Chern-Simons States at Genus One

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.