The GKZ system for the Hesse pencil of elliptic curves has more solutions than the period integrals. In this work we give different realizations and interpretations of the extra solution, in terms of oscillating integral, Eichler integral, chain integral on the elliptic curve, limit of a period of a certain compact Calabi-Yau threefold geometry, etc. We also highlight the role played by the orbifold singularity on the moduli space and its relation to the GKZ system.

We verify a long-standing conjecture of Nazarov, Treil, and Volberg.

Let $\mathbb {D}\subset \mathbb {C}$ be the unit disk and let $\gamma_{1},\gamma _{2}:[0,T]\to\overline{\mathbb {D}}\setminus\{0\}$ be parametrizations of two slits $\Gamma_{1}:=\gamma(0,T], \Gamma_{2}:=\gamma_{2}(0,T]$ such that $\Gamma_{1}$ and $\Gamma_{2}$ are disjoint.