Mathematical Physics

The AGT correspondence and its extensions posit geometric constructions of vertex algebras and their modules from cohomology of variants of moduli of sheaves on surfaces. Physically, the correspondence has found an explanation through the holomorphic-topological twist of the six dimensional N=(2,0) superconformal field theories. In this talk, I’ll propose a variant of the AGT correspondence coming from the so-called minimal twist of these theories. Instead of vertex algebras, the natural algebras appearing will be holomorphic factorization algebras in three complex dimensions. From this data, I will explain how one extracts an associative algebra and a module which conjecturally agrees with a quantization of moduli of Higgs sheaves on surfaces. In examples, the pair will admit a Hodge-deRham deformation to the Heisenberg algebra and its action on cohomology of Hilbert schemes of surfaces, constructed in work of Grojnowski-Nakajima.

The N=4 chiral algebra is a vertex operator algebra generated by the Schur subsector of the 4D N=4 SYM theory, and it can be obtained by applying the holomorphic-topological twist to the theory in the presence of the Omega-background. For the first part of the talk, I will discuss how the N=4 chiral algebra arises from the worldsheet perspective, using a conjectured worldsheet CFT dual of the 4D N=4 U(N) SYM at the free point. Then for the second part of the talk, I will view the N=4 chiral algebra as an abstract VOA and try to determine its structure using only the associativity of the OPE. We find that the algebra is uniquely characterized by the central charge (which can take an arbitrary value), without any additional free parameter. Furthermore, the truncation pattern of the OPE coefficients suggests that the algebra cannot arise from the symmetric orbifold.

Dimer models are of interest from many perspectives across geometry, combinatorics, and mathematical physics. In this talk, we explain how one throughline among these perspectives --- the spectral relationship between dimer models in T^2 and line bundles on curves in (C*)^2 --- may be understood as part of a more general mirror relationship between simplicial complexes in T^n and coherent sheaves on (C*)^n. We refer to the complexes arising this way as tropical Lagrangian coamoebae, as from a symplectic point of view they are in a sense dual objects to tropical varieties. This is joint work with Chris Kuo.

Vertex algebras are equivalent to translation-equivariant chiral algebras on $\mathbb{A}^1$, in the sense of Beilinson and Drinfeld. In this talk we give an algebraic construction of a chiral algebra on $\mathbb{A}^n$; this can be seen as an algebraic construction of a higher-dimensional vertex algebra. We introduce a model, in dg commutative algebras, of the derived algebra of functions on the configuration space of $k$ distinct labelled marked points in $\mathbb{A}^n$. Working in this model -- which we call the polysimplicial model-- we obtain a dg operad of chiral operations on a degree-shifted copy of the canonical sheaf. We prove that there is a quasi-isomorphism, to this dg operad, from the Lie-infinity operad. This result makes the shifted canonical sheaf into a first example of a homotopy polysimplicial chiral algebra on $\mathbb{A}^n$, in a sense which generalizes to higher dimensions Malikov and Schechtman's notion of a homotopy chiral algebra. This is joint work with Charles Young and Laura Felder.

The cohomological Hall algebra is an associative algebra associated to an abelian category, and models an algebra of BPS states in 4d HT theories. In theories which admit a certain duality we model orientifold lines as orthosymplectic modules for the CoHA and describe the basic structure of these modules using moduli stacks of objects with classical type stabilizer groups.

I'll discuss the application of Tannakian methods to 4d holomorphic-topological QFT's, using boundary conditions and interfaces to represent the (braided meromorphic tensor) category of line operators. This leads to an explicit geometric construction of Yangian-like algebras, generalizing results of Costello-Yamazaki-Witten in 4d Chern-Simons. By applying these methods to twisted 4d N=2 theories, I'll explain how one recovers and extends some expected relations between cohomological Hall algebras of BPS states and line operators --- in particular, representing line operators as modules for "BPS Yangians."

The KP equation is a well-known classical integrable system that describes shallow water waves in 2+1 dimensions. I will show that it is described by a 5d non-commutative Chern-Simons theory on mini-twistor correspondence space. The 2d chiral CFT living on a holomorphic surface defect is identified with W-algebra hidden symmetry of the KP equation. Based on work done in collaboration with David Skinner, Simon Heuveline and Surya Raghavendran.

In four-dimensional N=2 supersymmetric quantum field theories, half-BPS line defects survive the HT twist and are expected to form a monoidal category equipped with renormalized r-matrices. The Grothendieck group of this category is anticipated to coincide with the cluster algebra associated to the BPS quiver Q. Starting from such a quiver, we propose a candidate for such a category - roughly speaking, to each possible framing we associate a bimodule over the Cohomological Hall algebra of Q, and consider the subcategory they generate. We conjecture that this construction satisfies the expected properties. In this talk, I will present examples where this approach works and explain why we expect the RG functors to play the role of "categorified quantum torus charts".

Kac-Moody localization is a procedure that constructs D-modules on the moduli space of G-bundles on a curve from representation-theoretic data. This construction plays a central role in the geometric Langlands theory. I will describe a factorizable version of this construction and discuss some (potential) applications in the (relative) geometric Langlands program.