$SU(2)$ WZNW model at higher genera from the gauge field functional integral

We compute the gauge field functional integral giving the scalar product of the $SU(2)$ Chern–Simons theory states on a Riemann surface of genus $>1$. The result allows us to express the higher genera partition functions of the $SU(2)$ WZNW conformal field theory by explicit finite-dimensional integrals. Our calculation may also shed new light on the functional integral of the Liouville theory.