About

Owen Gwilliam is an associate professor in mathematics at the University of Massachusetts, Amherst. Before joining UMass in the fall of 2018, he spent four years as a postdoctoral fellow at the Max Planck Institute for Mathematics in Bonn. His academic journey also includes an NSF postdoctoral position at UC Berkeley following graduate school at Northwestern. His research revolves around quantum field theory, focusing on applications of homotopical ideas to QFT and the use of QFT to explore geometry and representation theory. Gwilliam's work includes developing machinery to manipulate factorization algebras and investigating issues in nonperturbative quantum field theory. He has co-authored a two-volume book with Kevin Costello on how the observables of perturbative QFT are encoded in factorization algebras on spacetime, building upon Costello's earlier work on renormalization. His research also involves proving deformation quantization theorems for field theory and a factorization refinement of the Noether theorem. Gwilliam actively contributes to the academic community through organizing summer schools on physical mathematics, supervising graduate students and REU groups, and engaging in expository and research activities related to the Batalin-Vilkovisky formalism, free field theories, and index theorems.

Research topics

• Mathematical physics
• Pure mathematics
• Theoretical physics
• Geometry
• Mathematics
• Physics

Selected publications

• Holomorphic field theories and higher algebra

  Bulletin of the London Mathematical Society · 2025-07-21

  articleOpen access1st authorCorresponding

  Abstract Aimed at complex geometers and representation theorists, this survey explores higher dimensional analogs 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 analogs of the Hori–Tong dualities already studied by algebraic geometers.

  PublisherDOI

• “Factorization Algebra” for Encyclopedia of Mathematical Physics

  Encyclopedia of Mathematical Physics · 2024-10-03

  book-chapterSenior author
  PublisherDOI

• Quantization of topological-holomorphic field theories: local aspects

  Communications in Analysis and Geometry · 2024-01-01 · 1 citations

  article1st authorCorresponding
  PublisherDOI

• Twists of superconformal algebras

  arXiv (Cornell University) · 2024-03-28

  preprintOpen access

  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.

  PublisherOA PDFDOI

• Higher Deformation Quantization for Kapustin–Witten Theories

  Annales Henri Poincaré · 2024-02-16 · 3 citations

  article
  PublisherDOI

• Factorization algebra

  arXiv (Cornell University) · 2023-10-09

  preprintOpen accessSenior author

  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.

  PublisherOA PDFDOI

• Defects via factorization algebras

  Letters in Mathematical Physics · 2023-04-13 · 1 citations

  articleSenior author
  PublisherDOI

• A homological approach to the Gaussian Unitary Ensemble

  arXiv (Cornell University) · 2022-06-09 · 1 citations

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

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

  arXiv (Cornell University) · 2022-12-15 · 2 citations

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

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

  Advances in Mathematics · 2022-08-19 · 4 citations

  articleOpen access
  PublisherOA PDFDOI

Recent grants

• PostDoctoral Research Fellowship

  NSF · $150k · 2012–2016

• CAREER: Factorization Algebras in Quantum Field Theory

  NSF · $433k · 2021–2027

• Collaborative Research: Derived Differential Geometry and Field Theory

  NSF · $99k · 2018–2022

Frequent coauthors

• Kevin Costello

  33 shared

• Brian R. Williams

  Boston University

  18 shared

• Ryan Grady

  7 shared

• Chris Elliott

  Amherst College

  6 shared

• Kasia Rejzner

  5 shared

• Grégory Ginot

  Université Sorbonne Paris Nord

  4 shared

• Eugene Rabinovich

  University of Notre Dame

  4 shared

• Mahmoud Zeinalian

  Lehman College

  4 shared