Brian R. Williams

I am a graduate student at Northwestern University in my final year the PhD program. I am interested in homotopical ideas in quantum field theory. More specifically, I study BV-quantization in the context of perturbative QFT with an emphasis on higher dimensional holomorphic field theories. Here is my CV.

Papers

The holomorphic bosonic string. With Owen Gwilliam.

This is the first part of a series of two papers where we construct the holomorphic sector of the bosonic string. In this part, we produce a one loop exact quantization for the bosonic string propogating in flat space. We find the the Weyl anomaly cancellation condition, and show how the factorization algebra recovers BRST cohomology.

Homotopy RG flow and the non-linear $\sigma$-model. With Ryan Grady.

The purpose of this note is to give a mathematical treatment to the low energy effective theory of the two-dimensional sigma model. Perhaps surprisingly, our low energy effective theory encodes much of the topology and geometry of the target manifold. In particular, we relate the beta-function of our theory to the Ricci curvature of the target, recovering the physical result of Friedan.

The Virasoro vertex algebra and factorization algebras on Riemann surface. Letters in Mathematical Physics. Published, awaiting volume identifier. doi:10.1007/s11005-017-0982-7.

We construct the Virasoro factorization algebra on an arbitrary Riemann surface. Locally, we show that this factorization algebra recovers the ordinary Virasoro vertex algebra, and that the factorization homology coincides with the conformal blocks. We exhibit an application of the quantum Noether theorem" to obtain the free field realization of the Virasoro factorization algebra in the beta-gamma factorization algebra.

Chiral differential operators via Batalin-Vilkovisky quantization. With Vassily Gorbounov and Owen Gwilliam. Submitted.

We show that the local observables of the curved beta gamma system encode the sheaf of chiral differential operators using the machinery in the book Factorization algebras in quantum field theory", by Kevin Costello and Owen Gwilliam. Our approach is in the spirit of Gelfand-Kazhdan formal geometry.

Asymptotic freedom in the BV-formalism. With Chris Elliott and Philsang Yoo. Journal of Geometry and Physics. Published, awaiting volume identifier. doi:10.1016/j.geomphys.2017.08.009.

We define the beta-function of a perturbative quantum field theory in the mathematical framework introduced by Costello – combining perturbative renormalization and the BV formalism – as the cohomology class of a certain functional measuring scale dependence of the effective interaction. We show that the one-loop beta-function is a well-defined element of the obstruction-deformation complex for translation-invariant and classically scale-invariant theories, and furthermore that it is locally constant as a function on the space of classical interactions and computable as a rescaling anomaly, or as the logarithmic one-loop counterterm. We compute the one-loop beta-function in first-order Yang–Mills theory, recovering the famous asymptotic freedom for Yang–Mills in a mathematical context.

Projects
• Symmetries of holomorphic field theories.

The approach to studying QFT via the BV-formalism developed by Costello-Gwilliam has great potential to understand, as well as classify, local symmetries of field theory. In general, one expects a local Lie algebra of symmetries to quantize to a symmetry by a certain factorization algebra (this is an instance of Noethers theorem written in a fancy way). Moreover, as Gwilliam points out in his thesis, anomalies to quantizing certain symmetries computed in this formalism should lead to a (family) index theorem over the classifying space of the Lie algebra of symmetries. I am studying some examples of this related to holomorphic theories which come from twists of supersymmetric field theories in various dimensions. The results are obtained in a local-to-global" way by considering the global sections of the factorization algebra of observables of the QFT. For the case of gauge theory this factorization algebra is related to a higher dimensional version of the Kac-Moody vertex algebra. In the case that the theory is diffeomorphism invariant the factorization algebre encoding the symmetries is related to a higher analogs of the Virasoro vertex algebra.

• Higher dimensional chiral differential operators.

Starting from the elliptic moduli problem of holomorphic maps from a complex d-fold to another complex manifold, one can construct a classical field theory which is a generalization of the holomorphic sigma model in complex dimension d = 1. I study the quantization of such a theory for arbitrary target and obtain a factorization algebra of higher dimensional chiral differential operators" that exists on any complex manifold whose (d+1)st component of the Chern character vanishes. I also study the theory over the moduli of complex structures on the source, which leads to a higher dimensional version of the holomorphic string, and hence deserves to be called higher holomorphic gravity".

• A Yangian action on CDO's

This is joint with Owen Gwilliam. Costello has shown how the Yangian quantum group appears from the observables of a certain four dimensional supersymmetric gauge theory, which we will hence call the Yangian theory". We show how to couple a certain two dimensional chiral defect to the bulk Yangian theory. This defect can be thought of as the holomorphic sigma model whose target is a complex manifold that has an action by holomorphic vector fields by the Lie algebra the Yangian theory is based on. We show that the observables on the defect are equivalent to an equivariant version of chiral differential operators. We then study the quantum coupling with the bulk Yangian theory. One consequence of this, which comes from studying line operators of the 4d theory, is an embedding of the K-theory of the Yangian into the Fourier modes of the sheaf of equivariant CDOs.

Miscellaneous notes

The syllabus for my qualifying topic on Sullivan's approach to rational homotopy theory.

Notes from a lecture series I gave on Observables of QFT in the BV-formalism" with Ryan Grady and Si Li at the conference "Factorization Algebras and Functorial Field Theory" at Oberwolfach. May 2016.

Here is a video of a lecture I gave at Perimiter Institute in Waterloo, Canada. Here is another video of a lecture I gave at the IBS Center for Geometry and Physics in Pohang, South Korea.