Kavli Affiliate: Todor E. Milanov
| First 5 Authors: Todor E. Milanov, Hsian-Hua Tseng, , ,
| Summary:
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 $k,mgeq 1$ 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 $D_1simeq {|z|leq epsilon}$ and
$D_2simeq{|z^{-1}|leq epsilon}$ 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 $mathcal{C}_{k,m}.$
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