The spaces of Laurent polynomials, $\mathbb{P}^1$-orbifolds, and integrable hierarchies

Abstract

Let $M_{k,m}$ be the space of Laurent polynomials in one variable $x^k + t_1 x^{k-1}+... t_{k+m}x^{-m},$ where $k,m\geq 1$ are fixed integers and $t_{k+m}\neq 0$. According to B. Dubrovin \cite{D}, $M_{k,m}$ can be equipped with a semi-simple Frobenius structure. In this paper we prove that the corresponding descendant and ancestor potentials of $M_{k,m}$ (defined by A. Givental) satisfy Hirota quadratic equations (HQE for short). Let $\mathcal{C}_{k,m}$ be the orbifold obtained from $\mathbb{P}^1$ by cutting small discs $D_1\simeq \{|z|\leq \epsilon\}$ and $D_2\simeq\{|z^{-1}|\leq \epsilon\}$ around $z=0$ and $z=\infty$ and gluing back the orbifolds $D_1/\mathbb{Z}_k$ and $D_2/\mathbb{Z}_m$ in the obvious way. We show that the orbifold quantum cohomology of $\mathcal{C}_{k,m}$ coincides with $M_{k,m}$ as Frobenius manifolds. Modulo some yet-to-be-clarified details, this implies that the descendant (respectively the ancestor) potential of $M_{k,m}$ is a generating function for the descendant (respectively ancestor) orbifold Gromov--Witten invariants of $\mathcal{C}_{k,m}$. There is a certain similarity between our HQE and the Lax operators of the Extended bi-graded Toda hierarchy, introduced by G. Carlet in \cite{car}. Therefore, it is plausible that our HQE characterize the tau-functions of this hierarchy and we expect that the Extended bi-graded Toda hierarchy governs the Gromov--Witten theory of $\mathcal{C}_{k,m}.$