Moonshine

I am interested in understanding the role of (super)conformal field theories (CFT) in the representation theory of finite sporadic groups. 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.

Four-Manifold Invariants

I am interested in the application of supersymmetric quantum field theories in discovering new 4-manifold invariants upto diffeomorphism. A classic example is the Seiberg-Witten invariants discovered using the N=2 Yang-Mills theory. The physical interpretation of Donaldson invariants as a (topological )supersymmetic quantum field theory was given by Witten. Thus using various supersymmetric theories in 4d, one can construct new invariants of 4-manifolds upto diffeomorphism.

Celestial Holography

The new observation that the Mellin transform of amplitudes of quantum field theories in flat 4d Minkowski space behaves as correlation functions of a CFT has led to the proposal of flat space holography which says that a 4d theory of quantum gravity is dual to a CFT living on the 2d boundary. This has proved to be fruitful in understanding the enhanced symmetries of the gravitational S-matrix and the IR physics in the bulk. My interests in celestial holography is in its applicabily to compute the asymptotic symmetries of a quantum gravity theory.

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 and their various generalisations.