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We introduce two-types of topologically twisted Chern-Simons-matter theorieson the direct product of circle and genus-h Riemann surface (S^1 \times\Sigma_h). The partition functions of first model agrees with the partitionfunctions of a generalizations of G/G gauged WZW model. We also find thatcorrelation functions of Wilson loops in first type Chern-Simons-matter theorycoincide with correlation functions of G elements in the generalization of G/Ggauged WZW model. The partition function of this model also has niceinterpretations as norms of eigen states of Hamiltonian in the quantumintegrable model (q-boson hopping model) and also as a geometric index over aparticular moduli space. In the second-type Chern-Simons-matter theory, thepartition function is related to integration over moduli space of Hitchinequation on Riemann surface.

Gauge/Bethe correspondence on $S^1 \times \Sigma_h$ and index over moduli space