Brian R. Williams

My research utilizes tools in geometry and homotopical algebra to uncover mathematical structures in quantum field theory and string theory.
Here is my CV.

  • Website for MA 721 (Differential topology I, fall 2023): here.
  • Website for New England Algebraic Topology and MAthematical Physics Seminar (November 11-12, 2023): here.
  • Website for topics class on Hilbert schemes and Nakajima quiver varieties (spring 2023): here.
  • Website for QFT reading seminar (Fall 2022): here.
  • Website for class on complex geometry: here.
  • Website for seminar on twisted holography and Koszul duality (fall 2021): here.
  • Website for twisted QFT seminar (organized with Tudor Dimofte, spring 2021): here.
  • Website for RTG seminar on Factorization homology of surfaces and quantum groups (fall 2019): here. (organized with Pavel Etingof and Valerio Toledano--Laredo.)
  • Website for RTG seminar on the Yangian and four-dimensional gauge theory (fall 2018): here. (This seminar took place at MIT and Northeastern University and was organized jointly with Pavel Etingof and Valerio Toledano--Laredo.)
  • This is a link to a list of old courses that I have taught.

  • Here is a list of (published or submitted) papers:
  • Enhanced symmetries in minimally twisted three-dimensional supersymmetric theories, pdf. With Niklas Garner and Surya Raghavendran. Submitted for publication. +
  • We show that the action of residual supersymmetries in holomorphic-topological twists of 𝒩 = 2 theories in three dimensions naturally extends to the action of certain infinite dimensional Lie superalgebras. We demonstrate this in a range of examples, including 𝒩 = 4 Yang-Mills theories and superconformal Chern-Simons theories, describing how the symmetries are implemented at the level of local operators.

  • Higgs and Coulomb branches from superconformal raviolo vertex algebras, pdf. With Niklas Garner and Surya Raghavendran. Submitted for publication. +
  • We propose a method for extracting the Higgs and Coulomb branches of a three-dimensional 𝒩 = 4 quantum field theory from the algebra of local operators in its holomorphic-topological twist using the formalism of raviolo vertex algebras. Our construction parallels that of the chiral ring and twisted chiral ring of an N = 2 superconformal vertex operator algebra.

  • Raviolo vertex algebras, pdf. With Niklas Garner. Submitted for publication. +
  • We develop the algebraic underpinnings for local operators in a partially holomorphic, partially topological three-dimensional quantum field theory. The term raviolo refers to the the configuration space of two points in three space (which is topologically a two-sphere) with its transversely holomorphic foliated structure.

  • Semi-chiral operators in 4d 𝒩 = 1 gauge theories, pdf. With Budzik, Gaiotto, Kulp, Wu, and Yu. Submitted for publication. +
  • We initiate the study of loop corrections to local operators in the holomorphic twist of four-dimensional supersymmetric gauge theory. Some corrections to indices and local operators of non-perturbative nature are also analyzed in the cases of SU(2) and SU(3) gauge theory.

  • Twisted eleven-dimensional supergravity, pdf. With Surya Raghavendran and Ingmar Saberi. To appear in Comm. Math. Phys. +
  • We construct an eleven-dimensional theory defined on products of Calabi--Yau fivefolds with one-manifolds whose free limit matches with that of the SU(5) twist of eleven-dimensional supergravity. We prove a number of results justifying that this is true at the fully interacting perturbative level. Surprisingly, on flat space there is an enhanced symmetry of this model by a central extension of the exceptional super Lie algebra E(5|10).

  • Superconformal algebras and holomorphic field theories, pdf. With Ingmar Saberi. Ann. Henri Poincaré 24 (2023), no. 2, 541–604. +
  • From the perspective of holomorphic QFT we provide a mathematical approach to the construction of vertex algebras from 4d 𝒩 = 2 superconformal theories using factorization algebras.

  • Constraints in the BV formalism: six-dimensional supersymmetry and its twists, pdf. With Ingmar Saberi. Adv. Math. 412 (2023), Paper No. 108806, 87 pp. +
  • The theory of constraints in the BV formalism is developed with an eye towards the ubiquitous 6d 𝒩 = (2,0) superconformal theory. We then describe the holomorphic and further topological twists of the abelian theory.

  • Quantization of topological-holomorphic field theories: local aspects, pdf. With Owen Gwilliam and Eugene Rabinovich. To appear in Communications in Analysis and Geometry. +
  • We construct perturbative, one-loop quantizations for mixed topological-holomorphic theories rigorously on foliated affine space and find a remarkable vanishing result about anomalies: the one-loop obstruction to quantization vanishes when the number of topological directions is greater than one.

  • Koszul duality in quantum field theory, pdf. With Natalie Paquette. Confluentes Math. 14 (2022), no. 2, 87–138. +
  • We review the appearance of Koszul duality in the physical problem of coupling QFTs to topological line defects, and illustrate the concept with some examples drawn from twists of various simple supersymmetric theories. The aim is to provide an elementary introduction for those interested in the appearance of Koszul duality in supersymmetric gauge theories with line defects and, ultimately, its generalizations to higher-dimensional defects and twisted holography.

  • Factorization algebras and abelian CS/WZW-type correspondences, pdf. With Owen Gwilliam and Eugene Rabinovich. Pure Appl. Math. Q. 18 (2022), no. 4, 1485–1553. +
  • We develop a method of quantization for free field theories on manifolds with boundary where the bulk theory is topological in the direction normal to the boundary and a local boundary condition is imposed. At the level of observables, the construction produces a stratified factorization algebra. The factorization algebra on the boundary stratum enjoys a perturbative bulk-boundary correspondence with this bulk factorization algebra. A central example is the factorization algebra version of the abelian Chern–Simons/Wess–Zumino–Witten correspondence, but we examine higher dimensional generalizations that are related to holomorphic truncations of string theory andM-theory and involve intermediate Jacobians.

  • A taxonomy of twists of supersymmetric Yang-Mills theory, pdf. With Chris Elliott and Pavel Safronov. Selecta Math. (N.S.) 28 (2022), no. 4, Paper No. 73, 124 pp. +
  • We give a complete classification of twists of supersymmetric Yang–Mills theories in dimensions 2 through 10. We formulate supersymmetric Yang–Mills theory classically using the BV formalism, and then we construct an action of the supersymmetry algebra using the language of L-infinity algebras. For each orbit in the space of square-zero supercharges in the supersymmetry algebra, under the action of the spin group and the group of R-symmetries, we give a description of the corresponding twisted theory. These twists can be described in terms of mixed holomorphic-topological versions of Chern–Simons and BF theory.

  • Perspectives on the pure spinor superfield formalism , pdf. With Eager, Hahner, and Saberi. J. Geom. Phys. 180 (2022), Paper No. 104626, 47 pp. +
  • In this note, we study, formalize, and generalize the pure spinor superfield formalism from a rather nontraditional perspective. We show how the pure spinor superfield formalism can be viewed as constructing a supermultiplet out of the input datum of an equivariant graded module over the ring of functions on the nilpotence variety. We use homotopy transfer to relate these multiplets to more standard component-field formulations. Physical properties of the resulting multiplets can then be understood in terms of algebrogeometric properties of the nilpotence variety.

  • Holomorphic Poisson field theories, pdf. With Chris Elliott. High. Struct. 5 (2021), no. 1, 282–309. +
  • We construct a class of QFTs depending on the data of a holomorphic Poisson structure. The main technical tool relies on a characterization of deformations and anomalies of such theories in terms of the Gelfand-Fuchs cohomology of formal Hamitlonian vector fields. In the symplectic case such theories are topological in a weak sense, which we refer to as `de Rham topological'. While infinitesimal translations act homotopically trivially, we show that the space of observables of such a theory does not define an En algebra. Additionally, we will highlight a conjectural relationship to theories of supergravity in four and five dimensions.

  • Higher Kac-Moody algebras and symmetries of holomorphic field theories, pdf. With Owen Gwilliam. Adv. Theor. Math. Phys. 25 (2021), no. 1, 129–239. +
  • We introduce a higher dimensional generalization of the affine Kac-Moody algebra using the language of factorization algebras. In particular, on any complex manifold there is a factorization algebra of `currents' associated to any Lie algebra. We classify local cocycles of these current algebras, and compare them to central extensions of higher affine algebras proposed by Faonte-Hennion-Kapranov. A central goal of this paper is to witness higher Kac-Moody algebras as symmetries of a class of holomorphic quantum field theories. In particular, we prove a generalization of the free field realization of an affine Kac-Moody algebra and also develop the theory of q-characters for this class of algebras in terms of factorization homology. Finally, we exhibit the `large N' behavior of higher Kac-Moody algebras and their relationship to symmetries of non-commutative field theories.

  • Toroidal prefactorization algebras associated to holomorphic fibrations and a relationship to vertex algebras, pdf. With Matt Szczesny and Jackson Walters. Adv. Math. 386 (2021), Paper No. 107799, 42 pp. +
  • This paper is devoted to developing a functorial treatment of toroidal vertex algebras and their geometric realizations via factorization algebras

  • Chiral differential operators via quantization of the holomorphic sigma model, pdf. With Owen Gwilliam and Vassily Gorbounov. Astérisque(2020), no. 419, ix + 210 pp. ISBN: 978-2-85629-920-3 +
  • The curved beta-gamma system is a nonlinear sigma model with a Riemann surface as the source and a complex manifold as the target. Its classical solutions pick out the holomorphic maps. Physical arguments identify its algebra of operators with a vertex algebra known as the chiral differential operators (CDOs). We verify these claims mathematically by constructing and quantizing rigorously this system using machinery developed by Kevin Costello and the second author, which combine renormalization, the Batalin-Vilkovisky formalism, and factorization algebras. Furthermore, we find that the factorization algebra of quantum observables of the curved beta-gamma system encodes the sheaf of chiral differential operators. In this sense our approach provides deformation quantization for vertex algebras. As in many approaches to deformation quantization, a key role is played by Gelfand-Kazhdan formal geometry.

  • Twisted characters and holomorphic symmetries, pdf. With Ingmar Saberi. Lett. Math. Phys. 110 (2020), no. 10, 2779–2853. +
  • We develop the idea of a local character for an arbitrary holomorphic field theory and compare it to the supersymmetric Witten index for four-dimensional supersymmetric gauge theories.

  • Renormalization for holomorphic field theories, pdf. Comm. Math. Phys. 374 (2020), no. 3, 1693–1742. +
  • The concept of a holomorphic field theory is defined in a version of the Batalin-Vilkovisky formalism. On this basis, the quantization and renormalization of holomorphic field theories are explored in a theory-independent way. The main result is that holomorphic field theories have trivial RG flow, and are finite, to first order in perturbation theory.

  • The holomorphic bosonic string, pdf. With Owen Gwilliam. American Mathematical Society, Providence, RI, 2018, 213–258. ISBN: 978-1-4704-4243-9 +
  • We analyze a holomorphic version of the bosonic string in the formalism of quantum field theory developed by Costello and collaborators, which provides a powerful combination of renormalization theory and the Batalin-Vilkovisky formalism. Our focus here is on the case in which the target space-time is a vector space. We identify the critical dimension as an obstruction to satisfying the quantum master equation, and when the obstruction vanishes, we construct a one-loop exact quantization. Moreover, we show how the factorization algebra of observables recovers the BRST cohomology of the string and use this perspective to give a new construction of its Gerstenhaber structure. Finally, we show how the factorization homology along closed manifolds encodes the determinant line bundle over the moduli space of Riemann surfaces. An auxiliary goal of this paper is to give an exposition of this QFT formalism with the holomorphic bosonic string theory as the running example.

  • Homotopy RG flow and the non-linear sigma model, pdf. With Ryan Grady. American Mathematical Society, Providence, RI, 2018, 213–258. ISBN: 978-1-4704-4243-9 +
  • 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, this 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.

  • Asymptotic freedom in the BV formalism, pdf. With Chris Elliott and Philsang Yoo. J. Geom. Phys. 123 (2018), 246–283. +
  • 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.

  • The Virasoro vertex algebra and factorization algebras on Riemann surfaces, pdf. Lett. Math. Phys. 107 (2017), no. 12, 2189–2237. +
  • This paper focuses on the connection of holomorphic two-dimensional factorization algebras and vertex algebras which has been made precise in the book of Costello and Gwilliam. We provide a construction of the Virasoro vertex algebra starting from a local Lie algebra on the complex plane. Moreover, we discuss an extension of this factorization algebra to a factorization algebra on the category of Riemann surfaces. The factorization homology of this factorization algebra is computed as the correlation functions. We provide an example of how the Virasoro factorization algebra implements conformal symmetry of the beta-gamma system using the method of effective BV quantization.