The Determinant of a Hypergeometric Period Matrix

We consider a function $U=e^{-f_0}\prod_j^N f_j^{\alpha_j}$ on a real affine space, here $f_0,..,f_N$ are linear functions, $\alpha_1, ...,\alpha_N$ complex numbers. The zeros of the functions $f_1, ..., f_N$ form an arrangement of hyperplanes in the affine space. We study the period matrix of the hypergeometric integrals associated with the arrangement and the function $U$ and compute its determinant as an alternating product of gamma functions and critical points of the functions $f_0,..., f_N$ with respect to the arrangement. In the simplest example, $N=1, f_0=f_1=t$, the determinant formula takes the form $\int_0^\infty e^{-t} t^{\alpha -1} dt=\Gamma (\alpha).$ We also give a determinant formula for Selberg type exponential integrals. In this case the arangements of hyperplanes is special and admits a symmetry group, the period matrix is decomposed into blocks corresponding to different representations of the symmetry group on the space of the hypergeometric integrals associated with the arrangement. We compute the determinant of the block corresponding to the trivial representation.