Z α χ + (2) on R × S and pushing forward along the projection map produces precisely thecomplex Z α U (2), under the assumption that the radius of the compactification circle is one. To see this, We thank K. Costello for suggesting this definition to us, independently of dimensional reduction. ote that we must use the derived pushforward, so that π ∗ Z = H ∗ ( S , Z ). Making sense of the maps reducesto understanding the map induced on the sheaf cohomology of the circle from the map of sheaves(132) Z (2 πi ) p α −−−−−→ C . 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