Mathematical Physics

I will describe a correspondence between hyperkähler/quaternion-Kähler geometry and two dimensional chiral algebras that arises from twistor theory. As an application, I will explain how to use correlators of these chiral algebras to compute gravitational scattering amplitudes in four dimensional general relativity.

Zoom Link: https://pitp.zoom.us/j/98706048161?pwd=a3k1U3l2UXJlUjlqUElkOEZhZGwwUT09

In recent work, Aganagic proposed a categorification of quantum link invariants based on a category of A-branes, which is solvable explicitly. The theory is a generalization of Heegaard-Floer theory from gl(1|1) to arbitrary Lie algebras. I will describe in the detail the two simplest cases: the su(2) theory, categorifying the Jones polynomial, and the gl(1|1) theory, categorifying the Alexander polynomial. I will give an explicit algorithm for computing link homologies in these cases. I will also briefly describe the generalization to other simple Lie algebras and to Lie superalgebras of type gl(m|n). This talk is based on work to appear with Mina Aganagic and Miroslav Rapcak.

Zoom Link: TBD

This talk is based on joint work with Owen Gwilliam and Brian Williams. We define factorization homology of factorization envelopes valued in a collection of generalized Weyl modules supported on a cycle in a smooth complex projective variety X. When X is a smooth projective curve, and the cycle a collection of points, we recover the space of coinvariants studied in CFT. 

Zoom link:  https://pitp.zoom.us/j/97473973419?pwd=QjRQUklac2lTRlduSVc3S0FKQW9IQT09

The advent of topological materials has brought with it unexpected new connections between physics and pure mathematics. In particular, algebraic topology has played a significant role in the classification of topological materials. In this talk, I will offer a brief look at an emerging chapter in this story in which algebraic geometry — in particular the algebraic geometry of moduli spaces associated with complex curves — is used to anticipate new forms of quantum matter arising from 2-dimensional hyperbolic lattices.  In the process, I will explain my recent joint works with each of J. Maciejko, E. Kienzle, and A. Nagy that establishes an electronic band theory for 2-dimensional hyperbolic matter.

Zoom link:  https://pitp.zoom.us/j/98074477672?pwd=bmVScWx1M09EaGx2ZXZrRit6NXF5dz09

I will first review the construction of quiver Yangians for the quiver and superpotential from string theory on general toric Calabi-Yau threefolds; they serve as BPS algebras of these systems and their characters reproduce the unrefined BPS indices. I will then explain how to generalize the construction to arbitrary quivers and how to incorporate refined BPS indices. The entire construction allows for straightforward generalizations to trigonometric, elliptic, and generalized cohomologies. Time permitting, I will discuss some applications, such as on BPS counting and on Gauge/Bethe correspondence. 

Zoom link:  https://pitp.zoom.us/j/96812116687?pwd=RjlsSVhDaUVBREtUUnd5S04rQjhpZz09

Cluster structure on a quantum group allows one to work with its positive representations, a special class of modules similar in spirit to principal series representations but closed under tensor multiplication. On the other hand, cluster techniques proved inadequate for the study of finite-dimensional representation theory. I will discuss how one can reconcile positive and finite-dimensional representations into one theory by studying moduli spaces of local systems with non-generic monodromies.

Zoom link:  https://pitp.zoom.us/j/97677905324?pwd=Tkxlei9wN1llYVpncHA2cnpYKzlhUT09

I will describe a general categorical approach to constructing Hamiltonian actions on moduli spaces. In particular cases, this specializes to give a ``universal" Hitchin integrable system as well as the Calogero-Moser system.  Moreover, I will describe a generalization to higher dimensions of a classical result of Goldman which says that the Goldman Lie algebra of free loops on a surface acts by Hamiltonian vector fields on the character variety of the surface.  A key input is a description of deformations of Calabi-Yau structures, which is of independent interest. This is joint work with Chris Brav.

Zoom link:  https://pitp.zoom.us/j/92929253744?pwd=WGFNQmRJck5NdzFFdU8xcXRlN3RRQT09

There is a by now standard procedure for calculating twisted spin, spin^c, and string bordism groups for applications in physics: realize the twist as arising from a vector bundle, which allows one to split the corresponding Thom spectrum and greatly simplify the Adams spectral sequence computation. Not all twists arise from vector bundles, but Matthew Yu and I noticed that if you ignore this fact and pretend that everything is OK, you still get the right answer! In this talk, I'll discuss a theorem Matthew and I proved explaining this, by calculating the input to Baker-Lazarev's version of the Adams spectral sequence. Then I will discuss applications to anomalies of some quantum field theories and supergravity theories.

Zoom link:  https://pitp.zoom.us/j/93508575689?pwd=YVV6VlRwL1RGSG55V0cwTzdpUWROUT09

Symplectic duality predicts that symplectic singularities should come in pairs. For example, Nakajima quiver varieties are conjecturally dual to BFN Coulomb branches (of the corresponding quiver theories). Another family of potentially symplectically dual pairs was described recently in the works of Losev, Mason-Brown, and Matvieievskyi: they describe symplectically duals to Slodowy slices to nilpotent orbits.

In this talk, we will discuss the Hikita-Nakajima conjecture that relates the geometry of symplectically dual varieties. We will restrict to the cases of certain quiver varieties and Slodowy slices and discuss the picture in these cases.

Based on the joint work with Pavel Shlykov (arXiv:2202.09934) and the work in progress with Do Kien Hoang and Dmytro Matvieievskyi.

Zoom link:  https://pitp.zoom.us/j/98651907502?pwd=ODA1K3NKVHFLdkp6TEtaSnJXdThVZz09

Trace map on deformation quantized algebra leads to the algebraic index theorem. We investigate a two-dimensional chiral analogue of the algebraic index theorem via the theory of chiral algebras developed by Beilinson and Drinfeld. We construct a trace map on the elliptic chiral homology of the free beta gamma-bc system using the BV quantization framework. As an example, we compute the trace evaluated on the unit constant chiral chain and obtain the formal Witten genus in the Lie algebra cohomology. This talk is based on joint work with Si Li.

Zoom link:  https://pitp.zoom.us/j/91920988875?pwd=dUE1WDI5Q21KWTUxTGNsM21nblh5Zz09

On smooth schemes, every coherent sheaf admits a finite resolution by vector bundles, but on singular schemes, this is no longer true. The Arinkin-Gaitsgory singular support of coherent sheaves is an invariant of coherent sheaves on certain singular spaces that measures how far a particular coherent sheaf is from having such a resolution. In this talk, I will explain how the Arinkin-Gaitsgory theory of singular support decategorifies to a notion of singular support for chains on the associated complex analytic space of our scheme, measuring the difference between cohomology and Borel-Moore homology on singular spaces. In order to do so, we take advantage of the relationship between coherent sheaves and certain categories of matrix factorizations, also know as D-branes in Landau-Ginzburg models.

Zoom link: https://pitp.zoom.us/j/95698955865?pwd=Rm9ld3FUK3hiWGUzenBuZnQyTTRYZz09

The coordinate functions on a Poisson variety are log-canonical if the Poisson bracket of two coordinate functions equals a constant times the product of these functions. We consider the symplectic groupoid of unipotent upper-triangular matrices equipped with canonical Poisson bracket. We described a system of log-canonical coordinates and the corresponding cluster structure. As a bonus, we discovered a system of log-canonical coordinates on Teichmueller space of closed genus 2 surfaces. This is joint work with L. Chekhov.

Zoom link:  https://pitp.zoom.us/j/94716952708?pwd=R2RiQWRpcHFMYlJLMlB0UjlPVGZkQT09