2d Conformal Field Theory

2d conformal field theories provide one of the most well understood quantum field theories because of their enhanced symmetries. I am interested in understanding the algebraic structures arising from 2d CFTs. More precisely, I am interested in the theory of Vertex Operator Algebras (VOAs) and their application in Physics. VOAs were originally defined to prove the Moonshine conjecture, a deep relation between the representation theory of finite sporadic simple groups and modular forms. The first example of such a connection goes back to the construction of the Monster CFT using vertex operator algebras by Frenkel, Lepowsky and Meurmann on which the Monster group (the largest finite simple group) acts. This was later used by Borcherds to prove the Monstrous Moonshine conjecture. There are now many known examples of Moonshine for other sporadic groups although the underlying (super)vertex operator algebra structure is not known. Prominent examples include the Mathieu Moonshine of Eguchi, Ouguri and Tachikawa for the Mathieu group and the Umbral Moonshine of Cheng, Duncan and Harvey. VOA theory is most useful in describing rational 2d CFTs, for example Wess-Zumino-Witten models and their gaugings. I am also interested in the generalizations of VOA to non-chiral VOAs which can potentially describe the algebraic structure of non-rational 2d CFTs.

Four-Manifold Invariants

I am interested in the application of supersymmetric quantum field theories in understanding 4-manifold topology. A classic example is the Seiberg-Witten invariants discovered using the $\mathcal{N}=2$ Yang-Mills theory. The physical interpretation of Donaldson invariants as the path integral in a (topological)supersymmetric quantum field theory was given by Witten. Thus using various supersymmetric theories in 4d, one can (hope to) construct new invariants of 4-manifolds. So far all calculations have resulted in known invariants. I am also interested in understanding various algebro-geometric construction of 4-manifold invariants in terms of partition functions of supersymmetric theories.

Black Holes In String Theory

String theory provides the most promising candidate for quantum gravity. Black holes in string theory are one of the most important objects which can be used to test the consistency of string theory. One of the most famous results in this area is the understanding of the quantum structure of black holes in terms of microstates in string theory. I am interested in further testing this in various compactifications of string theory. Microstate counting of black holes also leads to a rich mathematical structure involving Siegel modular forms and mock modular forms.

Modular Forms

I am interested in the theory of modular forms because of their wide applications in physics. From a purely mathematical point of view, I am interested in the classical theory of modular forms, Jacobi forms, Harmonic Maass forms and Siegel modular forms.