Chern-Simons theory, Ehrhart polynomials, and representation theory

Abstract

The Hilbert space of level q Chern-Simons theory of gauge group G of the ADE type quantized on T2 can be represented by points that lie on the weight lattice of the Lie algebra g up to some discrete identifications. Of special significance are the points that also lie on the root lattice. The generating functions that count the number of such points are quasi-periodic Ehrhart polynomials which coincide with the generating functions of SU(q) representation of the ADE subgroups of SU(2) given by the McKay correspondence. This coincidence has roots in a string/M theory construction where D3(M5)-branes are put along an ADE singularity. Finally, a new perspective on the McKay correspondence that involves the inverse of the Cartan matrices is proposed.