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 $k,m\ge 1$ are fixed integers and $t_{k+m}\ne 0$. 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 $D_1\simeq \{|z|\le \epsilon \}$ and $D_2\simeq \{|z^{-1}|\le \epsilon \}$ 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 ${\mathcal{C}}_{k,m}.$

v2: Typos and mistakes fixed, to appear in Journal fuer die reine und angewandte Mathematik (Crelle's Journal). v3: Mistakes fixed and references updated

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