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arXiv · hep-th/9211003

Chern-Simons States at Genus One

Fernando Falceto·Krzysztof Gawedzki
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Abstract

We present a rigorous analysis of the Schrödinger 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.

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Fernando Falceto, Krzysztof Gawedzki. 1992-11-02. Chern-Simons States at Genus One. https://doi.org/10.1007/bf02099984

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