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Given a supermanifold equipped with an odd distribution of maximal dimension and constant symbol, we construct the formal moduli problem of deformations of the distribution. This moduli problem is described by a local super dg Lie algebra that provides both a resolution of the structure-preserving vector fields on superspace and a derived enhancement of superconformal symmetry. Applying our construction in standard physical examples returns the conformal supergravity multiplet in every known example, in any dimension and with any amount of supersymmetry$\unicode{x2014}$whether or not a superconformal algebra exists. We discuss new examples related to twisted supergravity, higher Virasoro algebras, and exceptional super Lie algebras. The compatibility of our techniques with twisting also leads to a computation of every twist of the stress tensor multiplet of a superconformal theory, including universal operator product expansions. Our approach uses a derived model for the space of functions constant along the distribution, which is applicable even when the distribution is non-involutive; we construct other natural multiplets, such as K\"ahler differentials, that appear naturally through this lens on superspace geometry., Comment: 68 pages, 7 tables. Comments welcome! v2: updated funding information

Topological field theories and holomorphic field theories naturally appear in both mathematics and physics. However, there exist intriguing hybrid theories that are topological in some directions and holomorphic in others, such as twists of supersymmetric field theories or Costello's 4-dimensional Chern-Simons theory. In this paper, we rigorously prove the ultraviolet (UV) finiteness for such hybrid theories on flat space, and present two significant vanishing results regarding anomalies., Comment: 51 pages, no figures. An error in the proof has been fixed. Corrected several typos. Comments welcome

We compute the local cohomology of vector fields on a manifold. In the smooth case this recovers the diagonal cohomology studied in work of Losik, Guillemin, Fuks and others. In the holomorphic case this cohomology has recently appeared in work of Hennion and Kapranov in their study of the Lie algebra cohomology of vector fields on a complex manifold. Additionally, we construct explicit representatives for cocycles in Gelfand--Fuks cohomology via descent., Comment: 30 pages, comments are very welcome!

In four-dimensional conformal field theory, the numbers a and c are defined as coefficients of particular terms in the operator product expansion (OPE) of the energy-momentum tensor. With supersymmetry there are relations between these coefficients and mixed R-symmetry and gravitational anomalies. In this paper we prove a relationship between these coefficients and anomalies to holomorphic reparametrization symmetry at the level of the holomorphic twist., Comment: 42 pages, comments welcome!

In this work, we revisit and elaborate on twisted holography for AdS$_3 \times S^3 \times X$ with $X= T^4$, K3, with a particular focus on K3. We describe the twist of supergravity, identify the corresponding (generalization of) BCOV theory, and enumerate twisted supergravity states. We use this knowledge, and the technique of Koszul duality, to obtain the $N \rightarrow \infty$, or planar, limit of the chiral algebra of the dual CFT. The resulting symmetries are strong enough to fix planar 2 and 3-point functions in the twisted theory or, equivalently, in a 1/4-BPS subsector of the original duality. This technique can in principle be used to compute corrections to the chiral algebra perturbatively in $1/N$., Comment: 91 pages, 9 figures (v2: minor corrections and additions)

We propose a method for extracting the Higgs and Coulomb branches of a three-dimensional N = 4 quantum field theory from the algebra of local operators in its holomorphic-topological twist using the formalism of raviolo vertex algebras. Our construction parallels that of the chiral ring and twisted chiral ring of an N = 2 superconformal vertex operator algebra., Comment: 45 pages; feedback welcome

We show that the action of residual supersymmetries in holomorphic-topological twists of $N = 2$ theories in three dimensions naturally extends to the action of certain infinite dimensional Lie superalgebras. We demonstrate this in a range of examples, including $N = 4$ Yang-Mills theories and superconformal Chern-Simons theories, describing how the symmetries are implemented at the level of local operators., Comment: 41 pages, feedback welcome!

We develop an algebraic structure modeling local operators in a three-dimensional quantum field theory which is partially holomorphic and partially topological. The geometric space organizing our algebraic structure is called the raviolo (or bubble) and replaces the punctured disk underlying vertex algebras; we refer to this structure as a raviolo vertex algebra. The raviolo has appeared in many contexts related to three-dimensional supersymmetric gauge theory, especially in work on the affine Grassmannian. We prove a number of structure theorems for raviolo vertex algebras and provide simple examples that share many similarities with their vertex algebra counterparts., Comment: 78 pages. Comments and feedback are welcomed!

We discuss the properties of quarter-BPS local operators in four dimensional ${\cal N}=1$ supersymmetric Yang-Mills theory using the formalism of holomorphic twists. We study loop corrections both to the space of local operators and to algebraic operations which endow the twisted theory with an infinite symmetry algebra. We classify all single-trace quarter-BPS operators in the planar approximation for $SU(N)$ gauge theory and propose a holographic dual description for the twisted theory. We classify perturbative quarter-BPS operators in $SU(2)$ and $SU(3)$ gauge theories with sufficiently small quantum numbers and discuss possible non-perturbative corrections to the answer. We set up analogous calculations for some theories with matter., Comment: v1:55+20 pages, 61 footnotes, comments welcome; v2: published version

We present a conjectural description of the space of local operators on a stack of finitely many fivebranes in $M$ theory at the level of the holomorphic twist. Our approach is through the lens of twisted holography and utilizes a description of the minimal twist of eleven-dimensional supergravity. We find that the spaces of local operators are modules for the exceptional linearly compact super Lie algebra $E(3|6)$. From the conjectural description of local operators we deduce closed formulas for the superconformal index of six-dimensional $\mathcal{N}=(2,0)$ theories of type $A_{N-1}$., Comment: Comments and feedback are welcome!

We construct a fully interacting holomorphic/topological theory in eleven dimensions that is defined on products of Calabi-Yau fivefolds with real one-manifolds. The theory describes a particular deformation of the cotangent bundle to the moduli space of Calabi-Yau structures on the fivefold. Its field content matches the holomorphic (or minimal) twist of the eleven-dimensional supergravity multiplet recently computed by the second two authors, and we offer numerous consistency checks showing that the interactions correctly describe interacting twisted eleven-dimensional supergravity at the perturbative level. We prove that the global symmetry algebra of our model on flat space is an $L_\infty$ central extension of the infinite-dimensional simple exceptional super Lie algebra $E(5,10)$, following a recent suggestion of Cederwall in the context of the relevant pure spinor model. Twists of superconformal algebras map to the fields of our model on the complement of a stack of M2 or M5 branes, laying the groundwork for a fully holomorphic version of twisted holography in this context., Comment: 58 pages. Comments welcome!

In this note, we study, formalize, and generalize the pure spinor superfield formalism from a rather nontraditional perspective. To set the stage, we review the notion of a multiplet for a general super Lie algebra, working in the context of the BV and BRST formalisms. Building on this, we explain how the pure spinor superfield formalism can be viewed as constructing a supermultiplet out of the input datum of an equivariant graded module over the ring of functions on the nilpotence variety. We use the homotopy transfer theorem and other computational techniques from homological algebra to relate these multiplets to more standard component-field formulations. Physical properties of the resulting multiplets can then be understood in terms of algebrogeometric properties of the nilpotence variety. We illustrate our discussion with many examples in various dimensions., Comment: v2: updated grant information. 66 pages, 27 tables. Comments welcome!

We formulate a twisted version of the conjectured duality between heterotic and type I string theories. Our formulation relates the chiral part of the heterotic string with a type I topological B-model on a Calabi-Yau five-fold. We provide a non-trivial check of this duality by showing that certain infinite-dimensional Lie algebras of global gauge transformations built from each theory are isomorphic. Matching the structure constants of the Lie algebras involves a detailed analysis of one-loop corrections on the type I side.

In this article, we introduce basic aspects of the algebraic notion of Koszul duality for a physics audience. We then review its appearance in the physical problem of coupling QFTs to topological line defects, and illustrate the concept with some examples drawn from twists of various simple supersymmetric theories. Though much of the content of this article is well-known to experts, the presentation and examples have not, to our knowledge, appeared in the literature before. Our aim is to provide an elementary introduction for those interested in the appearance of Koszul duality in supersymmetric gauge theories with line defects and, ultimately, its generalizations to higher-dimensional defects and twisted holography., Comment: v3: minor typos corrected. 57 pages, 4 figures

We pursue a uniform quantization of all twists of 4-dimensional N = 4 supersymmetric Yang-Mills theory, using the BV formalism, and we explore consequences for factorization algebras of observables. Our central result is the construction of a one-loop exact quantization on $\mathbb R^4$ for all such twists and for every point in a moduli of vacua. When an action of the group SO(4) can be defined - for instance, for Kapustin and Witten's family of twists - the associated framing anomaly vanishes. It follows that the local observables in such theories can be canonically described by a family of framed $\mathbb E_4$ algebras; this structure allows one to take the factorization homology of observables on any oriented 4-manifold. In this way, each Kapustin-Witten theory yields a fully extended, oriented 4-dimensional topological field theory \`a la Lurie and Scheimbauer., Comment: 54 pages, 3 figures. Comments welcome!

We show that a family of topological twists of a supersymmetric mechanics with a K\"ahler target exhibits a Batalin--Vilkovisky quantization. Using this observation we make a general proposal for the Hilbert space of states after a topological twist in terms of the cohomology of a certain perverse sheaf. We give several examples of the resulting Hilbert spaces including the categorified Donaldson--Thomas invariants, Haydys--Witten theory and the 3-dimensional A-model.

In both mathematics and physics, topological field theories and holomorphic field theories appear naturally, but there are interesting theories that are hybrids -- looking topological in some directions and holomorphic in others -- such as twists of supersymmetric field theories or Costello's 4-dimensional Chern--Simons theory. In this paper we construct perturbative, one-loop quantizations rigorously on the model manifold $\mathbb{R}^m \times \mathbb{C}^n$, and find a remarkable vanishing result about anomalies: the one-loop obstruction to quantization on $\mathbb{R}^m\times \mathbb{C}^n$ vanishes when $m\geq 1$. A concrete consequence of our results is the existence of exact and finite quantizations at one-loop for twists of pure $\mathcal{N} =2$ four-dimensional supersymmetric Yang--Mills theory.

We study twists of supergravity theories and supersymmetric field theories, using a version of the pure spinor superfield formalism. Our results show that, just as the component fields of supersymmetric multiplets are the vector bundles associated to the equivariant Koszul homology of the variety of square-zero elements in the supersymmetry algebra, the component fields of the holomorphic twists of the corresponding multiplets are the holomorphic vector bundles associated to the equivariant Koszul homology of square-zero elements in the twisted supersymmetry algebra. The BRST or BV differentials of the free multiplet are induced by the brackets of the corresponding super Lie algebra in each case. We make this precise in a variety of examples; applications include rigorous computations of the minimal twists of eleven-dimensional and type IIB supergravity, in the free perturbative limit. The latter result proves a conjecture by Costello and Li, relating the IIB multiplet directly to a presymplectic BV version of minimal BCOV theory., Comment: 47 pages, no figures. Comments welcome!

We formulate the abelian six-dimensional $\mathcal{N}=(2,0)$ theory perturbatively, in a generalization of the Batalin-Vilkovisky formalism. Using this description, we compute the holomorphic and non-minimal twists at the perturbative level. This calculation hinges on the existence of an $L_\infty$ action of the supersymmetry algebra on the abelian tensor multiplet, which we describe in detail. Our formulation appears naturally in the pure spinor superfield formalism, but understanding it requires developing a presymplectic generalization of the BV formalism, inspired by Dirac's theory of constraints. The holomorphic twist consists of symplectic-valued holomorphic bosons from the $\mathcal{N}=(1,0)$ hypermultiplet, together with a degenerate holomorphic theory of holomorphic coclosed one-forms from the $\mathcal{N}=(1,0)$ tensor multiplet, which can be interpreted as representing the intermediate Jacobian. We check that our formulation and our results match with known ones under various dimensional reductions, as well as comparing the holomorphic twist to Kodaira-Spencer theory. Matching our formalism to five-dimensional Yang-Mills theory after reduction leads to some issues related to electric-magnetic duality; we offer some speculation on a nonperturbative resolution., Comment: v2: updated grant information. 82 pages, seven figures. Comments appreciated!

We construct a class of quantum field theories depending on the data of a holomorphic Poisson structure on a piece of the underlying spacetime. The main technical tool relies on a characterization of deformations and anomalies of such theories in terms of the Gelfand-Fuchs cohomology of formal Hamiltonian vector fields. In the case that the Poisson structure is non-degenerate such theories are topological in a certain weak sense, which we refer to as "de Rham topological". While the Lie algebra of translations acts in a homotopically trivial way, we will show that the space of observables of such a theory does not define an E_n-algebra. Additionally, we will highlight a conjectural relationship to theories of supergravity in four and five dimensions., Comment: 28 pages, comments welcome

We give a complete classification of twists of supersymmetric Yang--Mills theories in dimensions $2\leq n \leq 10$. We formulate supersymmetric Yang--Mills theory classically using the BV formalism, and then we construct an action of the supersymmetry algebra using the language of $L_\infty$ algebras. For each orbit in the space of square-zero supercharges in the supersymmetry algebra, under the action of the spin group and the group of R-symmetries, we give a description of the corresponding twisted theory. These twists can be described in terms of mixed holomorphic-topological versions of Chern--Simons and BF theory.

We develop a method of quantization for free field theories on manifolds with boundary where the bulk theory is topological in the direction normal to the boundary and a local boundary condition is imposed. Our approach is within the Batalin-Vilkovisky formalism. At the level of observables, the construction produces a stratified factorization algebra that in the bulk recovers the factorization algebra developed by Costello and Gwilliam. The factorization algebra on the boundary stratum enjoys a perturbative bulk-boundary correspondence with this bulk factorization algebra. A central example is the factorization algebra version of the abelian Chern-Simons/Wess-Zumino-Witten correspondence, but we examine higher dimensional generalizations that are related to holomorphic truncations of string theory and $M$-theory and involve intermediate Jacobians., Comment: Incorporated referee suggestions

We examine Chern-Simons theory as a deformation of a 3-dimensional BF theory that is partially holomorphic and partially topological. In particular, we introduce a novel gauge that leads naturally to a one-loop exact quantization of this BF theory and Chern-Simons theory. This approach illuminates several important features of Chern-Simons theory, notably the bulk-boundary correspondence of Chern-Simons theory with chiral WZW theory. In addition to rigorously constructing the theory, we also explain how it applies to a large class of closely related 3-dimensional theories and some of the consequences for factorization algebras of observables., Comment: 35 pages, 1 figure

We show that four-dimensional superconformal algebras admit an infinite-dimensional derived enhancement after performing a holomorphic twist. The type of higher symmetry algebras we find are closely related to algebras studied by Faonte-Hennion-Kapranov, Hennion-Kapranov, and the second author with Gwilliam in the context of holomorphic QFT. We show that these algebras are related to the two-dimensional chiral algebras extracted from four-dimensional superconformal theories by Beem and collaborators; further deforming by a superconformal element induces the Koszul resolution of a plane in $\mathbb{C}^2 \cong \mathbb{R}^4$. The central charges at the level of chiral algebras arise from central extensions of the higher symmetry algebras., Comment: 59 pages, two figures. v. 3: major edits to prepare for journal submission

We consider holomorphic twists of arbitrary supersymmetric theories in four dimensions. Working in the BV formalism, we rederive classical results characterizing the holomorphic twist of chiral and vector supermultiplets, computing the twist explicitly as a family over the space of nilpotent supercharges in minimal supersymmetry. The BV formalism allows one to work with or without auxiliary fields, according to preference; for chiral superfields, we show that the result of the twist is an identical BV theory, the holomorphic $\beta\gamma$ system with superpotential, independent of whether or not auxiliary fields are included. We compute the character of local operators in this holomorphic theory, demonstrating agreement of the free local operators with the usual index of free fields. The local operators with superpotential are computed via a spectral sequence, and are shown to agree with functions on a formal mapping space into the derived critical locus of the superpotential. We consider the holomorphic theory on various geometries, including Hopf manifolds and products of arbitrary pairs of Riemann surfaces, and offer some general remarks on dimensional reductions of holomorphic theories along the $(n-1)$-sphere to topological quantum mechanics. We also study an infinite-dimensional enhancement of the flavor symmetry in this example, to a recently-studied central extension of the derived holomorphic functions with values in the original Lie algebra that generalizes the familiar Kac--Moody enhancement in two-dimensional chiral theories.

We introduce a higher dimensional generalization of the affine Kac-Moody algebra using the language of factorization algebras. In particular, on any complex manifold there is a factorization algebra of "currents" associated to any Lie algebra. We classify local cocycles of these current algebras, and compare them to central extensions of higher affine algebras recently proposed by Faonte-Hennion-Kapranov. A central goal of this paper is to witness higher Kac-Moody algebras as symmetries of a class of holomorphic quantum field theories. In particular, we prove a generalization of the free field realization of an affine Kac-Moody algebra and also develop the theory of q-characters for this class of algebras in terms of factorization homology. Finally, we exhibit the "large N" behavior of higher Kac-Moody algebras and their relationship to symmetries of non-commutative field theories., Comment: All around improvements to exposition. Added Section 5: Large N Limits

We introduce the concept of a holomorphic field theory on any complex manifold in the language of the Batalin-Vilkovisky formalism. When the complex dimension is one, this setting agrees with that of chiral conformal field theory. Our main result concerns the behavior of holomorphic theories under renormalization group flow. Namely, we show that holomorphic theories are one-loop finite. We use this to completely characterize holomorphic anomalies in any dimension. Throughout, we compare our approach to holomorphic field theories to more familiar approaches including that of supersymmetric field theories.

The purpose of this note is two give a mathematical treatment to the low energy effective theory of the two-dimensional sigma model. Perhaps surprisingly, our low energy effective theory encodes much of the topology and geometry of the target manifold. In particular, we relate the $\beta$-function of our theory to the Ricci curvature of the target, recovering the physical result of Friedan., Comment: Fixed reference

We show that the local observables of the curved beta gamma system encode the sheaf of chiral differential operators using the machinery of the book "Factorization algebras in quantum field theory", by Kevin Costello and the second author, which combines renormalization, the Batalin-Vilkovisky formalism, and factorization algebras. Our approach is in the spirit of deformation quantization via Gelfand-Kazhdan formal geometry. We begin by constructing a quantization of the beta gamma system with an n-dimensional formal disk as the target. There is an obstruction to quantizing equivariantly with respect to the action of formal vector fields on the target disk, and it is naturally identified with the first Pontryagin class in Gelfand-Fuks cohomology. Any trivialization of the obstruction cocycle thus yields an equivariant quantization with respect to an extension of formal vector fields by the closed 2-forms on the disk. By results in the book listed above, we then naturally obtain a factorization algebra of quantum observables, which has an associated vertex algebra easily identified with the formal beta gamma vertex algebra. Next, we introduce a version of Gelfand-Kazhdan formal geometry suitable for factorization algebras, and we verify that for a complex manifold with trivialized first Pontryagin class, the associated factorization algebra recovers the vertex algebra of CDOs on the complex manifold., Comment: Some minor errors and typos fixed. Added section on the formal construction of the Witten genus. Added section on the conformal anomaly and realization of the Virasoro vertex algebra in the case of a curved target. Added a further comparisons to and interpretations of physical aspects of the non-linear sigma model. This is the final, published version

This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the forthcoming book of Costello-Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as are the correlation functions. We provide an example of how the Virasoro factorization algebra implements conformal symmetry of the beta-gamma system using the method of effective BV quantization.