The topological chiral homology of the spherical category

Beraldo, Dario

doi:10.1112/topo.12098

Cited by 11 publications

(14 citation statements)

References 23 publications

(81 reference statements)

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“…It can be constructed using the formalism of [76], and it is a motivating example of Toën's "brane operations" construction [83] (though the compatibility of the two constructions is not currently documented). The factorization homology of this 3-disc structure (i.e., the structure of "line operator Ward identities" for the geometric Langlands program) is calculated in [85]. See also [82,84].…”

Section: Line Operatorsmentioning

confidence: 99%

“…The existence of a 3-disc structure on the spherical category was first observed by Lurie in 2005. It is mentioned in[82,83,84] -a related construction appears in[76] -and the factorization homology of this 3-disc structure is described in[85] 6. It's important to note that the entire disc algebra structure (not just the L ∞ part) is important for the deformation problem -d-disc algebras[82,18] define enhanced "slightly noncommutative" formal moduli spaces, whose rings of functions are themselves d-disc algebras.…”

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confidence: 99%

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Secondary Products in Supersymmetric Field Theory

Beem

Ben‐Zvi

Bullimore

et al. 2020

Ann. Henri Poincaré

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The product of local operators in a topological quantum field theory in dimension greater than one is commutative, as is more generally the product of extended operators of codimension greater than one. In theories of cohomological type these commutative products are accompanied by secondary operations, which capture linking or braiding of operators, and behave as (graded) Poisson brackets with respect to the primary product. We describe the mathematical structures involved and illustrate this general phenomenon in a range of physical examples arising from supersymmetric field theories in spacetime dimension two, three, and four. In the Rozansky-Witten twist of three-dimensional N = 4 theories, this gives an intrinsic realization of the holomorphic symplectic structure of the moduli space of vacua. We further give a simple mathematical derivation of the assertion that introducing an Ω-background precisely deformation quantizes this structure. We then study the secondary product structure of extended operators, which subsumes that of local operators but is often much richer. We calculate interesting cases of secondary brackets of line operators in Rozansky-Witten theories and in four-dimensional N = 4 super Yang-Mills theories, measuring the noncommutativity of the spherical category in the geometric Langlands program.

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Section: Line Operatorsmentioning

confidence: 99%

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confidence: 99%

Secondary Products in Supersymmetric Field Theory

Beem

Ben‐Zvi

Bullimore

et al. 2020

Ann. Henri Poincaré

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“…1.10.3. For generalizations of these computations to the topological setting, the reader may consult [Ber19b].…”

Section: H For Harishmentioning

confidence: 99%

Sheaves of categories with local actions of Hochschild cochains

Beraldo¹

2019

Compositio Math.

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The notion of Hochschild cochains induces an assignment from Aff, affine DG schemes, to monoidal DG categories. We show that this assignment extends, under appropriate finiteness conditions, to a functor H : Aff → Alg bimod (DGCat), where the latter denotes the category of monoidal DG categories and bimodules. Any functor A : Aff → Alg bimod (DGCat) gives rise, by taking modules, to a theory of sheaves of categories ShvCat A .In this paper, we study ShvCat H . Vaguely speaking, this theory categorifies the theory of D-modules, in the same way as Gaitsgory's original ShvCat categorifies the theory of quasi-coherent sheaves. We develop the functoriality of ShvCat H , its descent properties and the notion of H-affineness.We then prove the H-affineness of algebraic stacks: for Y a stack satisfying some mild conditions, the ∞-category ShvCat H (Y) is equivalent to the ∞-category of modules for H(Y), the monoidal DG category of higher differential operators. The main consequence, for Y quasi-smooth, is the following: if C is a DG category acted on by H(Y), then C admits a theory of singular support in Sing(Y), where Sing(Y) is the space of singularities of Y.As an application to the geometric Langlands program, we indicate how derived Satake yields an action of H(LSǦ) on D(Bun G ), thereby equipping objects of D(Bun G ) with singular support in Sing(LSǦ). 1.1.2. Tannaka duality. For Y an algebraic stack satisfying mild conditions, Tannaka duality ([Lur18, Chapter 9]) allows to "recover" Y from the symmetric monoidal DG category QCoh(Y). On the other hand, the DG algebra H * (Y, O Y ) does not recover Y, unless Y is an affine DG scheme.

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“…Beraldo [Ber19] pushes this further and views Sph G as an E 3 -monoidal category, using the 3-dimensional pair of pants, so the Hochschild cochains will have the structure of an E 4 -algebra. Beraldo then computes the factorization homology of Sph G on a d-manifold M , obtaining an E 3−d -monoidal category that he identifies with the category H(LS Betti G (M )) defined in [Ber21].…”

Section: Introductionmentioning

confidence: 99%

Higher Deformation Quantization for Kapustin-Witten Theories

Elliott¹,

Gwilliam²,

Williams³

2021

Preprint

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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 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 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 à la Lurie and Scheimbauer. Contents 1. Introduction 2.1. Classical BV Formalism 2.2. Quantum BV formalism 2.3. Equivariant BV formalism 3. Twists of 4d N = 4 Super Yang-Mills Theory 3.1. Summary of Twists 3.2. Anomaly Vanishing 4. Perturbation Theory Around a Vacuum 4.1. Moduli of Vacua 4.2. Families of Classical Field Theories over [g * /G] 4.3. Families of Quantum Field Theories over [g * /G]

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The topological chiral homology of the spherical category

Cited by 11 publications

References 23 publications

Secondary Products in Supersymmetric Field Theory

Secondary Products in Supersymmetric Field Theory

Sheaves of categories with local actions of Hochschild cochains

Higher Deformation Quantization for Kapustin-Witten Theories

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