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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.
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, BPS Invariants, and Wall-Crossing in Holomorphic-topological Field Theories
My research explores the origins of quantum groups and their representations in holomorphic-topological field theories, including four- and five-dimensional Chern–Simons theories. Their defects realize algebraic operations physically: fusion encodes tensor products, while intersections and exchanges produce intertwiners and R-matrices. A central result identifies Miura operators with particular R-matrices, connecting W- and Y-algebras and their q-deformations to representations of affine Yangians and quantum toroidal algebras. This connection is realized through intersecting branes and quantum interactions of line and surface defects. My ongoing work investigates how these algebraic structures extend in the presence of boundary conditions, interfaces, and compactification.
Geometric engineering brings derived geometry into this picture through moduli spaces of sheaves on Calabi–Yau varieties and representations of quivers with potential. A central focus is the relationship between stable envelopes, cohomological Hall algebras, and Yangian-type algebras. Stable envelopes provide geometric constructions of R-matrices and tensor-product structures, while cohomological Hall algebras organize operations defined by extensions of objects in the underlying category. I am studying how these approaches fit together, with particular attention to framed moduli spaces and the representations they support.
Wall-crossing adds a further layer to this picture. Varying stability can change the spectrum of stable objects, while changes of heart can lead to different quiver descriptions of the same underlying Calabi–Yau category. My ongoing work explores how stable envelopes, Hall-algebra constructions, and quantum-group representations behave under these changes, including their relation to quiver mutations and reflection operations. Holomorphic-topological field theories provide a physical framework for investigating these relationships through defects and interfaces.
A related direction concerns enumerative geometry: I use supersymmetric localization to study defect partition functions, their proposed interpretation as generating functions of refined open BPS invariants, and the quantum mirror curves governing their Nekrasov–Shatashvili limits.
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