SCHRÖDINGER REPRESENTATION FOR ABELIAN CHERN-SIMONS THEORIES ON NON-TRIVIAL SPACE-TIMES

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Abstract
We consider the U(1) Chern-Simons theory on three-manifolds of the form R×Σ, where Σ is a compact Riemann surface of genus g, and quantize the theory in the functional Schrödinger representation. Imposing gauge invariance at the quantum level requires the quantization of the overall normalization parameter k of the action and restricts the Hilbert space to a finite kg-dimensional space of functions on a g-dimensional configuration space. Gauge transformations are realized with a 1-cocycle and the theory is characterized by 2g vacuum angles. We consider the gauge- and coordinate-invariant fixed time operators of the theory, the quantum holonomy operators for closed curves C on Σ, and show that their eigenvalues on physical states are not completely determined by the homology class of C. Rather, there is an additional phase contribution fixed by the oriented self-intersection number of C.

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