Abstract
We constructN=2 supersymmetric quantum Hamiltonians with meromorphic superpotentials on compact Riemann surfaces and investigate the topological properties of these Hamiltonians.L 2-cohomology groups for supercharge (a deformed\(\bar \partial \) operator) are considered and the Witten index for the supersymmetric Hamiltonian with meromorphic superpotential is calculated in terms of Euler characteristic of the Riemann surface and the degree of a divisor of poles for the differential of the superpotential.
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