Gromov--Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda Hierarchies

Abstract

We construct an integrable hierarchy in the form of Hirota quadratic equations (HQE) that governs the Gromov--Witten (GW) invariants of the Fano orbifold projective curve ${\mathbb{P}}_{a_1,a_2,a_3}^1$. The vertex operators in our construction are given in terms of the $K$-theory of ${\mathbb{P}}_{a_1,a_2,a_3}^1$ via Iritani's $\Gamma$-class modification of the Chern character map. We also identify our HQEs with an appropriate Kac--Wakimoto hierarchy of ADE type. In particular, we obtain a generalization of the famous Toda conjecture about the GW invariants of ${\mathbb{P}}^1$ .

v3: 62 pages, 1 figure, major revision: the correspondence under mirror symmetry between Iritani's Gamma-integral structure and the natural integral structure in LG model is proved in the case of ${\mathbb{P}}_{a_1,a_2,a_3}^1$ and is used to greatly improved the study of root systems; the main result is strengthened; materials in this paper are reorganized; the title of the paper is changed

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Cited by in corpus (13)

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• Extended affine Weyl groups of BCD type, Frobenius manifolds and their Landau-Ginzburg superpotentials
• On the extended multi-component Toda hierarchy
• Twisted logarithmic modules of vertex algebras
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• Frobenius manifolds and a new class of extended affine Weyl groups ${\tilde{W}}^{(k,k+1)}(A_l)$
• Variational Bihamiltonian Cohomologies and Integrable Hierarchies I: Foundations
• $E_8$ spectral curves
• The Period map for quantum cohomology of ${\mathbb{P}}^2$
• Hirota Quadratic Equations for the Gromov--Witten Invariants of ${\mathbb{P}}_{n-2,2,2}^1$
• Dressing operators in equivariant Gromov-Witten theory of ${\mathbb{CP}}^1$
• Dubrovin duality and mirror symmetry for ADE resolutions
• The extended D-Toda hierarchy