THE SPACES OF LAURENT POLYNOMIALS, P 1-ORBIFOLDS, AND INTEGRABLE HIERARCHIES
THE SPACES OF LAURENT POLYNOMIALS, P 1-ORBIFOLDS, AND INTEGRABLE HIERARCHIES
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发表时间：
2006
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通讯作者：
Hsian-Hua Tseng
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作者：
Hsian-Hua Tseng
令 Mk,m 为一个变量 x + t1x k−1 + 中的洛朗多项式空间。 。 。 tk+mx ，其中k、m ≥ 1 为固定整数且tk+m 6= 0。根据B. Dubrovin [D]，Mk,m 可以配备半简单Frobenius 结构。在本文中，我们证明了 Mk,m（定义如[G1]）相应的后代势和祖先势满足 Hirota 二次方程（简称 HQE）。令 Ck,m 为由 P 切割小圆盘 D1 ∼= {|z| 得到的环折≤ ϫ} 且 D2 ∼= {|z−1| ≤ ϫ} 围绕 z = 0 和 z = ∞ 并以明显的方式粘回轨道折叠 D1/Zk 和 D2/Zm。我们证明 Ck,m 的轨道量子上同调与作为 Frobenius 流形的 Mk,m 一致。对一些尚未澄清的细节取模，这意味着 Mk,m 的后代（分别是祖先）势是 Ck,m 的后代（分别是祖先）轨道 Gromov-Witten 不变量的生成函数。我们的 HQE 和扩展二级 Toda 层次结构的 Lax 算子之间有一定的相似性，由 G. Carlet 在 [Car] 中介绍。因此，我们的 HQE 描述了该层次结构的 tau 函数是合理的，并且我们期望扩展二阶 Toda 层次结构控制 Ck,m 的 Gromov-Witten 理论。
Let Mk,m be the space of Laurent polynomials in one variable x + t1x k−1 + . . . tk+mx , where k, m ≥ 1 are fixed integers and tk+m 6= 0. According to B. Dubrovin [D], Mk,m can be equipped with a semi-simple Frobenius structure. In this paper we prove that the corresponding descendant and ancestor potentials of Mk,m (defined as in [G1]) satisfy Hirota quadratic equations (HQE for short). Let Ck,m be the orbifold obtained from P by cutting small discs D1 ∼= {|z| ≤ ǫ} and D2 ∼= {|z−1| ≤ ǫ} around z = 0 and z = ∞ and gluing back the orbifolds D1/Zk and D2/Zm in the obvious way. We show that the orbifold quantum cohomology of Ck,m coincides with Mk,m as Frobenius manifolds. Modulo some yet-to-be-clarified details, this implies that the descendant (respectively the ancestor) potential of Mk,m is a generating function for the descendant (respectively ancestor) orbifold Gromov–Witten invariants of Ck,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 [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 Ck,m.