Chern-Simons States at Genus One
Fernando Falceto, Krzysztof Gawedzki

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 Chern-Simons theory on 3-manifold torusline, with insertions of Wilson lines. The quantum states, defined as gauge covariant holomorphic functionals of smooth -connections on the torus, are expressed by degree theta-functions satisfying additional conditions. The conditions are obtained by splitting the space of semistable -connections into nine submanifolds and by analyzing the behavior of states at four codimension 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…
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