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We study a problem related to Kontsevich's homological mirror symmetryconjecture for the case of a generic curve $\cal Y$ with bi-degree (2,2) in aproduct of projective lines ${\Bbb P}^{1} \times {\Bbb P}^{1}$. We calculatetwo differenent monodromy representations of period integrals for the affinevariety ${\cal X}^{(2,2)}$ obtained by the dual polyhedron mirror varietyconstruction from $\cal Y$. The first method that gives a full representationof the fundamental group of the complement to singular loci relies on thegeneralised Picard-Lefschetz theorem. The second method uses the analyticcontinuation of the Mellin-Barnes integrals that gives us a proper subgroup ofthe monodromy group. It turns out both representations admit a Hermitianquadratic invariant form that is given by a Gram matrix of a split generator ofthe derived category of coherent sheaves on on $\cal Y$ with respect to theEuler form.

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