Suresh Govindarajan

Bio: Suresh Govindarajan is an academic researcher from Indian Institute of Technology Madras. The author has contributed to research in topics: Superpotential & Siegel modular form. The author has an hindex of 21, co-authored 130 publications receiving 1500 citations. Previous affiliations of Suresh Govindarajan include University of Pennsylvania & Indian Institutes of Technology.

The Brylinski Filtration for Affine Kac-Moody Algebras and Representations of ${\mathscr{W}}$ -algebras

Abstract: We study the Brylinski filtration induced by a principal Heisenberg subalgebra of an affine Kac-Moody algebra $\mathfrak {g}$ , a notion first introduced by Slofstra. The associated graded space of this filtration on dominant weight spaces of integrable highest weight modules of $\mathfrak {g}$ has Hilbert series coinciding with Lusztig’s t-analog of weight multiplicities. For the level 1 vacuum module L(Λ0) of affine Kac-Moody algebras of type A, we show that the Brylinski filtration may be most naturally understood in terms of representations of the corresponding ${\mathscr{W}}$ -algebra. We show that the sum of dominant weight spaces of L(Λ0) in the principal vertex operator realization forms an irreducible Verma module of ${\mathscr{W}}$ and that the Brylinski filtration is induced by the Poincare-Birkhoff-Witt basis of this module. This explicitly determines the subspaces of the Brylinski filtration. Our basis may be viewed as the analog of Feigin-Frenkel’s basis of ${\mathscr{W}}$ for the ${\mathscr{W}}$ -action on the principal rather than on the homogeneous realization of L(Λ0).

$\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras.

Abstract: We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a $\widehat{sl(2)}$ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of $\widehat{sl(2)}$ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients of these vector-valued modular forms are related to multiplicities of roots appearing on the sum side of the Weyl-Kac-Borcherds denominator formulae.

Mathieu moonshine and Siegel Modular Forms

Abstract: A second-quantized version of Mathieu moonshine leads to product formulae for functions that are potentially genus-two Siegel Modular Forms analogous to the Igusa Cusp Form. The modularity of these functions do not follow in an obvious manner. For some conjugacy classes, but not all, they match known modular forms. In this paper, we express the product formulae for all conjugacy classes of M24 in terms of products of standard modular forms. This provides a new proof of their modularity.

Darcy’s experimental empirical relation and its extension

Abstract: Darcy’s law is widely used to describe the steady-state laminar incompressible single-phase fluid flow in a fully saturated porous medium at the macroscopic-scale. However, in reality, we will be dealing with transient non-laminar compressible multi-phase fluid flow through a saturated porous medium. In this context, it is important to understand the original framework of Darcy’s law; and subsequently, we need to understand clearly, under what circumstances the classical Darcy’s law was extended in order to consider the (a) the differential form of Darcy’s law; (b) the non-linear relation between pressure gradient and fluid velocity; (b) the transient nature of fluid flow; (c) the fluid flow through heterogeneous and anisotropic reservoirs or aquifers. It has been reemphasized from the present study that the presence of weak inertial effect along with the laminar fluid regime causes the ‘non-linear’ relation between the macroscopic pressure gradient and the macroscopic fluid velocity, while the strong inertial effect paves the way for the onset of transient nature of fluid flow.

The Topological Vertex

Abstract: We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kahler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum Kodaira-Spencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann surface as fermions related to the chiral boson by bosonization.

Moduli spaces of local systems and higher Teichmüller theory

Abstract: Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmuller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm.

The statistics of string/M theory vacua

Abstract: We discuss systematic approaches to the classification of string/M theory vacua, and physical questions this might help us resolve. To this end, we initiate the study of ensembles of effective Lagrangians, which can be used to precisely study the predictive power of string theory, and in simple examples can lead to universality results. Using these ideas, we outline an approach to estimating the number of vacua of string/M theory which can realize the Standard Model.