Mathematical Physics

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.