Owen Gwilliam

I am an assistant professor in mathematics at the University of Massachusetts, Amherst. In summer 2018 I finished four years as a postdoctoral fellow at the Max Planck Institute for Mathematics in Bonn.

Research

My research revolves around quantum field theory, focusing both on applications of homotopical ideas to QFT itself and applications of QFT to geometry and representation theory. My books and papers are described below.

• Inspired by the work of Beilinson-Drinfeld and Francis-Gaitsgory-Lurie on factorization algebras, Kevin Costello and I have developed a version of factorization algebras appropriate to perturbative QFT, building upon Costello's earlier work developing a renormalization machine. We have proved a kind of deformation quantization theorem for field theory, as well as a factorization refinement of the Noether theorem. We develop these ideas in a two-volume book ``Factorization Algebras in Quantum Field Theory'' to be published by Cambridge University Press, with the first volume currently available. The first volume introduces factorization algebras and develops examples, primarily from free field theories. (This is not as boring as it might sound: we recover vertex algebras in complex dimension 1 and a quantum group for abelian Chern-Simons theory in dimension 3.) The second volume develops interacting classical and quantum field theory using the Batalin-Vilkovisky formalism and proves the deformation quantization and Noether theorems. (We are editing it for publication in 2018.)

• My thesis shows how these ideas work in several simple contexts. The thesis has a few distinct pieces. First, I provide an expository introduction to the Batalin-Vilkovisky formalism. Second, I prove that the BV formalism provides a determinantal functor on perfect complexes. Third, I show by example how our factorization algebra procedure recovers vertex algebras, notably free bosons, free fermions, and affine Kac-Moody vertex algebras. Finally, I prove an index theorem based on our techniques. Some of my recent work is about extending and enhancing these results.

• In Duals and adjoints in higher Morita categories, Claudia Scheimbauer and I explored higher analogs of the Morita (bi)category that are of interest from the perspective of functorial field theories as well as factorization algebras. In a companion paper we use this machinery to explore a version of relative TFTs, focusing on low-dimensional examples where the consequences for algebra and category theory are easy to appreciate.

• In Chiral differential operators via Batalin-Vilkovisky quantization, written with Vassily Gorbounov and Brian Williams, we construct the curved βγ system using a combination of the BV formalism and Gelfand-Kazhdan formal geometry, modified to work with factorization algebras. We then show that the associated vertex algebra is the chiral differential operators. Our methods use BV techniques to realize mathematically the physical arguments given by Witten and Nekrasov recovering CDOs from the βγ system.

• Expanding on from the CDO project, Brian Williams and I wrote an exposition about the BV quantization of the holomorphic string with linear target space. It has a version of the dimensional anomaly and recovers the chiral sector of the standard bosonic string. We still need to write up the case of a curved target.

• With Kasia Rejzner, I have begun to explore the relationship between perturbative AQFT, particularly in conjunction with the BV formalism, and the BV/factorization package that I've developed with Costello. To get our bearings, we examined the case of the free scalar field in some detail.

• With Rune Haugseng, I carefully studied linear BV quantization and formulate it as a functor of infinity-categories and then as a map of derived stacks in the imaginatively-named Linear BV quantization as a functor of infinity-categories.

• With Dmitri Pavlov, I reexamined filtered derived categories from the perspective of model categories and infinity-categories, with an eye towards future work on D-modules and factorization algebras, in Enhancing the filtered derived category.

• With Ryan Grady, I have pursued several projects touching on derived geometry and QFT. In One-dimensional Chern-Simons Theory and the A-hat genus, we constructed a one-dimensional TFT whose partition function recovers the A-hat genus of a manifold. In L-infinity spaces and derived loop spaces, we clarified and further developed Costello's approach to derived geometry via L-infinity spaces. Finally, in Lie algebroids as L-infinity spaces, we showed that Lie algebroids (and associated constructions like representations up to homotopy) fit naturally into this version of derived geometry.

There are several other projects at various stages of development. If you'd like to know more, feel free to contact me.

Notes

• With Theo Johnson-Freyd, I wrote an expository introduction to the BV formalism, emphasizing how one would rediscover Feynman diagrams for purely homological reasons.

• Based on a lecture by Costello, I wrote up an explanation for how the βγ system arises from the usual two-dimensional sigma model. To be more precise, after rewriting the sigma model in the first-order formalism, one can look at scaling the metric on the target to infinity and then take the chiral sector.

• For an MPIM seminar on topological insulators, I made slides introducing some basic, relevant ideas for a mathematical audience. It might be helpful to others wanting to see a quick overview, but beware that in this seminar I was the blind leading the blind.

• I made slides for three talks in my last year at MPIM:
  • A survey of the BV/factorization package at a conference on mathematical QFT at York in September 2017
  • A discussion of Chern-Simons theory in the style of pAQFT for Fredenhagen's birthday conference in December 2017
  • A talk about a construction of quantum groups from perturbative quantization of Chern-Simons theory for the Fachbeirat in June 2018. It discusses how the stratified factorization algebras of Ayala-Francis-Tanaka and the higher Morita categories of Calaque-Scheimbauer and Haugseng provide a powerful language for describing topological defects of TFTs.

• I gave a lecture series in November 2017 at the Hausdorff Institute on my joint work with John Francis and Kevin Costello about how perturbative Chern-Simons theory recovers quantum groups. If you're interested in seeing video, please contact me.

Activities

With Claudia Scheimbauer, I taught a course on derived deformation theory and Koszul duality at the University of Bonn in 2016. Here is our website.

I've also been lucky to be an organizer of several conferences (at MPIM, the Simons Center, Banff, IBS-CGP, and Oberwolfach).