I am a fifth year graduate student at Northwestern University. 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.

Brian Williams. Forthcoming. The Virasoro vertex algebra and factorization algebras on Riemann surface, Letters in Mathematical Physics.

Brian Williams, Owen Gwilliam, and Vassily Gorbounov. Submitted. Chiral differential operators via Batalin-Vilkovisky quantization.

Brian Williams, Chris Elliott, and Philsang Yoo. Submitted. Asymptotic freedom in the BV-formalism.

- 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, 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.