On monodromy representation of period integrals associated to an algebraic curve with bi-degree (2,2)

TL;DR

This paper investigates monodromy representations of period integrals for a bi-degree (2,2) algebraic curve in ${f P}^1 imes {f P}^1$, connecting mirror symmetry, monodromy, and derived categories.

Contribution

It computes two distinct monodromy representations for the period integrals of a specific algebraic curve, linking them to Hermitian invariants and derived category structures.

Findings

Both monodromy representations admit a Hermitian quadratic invariant form.

The first method uses the generalized Picard-Lefschetz theorem for full monodromy.

The second method employs Mellin-Barnes integrals for a subgroup of the monodromy group.

Abstract

We study a problem related to Kontsevich's homological mirror symmetry conjecture for the case of a generic curve $\cal Y$ with bi-degree (2,2) in a product of projective lines ${\Bbb P}^{1} \times {\Bbb P}^{1}$. We calculate two differenent monodromy representations of period integrals for the affine variety ${\cal X}^{(2,2)}$ obtained by the dual polyhedron mirror variety construction from $\cal Y$. The first method that gives a full representation of the fundamental group of the complement to singular loci relies on the generalised Picard-Lefschetz theorem. The second method uses the analytic continuation of the Mellin-Barnes integrals that gives us a proper subgroup of the monodromy group. It turns out both representations admit a Hermitian quadratic invariant form that is given by a Gram matrix of a split generator of the derived category of coherent sheaves on on $\cal Y$ with respect to the Euler form.