arXiv subjects

Owen Gwilliam

Publications and source records attributed to Owen Gwilliam.

At least 19 recordsLinked to original sources

arXiv2026

Duality of generalized Maxwell theories as an equivalence in derived geometry

We propose a non-perturbative description of the moduli spaces encoding p-form generalized Maxwell theories in any dimension, using derived differential geometry. Our approach synthesizes the Batalin--Vilkovisky formalism with differential cohomology. Within this framework we formulate Dirac charge quantization and show how such charge-quantized moduli spaces exhibit abelian duality between generalized Maxwell theories of different types. We also describe the compactification of generalized Maxwell theories along closed Riemannian manifolds by computing the pushforward of the underlying sheaves of cochain complexes that model differential cohomology.

math-ph↗

arXiv2026

Chern-Simons factorization algebras and knot polynomials

This work identifies the Reshetikhin-Turaev invariant of links in terms of a trace map on factorization homology. In particular, to recover the knot invariants associated to Chern-Simons theories, we construct a filtered $\mathcal{E}_3$-algebra $\mathcal{A}^\lambda$ by BV quantization of Chern-Simons theory for a semi-simple Lie algebra ${\frak g}$ with invariant pairing~$\lambda$, and we prove that a finite-dimensional representation $V$ of the Drinfeld-Jimbo quantum group $U_\hbar{\frak g}$ defines a perfect $\mathcal{A}^\lambda$ module~$\mathcal{V}$. For any framed link $K$ in $\mathbb{R}^3$, we then prove that there is an equality \[\int_{K\subset\mathbb{R}^3}{\rm tr}(V) = Z_V(K\subset\mathbb{R}^3) \] between the factorization homology trace for $V$ and the Reshetikhin-Turaev link invariant determined by~$V$.

math.QA↗

arXiv2026

Multivariate Multicycle Codes for Complete Single-Shot Decoding

We introduce multivariate multicycle (MM) codes, a new family of quantum error-correcting codes (QECCs) that unifies bivariate bicycle, multivariate bicycle, abelian two-block group algebra, generalized bicycle, trivariate tricycle, and toric codes. MM codes are Calderbank-Shor-Steane (CSS) codes defined from length-$\textit{t+1}$ chain complexes with $\textit{$t \ge 4$}$. The chief advantage of these codes is that they possess metachecks and high confinement that permit complete single-shot decoding. We offer a framework that facilitates the construction of long-length chain complexes through the use of Koszul complex. In particular, obtaining explicit boundary maps (parity check and metacheck matrices) is particularly straightforward in our approach. This simple but very general parameterization of codes permitted us to efficiently perform a numerical search, where we identify several MM code candidates that demonstrate these capabilities at high rates and high code distances. Examples of new codes with parameters $[[n,k,d]]$ include $[[96, 12, 8]]$, $[[144, 12, 12]]$, $[[216, 12, 14]]$, $[[288, 12, 16]]$, $[[324, 12, 20]]$, $[[432, 12, 27]]$, $[[486, 24, 12]]$, $[[630, 70, 9]]$, and $[[648, 18, 23]]$. Notably, our codes achieve confinement profiles that surpass all known single-shot-decodable quantum CSS codes of practical blocksize. Our codes are also the first explicit instances of collapsed 5D through 9D higher dimensional QECCs, with check weights significantly lower than those of recent small instances of quantum Tanner codes.

quant-ph↗

arXiv2025

Holomorphic field theories and higher algebra

Aimed at complex geometers and representation theorists, this survey explores higher dimensional analogues of the rich interplay between Riemann surfaces, Virasoro and Kac-Moody Lie algebras, and conformal blocks. We introduce a panoply of examples from physics -- field theories that are holomorphic in nature, such as holomorphic Chern-Simons theory -- and interpret them as (derived) moduli spaces in complex geometry; no comfort with physics is presumed. We then describe frameworks for quantizing such moduli spaces, offering a systematic generalization of vertex algebras and conformal blocks via factorization algebras, and we explain how holomorphic field theories generate examples of these higher algebraic structures. We finish by describing how the conjecture of Seiberg duality predicts a surprising relationship between holomorphic gauge theories on algebraic surfaces and how it suggests analogues of the Hori-Tong dualities already studied by algebraic geometers.

math.AG↗

arXiv2025

Remarks on the locality of generalized global symmetries

We examine generalized global symmetries as a kind of compactly supported cohomology, and so are led to revisit questions about the locality of quantum field theory, following Segal. Physics naturally suggests a generalization of factorization algebras, aimed at capturing nonperturbative information, and we explain how higher group symmetries offer examples of this generalization, providing an extension of the nonabelian Poincar\'e duality of Salvatore and Lurie. Finally, we explore how continuous generalized symmetries and anomalies can be cast in this framework.

math-ph↗

arXiv2024

Twists of superconformal algebras

We take first steps toward a theory of ``conformal twists'' for superconformal field theories in dimension 3 to 6, extending the well-known analysis of twists for supersymmetric theories. A conformal twist is a square-zero odd element in the superconformal Lie algebra, and we classify all twists and describe their orbits under the adjoint action of the superconformal group. We work mostly with the complexified superconformal algebras, unless explicitly stated otherwise; real forms of the superconformal algebra may have important physical implications, but we only discuss these subtleties in a few special cases. Conformal twists can give rise to interesting subalgebras and protected sectors of operators in a superconformal field theory, with the Donaldson--Witten topological field theory and the vertex operator algebras of 4-dimensional N=2 SCFTs being prominent examples. To obtain mathematical precision, we explain how to extract vertex algebras and E_n algebras from a twisted superconformal field theory using factorization algebras.

math-ph↗

arXiv2023

The observables of a perturbative algebraic quantum field theory form a factorization algebra

We demonstrate that perturbative algebraic QFT methods, as developed by Fredenhagen and Rejzner, naturally yields a factorization algebras of observables for a large class of Lorentzian theories. Along the way we carefully articulate cochain-level refinements of multilocal functionals, building upon results about the variational bicomplex, and we lift existing results about Epstein-Glaser renormalization to these multilocal differential forms, results which may be of independent interest.

math-ph↗

arXiv2023

Factorization algebra

Factorization algebras are local-to-global objects living on manifolds, and they arise naturally in mathematics and physics. Their local structure encompasses examples like associative algebras and vertex algebras; in these examples, their global structure encompasses Hochschild homology and conformal blocks. In the setting of quantum field theory, factorization algebras articulate a minimal set of axioms satisfied by the observables of a theory, and they capture concepts like the operator product and correlation functions. In this survey article for the Encyclopedia of Mathematical Physics, 2nd Edition, we give the definitions and key examples, compare this approach with other approaches to mathematically formalizing field theory, describe key results, and explain how higher symmetries can be encoded in this framework.

math-ph↗

arXiv2022

Defects via Factorization Algebras

We provide a mathematical formulation of the idea of a defect for a field theory, in terms of the factorization algebra of observables and using the BV formalism. Our approach follows a well-known ansatz identifying a defect as a boundary condition along the boundary of a blow-up, but it uses recent work of Butson-Yoo and Rabinovich on boundary conditions and their associated factorization algebras to implement the ansatz. We describe how a range of natural examples of defects fits into our framework.

math-ph↗

arXiv2022

A homological approach to the Gaussian Unitary Ensemble

We study the Gaussian Unitary Ensemble (GUE) using noncommutative geometry and the homological framework of the Batalin-Vilkovisky (BV) formalism. Coefficients of the correlation functions in the GUE with respect to the rank $N$ are described in terms of ribbon graph Feynman diagrams that then lead to a counting problem for the corresponding surfaces. The canonical relations provided by this homological setup determine a recurrence relation for these correlation functions. Using this recurrence relation and properties of the Catalan numbers, we determine the leading order behavior of the correlation functions with respect to the rank $N$. As an application, we prove a generalization of Wigner's semicircle law and compute all the large $N$ statistical correlations for the family of random variables in the GUE defined by multi-trace functions.

math-ph↗

arXiv2022

Framed $\mathbb E_n$-Algebras from Quantum Field Theory

This paper addresses the following question: given a topological quantum field theory on $\mathbb R^n$ built from an action functional, when is it possible to globalize the theory so that it makes sense on an arbitrary smooth oriented $n$-manifold? We study a broad class of topological field theories -- those of AKSZ type -- and obtain an explicit condition for the vanishing of the framing anomaly, i.e., the obstruction to performing this globalization procedure. We also interpret our results in terms of identifying the observables as an algebra over the framed little $n$-disks operad. Our analysis uses the BV formalism for perturbative field theory and the notion of factorization homology.

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↗

arXiv2021

Large $N$ phenomena and quantization of the Loday-Quillen-Tsygan theorem

We offer a new approach to large $N$ limits using the Batalin-Vilkovisky formalism, both commutative and noncommutative, and we exhibit how the Loday-Quillen-Tsygan Theorem admits BV quantizations in that setting. Matrix integrals offer a key example: we demonstrate how this formalism leads to a recurrence relation that in principle allows us to compute all multi-point correlation functions. We also explain how the Harer-Zagier relations may be expressed in terms of this noncommutative geometry derived from the BV formalism. As another application, we consider the problem of quantization in the large $N$ limit and demonstrate how the Loday-Quillen-Tsygan Theorem leads us to a solution in terms of noncommutative geometry. These constructions are relevant to open topological field theories and string field theory, providing a mechanism that relates moduli of categories of branes to moduli of brane gauge theories.

math.QA↗

arXiv2021

Quantization of topological-holomorphic field theories: local aspects

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.

math-ph↗

arXiv2021

Factorization algebras and abelian CS/WZW-type correspondences

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.

math.QA↗

arXiv2020

Spontaneous symmetry breaking: a view from derived geometry

We examine symmetry breaking in field theory within the framework of derived geometry, as applied to field theory via the Batalin-Vilkovisky formalism. Our emphasis is on the standard examples of Ginzburg-Landau and Yang-Mills-Higgs theories and is primarily interpretive. The rich, sophisticated language of derived geometry captures the physical story elegantly, allowing for sharp formulations of slogans (e.g., for the Higgs mechanism, that the unstable ghosts eat the Goldstone modes). Rewriting these results in the BV formalism provides, as one nice payoff, a reformulation of 't Hooft's family of gauge-fixing conditions for spontaneously broken gauge theory that behaves well in the $\xi \to \infty$ limit.

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

A one-loop exact quantization of Chern-Simons theory

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.

math-ph↗