Twisted cohomology and homology groups associated to the Riemann-Wirtinger integral

The Riemann-Wirtinger integral is a function defined by a definite integral on a complex torus whose integrand is a power product of the exponential function and theta functions. It was found as a special solution of the system of differential equations which governs the monodromy-preserving deformation of Fuchsian differential equations on the complex torus [11], [12]. The main purpose of this paper is to give an interpretation to this integral as the pairing between twisted cohomology and homology groups with coefficients in rank-one local systems on the complex torus minus finitely many points. The local systems contain a parameter corresponding to the Jacobian of the torus. We also give an explicit description of the cohomology groups valid for any value of the parameter. This result shall be applied to studies on generalization of the Wirtinger integral, which is another integral representation of Gauss’ hypergeometric function in terms of the power product of theta functions [14].