TL;DR
This paper computes the scalar product of SU(2) Chern-Simons states on higher genus surfaces using free field techniques, leading to finite-dimensional integrals that relate to WZW conformal field theory and the KZB connection.
Contribution
It provides a new finite-dimensional integral formula for the scalar product of Chern-Simons states at higher genus, extending previous work and linking to conformal field theory.
Findings
Derived a finite-dimensional integral expression for the scalar product
Connected the scalar product to higher genus WZW partition functions
Suggested a relation to the hermitian metric of the KZB connection
Abstract
We compute, by free field techniques, the scalar product of the SU(2) Chern-Simons states on genus > 1 surfaces. The result is a finite-dimensional integral over positions of ``screening charges'' and one complex modular parameter. It uses an effective description of the CS states closely related to the one worked out by Bertram. The scalar product formula allows to express the higher genus partition functions of the WZW conformal field theory by finite-dimensional integrals. It should provide the hermitian metric preserved by the Knizhnik-Zamolodchikov-Bernard connection describing the variations of the CS states under the change of the complex structure of the surface.
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