Let be a 1-form on with poles of order , and let be the integrable connections on . Consider the twisted cohomology groups for the complexes , where is the vector space of rational -forms with poles at most at and consider which satisfies . An integral for some is called the hypergeometric integral, where is an element of twisted homology group , which is proved to be dual to by the pairing . The authors introduce the cohomological intersection pairing for -dimensional vector spaces and , and the homological intersection pairing for and For a choice of bases and of the groups and , define the four matrices of size
The main result of this paper is the following.
Theorem. We have twisted period relations:
which give quadratic relations among confluent hypergeometric integrals.