The spaces of Laurent polynomials, $\mathbb{P}^1$-orbifolds, and integrable hierarchies
| dc.creator | Milanov, Todor E. | |
| dc.creator | Tseng, Hsian-Hua | |
| dc.date | 2006-07-01 | |
| dc.date | 2007-09-04 | |
| dc.date.accessioned | 2026-07-07T09:51:09Z | |
| dc.date.available | 2026-07-07T09:51:09Z | |
| dc.description | 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 $D_1\simeq \{|z|\leq ε\}$ and $D_2\simeq\{|z^{-1}|\leq ε\}$ 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 $\mathcal{C}_{k,m}.$ | |
| dc.description | v2: Typos and mistakes fixed, to appear in Journal fuer die reine und angewandte Mathematik (Crelle's Journal). v3: Mistakes fixed and references updated | |
| dc.identifier | https://arxiv.org/abs/math/0607012 | |
| dc.identifier | http://arxiv.org/abs/math/0607012 | |
| dc.identifier | Journal fuer die reine und angewandte Mathematik (Crelle's Journal), Volume 2008, Issue 622, Pages 189--235 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/165187 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Mathematical Physics | |
| dc.subject | 14N35, 17B69, 32S30 | |
| dc.title | The spaces of Laurent polynomials, $\mathbb{P}^1$-orbifolds, and integrable hierarchies | |
| dc.type | text |