Chern-Simons States at Genus One.

摘要

The authors present a rigorous analysis of the Schrodinger picture quantization for the SU(2) Chern-Simons theory on 3-manifold torusxline, 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. The authors 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, they prove a recursion relation for the dimensions of the spaces of states which, together with the (unproven) absence of states for spins > 1/2 level implies the Verlinde dimension formula.