Elliptic orbifold lines and integrable hierarchies

We prove that the Gromov–Witten invariants of the elliptic orbifold lines ${\mathbf{P}}_{3,3,3}^1$, ${\mathbf{P}}_{2,4,4}^1$, and ${\mathbf{P}}_{2,3,6}^1$ satisfy a certain system of Hirota Quadratic (or Bilinear) Equations. Our result is the analogue of the so-called Toda conjecture in the Gromov-Witten theory of ${\mathbf{P}}^1$ or more precisely its non-extended version. A new feature in our constructions is a certain bilinear operator whose principal symbol can be expressed in terms of elliptic theta functions.