Research

Modern study of theoretical physics and mathematics reveals a deep, two-way exchange that continues to drive advances in both fields. My research probes the geometric and algebraic structures underlying exact solutions of supersymmetric field theories across diverse spacetime dimensions, uncovering unexpected correspondences among them motivated by their embedding in string and M-theory.

Gauge-theoretic Study of Geometric Langlands Correspondence

The geometric Langlands correspondence (GLC) relates flat connections for a Langlands dual group on a Riemann surface to certain sheaves on the moduli space of holomorphic bundles on the same surface. A remarkable physical perspective on this correspondence arises from the six-dimensional 𝒩=(0,2) superconformal field theory. In the conventional gauge-theoretic approach, one first compactifies this theory on a two-torus to obtain four-dimensional 𝒩=4 supersymmetric gauge theory. Upon further compactification on the Riemann surface on which one studies the GLC, the S-duality of the 𝒩=4 theory descends to the mirror symmetry of the resulting topological sigma models on the moduli spaces of Higgs bundles, leading to the GLC. My research explores a complementary viewpoint based on exchanging the order of compactifications: instead of reducing on the torus first, one compactifies the six-dimensional theory along the Riemann surface. This produces a four-dimensional 𝒩=2 theory of class 𝒮 associated with the surface, providing a new framework for studying the GLC.

In this framework, half-BPS surface defects are introduced to determine boundary conditions at the emergent boundaries of the two-dimensional worldsheet. Accordingly, sections of sheaves on the moduli space of holomorphic bundles on the Riemann surface are reformulated as partition functions of the 4d 𝒩=2 theory in the presence of these defects, with the boundary condition at infinity specified. The powerful method of supersymmetric localization then reduces this path integral to an integral of equivariant classes of bundles over the moduli space of framed torsion-free (or parabolic) sheaves. Although the geometry of the moduli space of Higgs bundles appears to become less transparent in this approach, it is in fact encoded in the contents of the 4d 𝒩=2 gauge theory in an intricate way. This unusual incarnation of the geometric structures is precisely what allows the 𝒩=2 gauge-theoretic formulation of the GLC to complement more traditional approaches and introduce new conceptual and computational tools.

I'm also pursuing the K-theoretic and elliptic variants of the program, by uplifting the framework to the 5d 𝒩=1 theory compactified on a circle and the 6d 𝒩=(1,0) theory compactified on an elliptic curve.

Geometric Representation Theory in Twisted String and M-theory

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