1 research outputs found
Chern-Simons States at Genus One
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 states for spins\s>{_1\over^2}level implies the Verlinde dimension
formula.Comment: 33 pages, IHES/P