Mathematical Physics

The Heisenberg algebra plays an important role in many areas of mathematics and physics. Khovanov constructed a categorical analogue of this algebra which emphasizes its connections to representation theory and combinatorics. Recently, Brundan, Savage, and Webster have shown that the Grothendieck group of this category is isomorphic to the Heisenberg algebra. However, applying an alternative decategorification functor called the trace to the Heisenberg category yields a richer structure: a W-algebra, an infinite-dimensional Lie algebra related to conformal field theory. In this talk, we will describe recent work extending this latter result to a generalized version of the Heisenberg category associated to an arbitrary graded Frobenius algebra F. The decategorifications of this Frobenius Heisenberg category conjecturally provide connections to alternative Heisenberg algebras and W-algebras associated to the algebra F. This is joint work with Alistair Savage (University of Ottawa).

This seminar will be a continued discussion on the topic of last week's seminar: 

I will describe some general mathematical structures expected to arise from field theories with boundary conditions in terms of factorization algebras, and outline some results and future directions in the study of boundary chiral algebras for 3d N=4 theories following the work of Costello and Gaiotto.

Given by Dylan Butson.

I will describe some general mathematical structures expected to arise from field theories with boundary conditions in terms of factorization algebras, and outline some results and future directions in the study of boundary chiral algebras for 3d N=4 theories following the work of Costello and Gaiotto.

Bernstein operators are vertex operators that create and annihilate Schur polynomials. These operators play a significant role in the mathematical formulation of the Boson-Fermion correspondence due to Kac and Frenkel. The role of this correspondence in mathematical physics has been widely studied as it bridges the actions of the infinite Heisenberg and Clifford algebras on Fock space. Cautis and Sussan conjectured a categorification of this correspondence within the framework of Khovanov's Heisenberg category. I will discuss how to categorify the Bernstein operators and settle the Cautis-Sussan conjecture, thus proving a categorical Boson-Fermion correspondence.

The elliptic quantum (toroidal) group U_{q,p}(g) is an elliptic and dynamical analogue of the Drinfeld realization 
of the affine quantum (toroidal) group U_q(g). I will discuss an interesting connection of its representations with 
a geometry such as an identification of the elliptic weight functions derived by using  the vertex operators with 
the elliptic stable envelopes in [Aganagic-  Okounkov ’16] and correspondence between the Gelfand-Tsetlin bases 
of a finite dimensional representation of U_{q,p} with the fixed point classes in the equivariant elliptic cohomology.

Mirković-Vilonen cycles are certain algebraic cycles in the affine Grassmannian that give rise to a particular weight basis (the MV basis) under the Geometric Satake equivalence. I will state a conjecture about the Weyl group action on weight-zero MV cycles and equivariant multiplicities. I can prove it for small coweights in type A. Equivalently, I show that the MV basis agrees with the Springer basis. I have similar results for the Ginzburg-Nakajima basis. A primary tool is work of Braverman, Gaitsgory and Vybornov.

Integrals of characteristic classes of tautological sheaves on the Hilbert scheme of points on a surface often arise in enumerative problems. We explain how to use the K-theoretic Donaldson-Thomas theory of toric Calabi-Yau threefolds to study K-theoretic versions of such expressions.

Namely, we explicate a precise relationship between K-theoretic Donaldson-Thomas theory and the refined topological vertex of Iqbal, Kosçaz and Vafa. Applying such results to specific toric threefolds, we deduce dualities satisfied by certain generating series that control integrals over the Hilbert scheme of points on a surface. We then explain how to use these dualities to evaluate certain Euler characteristics of tautological bundles on the Hilbert scheme of points on a general surface.

A 3d N=4 gauge theory admits two topological twists, which we'll simply call A and B. The two twists are exchanged by 3d mirror symmetry. It is known that local operators in the A (resp. B) twist include the Coulomb-branch (resp. Higgs-branch) chiral rings.  In this talk I will discuss the *line* operators preserved by the two twists, which in each case should have the structure of a braided tensor category.

Roughly speaking, one finds vortex lines in the A-twist and Wilson lines in the B-twist. I will propose a geometric identification of the corresponding categories, and explain how geometric calculations can be used to find the vector spaces of local operators at junctions of lines.

Combined with 3d mirror symmetry, this will lead to new dualities of braided tensor categories. [Joint work with N. Garner, M. Geracie, and J. Hilburn.]

The 8-vertex model and the XYZ spin chain have been found to emerge from gauge theories in various ways, such as 4d and 2d Nekrasov-Shatashvili correspondences, the action of surface operators on the supersymmetric indices of class-Sk theories, and correlators of line operators in 4d Chern-Simons theory.  I will explain how string theory unifies these phenomena.  This is based on my work with Kevin Costello [arXiv:1810.01970].

Positive geometries are real semialgebraic spaces that are
equipped with a meromorphic ``canonical form" whose residues reflect
the boundary structure of the space.  Familiar examples include
polytopes and the positive parts of toric varieties.  A central, but
conjectural, example is the amplituhedron of Arkani-Hamed and Trnka.
In this case, the canonical form should essentially be the tree
amplitude of N=4 super Yang-Mills.

I will talk about the definition and examples of positive geometries,
and discuss what is known about the geometry and combinatorics of the
amplituhedron.  The talk will be based on various joint works with
Arkani-Hamed, Bai, Galashin, and Karp.