Chern-Simons States at Genus One

TL;DR

This paper rigorously analyzes the quantization of SU(2) Chern-Simons theory on a torus times a line, describing quantum states as theta-functions and deriving the Verlinde dimension formula through geometric and analytical methods.

Contribution

It provides a detailed construction of quantum states as theta-functions and introduces a recursion relation for their dimensions, connecting to the Verlinde formula.

Findings

States are expressed as degree 2k theta-functions with specific conditions.

Constructs the Knizhnik-Zamolodchikov-Bernard connection for different complex structures.

Proves a recursion relation for state space dimensions leading to the Verlinde formula.

Abstract

We present a rigorous analysis of the Schr\"{o}dinger picture quantization for the $SU(2)$ Chern-Simons theory on 3-manifold torus$\times$line, with insertions of Wilson lines. The quantum states, defined as gauge covariant holomorphic functionals of smooth $su(2)$-connections on the torus, are expressed by degree $2k$ theta-functions satisfying additional conditions. The conditions are obtained by splitting the space of semistable $su(2)$-connections into nine submanifolds and by analyzing the behavior of states at four codimension $1$ strata. We construct the Knizhnik-Zamolodchikov-Bernard connection allowing to compare the states for different complex structures of the torus and different positions of the Wilson lines. By letting two Wilson lines come together, we prove a recursion relation for the dimensions of the spaces of states which, together with the (unproven) absence of states for spins$\s>{_1\over^2}$level implies the Verlinde dimension formula.