Notes on twisted homology and cohomology groups for the Wirtinger integral

The Wirtinger integral is one of the integral representations of the Gauss hypergeometric function. Its integrand is given by a product of complex powers of theta functions. We study the structure of the twisted homology and cohomology groups associated with this integral. Using the involution on the complex torus, we show that these groups decompose into eigenspaces which are orthogonal with respect to the intersection forms. Each eigenspace is related to the twisted (co)homology group associated with the Euler-type integral representation of the Gauss hypergeometric function. We also show that the corresponding intersection matrices admit simple forms.