Research

Here is an up-to-date list of publications of mine courtesy INSPIRE and an atom feed from the arXiv.

I plan on adding notes on various topics that I am currently working on. I will try to make the notes accessible to graduate students though no particular attempt will be made at pedagogy.

• Counting of BPS states in field theory and string/M theory. In a series of papers, arXiv:0807.4451, arXiv:0907.1410 and arXiv:1006.3472, my student Gopala Krishna and I showed the existence of generalized Kac-Moody algebras (some new and some originally discovered by Gritsenko and Nikulin) whose denominator formulae are genus-two modular forms (some constructed by us) that appearing in the counting of 1/4 BPS dyons in CHL string theory ]
• Mathieu moonshine: Serendipitously, I stumbled on a moonshine for the sporadic simple group $M_{24}$ — it first appeared in a paper (arXiv:0907.1410) with my then student, Gopala Krishna. It mapped conjugacy classes of $M_{24}$ to Siegel modular forms. Then, we didn't know how to show the action of $M_{24}$ explicitly. A new and seemingly different (to those authors at least!) result arXiv:1004.0956 , was the appearance of $M_{24}$ in the elliptic genus of K3. In arXiv:1106.5715, I have shown that the Siegel modular form implies an infinite number of moonshines that subsumes all existing moonshines for the group $M_{24}$. Further, I am able to show how the Siegel modular form arises as a trace over a huge $M_{24}$-module. In another work (arXiv:1012.5732), I have also shown that there is a moonshine for another sporadic simple group, $M_{12}$.
• Enumerating numbers of solid partitions This is a project involving undergraduates to compute the number of solid partitions of a positive integer. As of June 2011, we have added eighteen new numbers — this is sequence number A000293 in the Online Encyclopedia of Integer Sequences maintained by N.J.A. Sloane.
• Refined counting of higher dimensional partitions:. I have been able to show the existence of structures in the higher dimensional partitions if one treats all dimensions on an equal footing. This work (arXiv:1203.4419) shows how a clever combination of these structures with some exact enumerations provides a way to determine partitions in any dimension for integers $\leq 26$. See my Online Partition Generator.
• Asymptotics of higher-dimensional partitions: I discovered by accident that a one-parameter fit (see plot below) to the numbers of solid partitions worked much better than it had a right to! Understanding this result led to the paper arXiv:1105.6231 written with a couple of IITM undergraduates (Srivatsan and Naveen). Using Monte Carlo simulations, in collaboration with Nicolas Destainville,we have shown that the conjecture is false albeit by a small amount. arXiv:1406.5605
• Ricci-Flat metrics: I have been involved in a project on finding explicit Ricci-Flat metrics for the resolution of singular spaces such as orbifolds of the form $\mathbb{C}^3/\Gamma$, where $\Gamma$ is any discrete sub-group of $SU(3)$ or cones over spaces such as $Y^{p,q}$ and $L^{p,q,r}$.
• Leigh-Strassler deformed $\mathcal{N}=4$ SYM theory: Chiral primaries and spin-chains; find the gravity dual to LS theory. Along with my student Pramod, using the spin-chain approach we have verified a conjecture on the spectrum of chiral primaries up to dimension 9 operators. This will be appearing soon.