arXiv subjects

Brian R Williams

Publications and source records attributed to Brian R Williams.

6 recordsLinked to original sources

arXiv2024

The local cohomology of vector fields

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.

math.DG↗

arXiv2024

A complex geometric perspective on a,c anomalies

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.

math-ph↗

arXiv2021

Higher Deformation Quantization for Kapustin-Witten Theories

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 à la Lurie and Scheimbauer.

math-ph↗

arXiv2020

Chiral differential operators via Batalin-Vilkovisky quantization

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.

math.QA↗

arXiv2020

Holomorphic Poisson Field Theories

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.

math-ph↗

arXiv2017

The Virasoro vertex algebra and factorization algebras on Riemann surfaces

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.

math.QA↗