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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


boardpic 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.