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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Gromov–Witten theory of Fano orbifold curves, Gamma integral structures and ADE-Toda hierarchies

Todor Milanov, Yefeng Shen and Hsian-Hua Tseng

Geometry & Topology 20 (2016) 2135–2218
Abstract

We construct an integrable hierarchy in the form of Hirota quadratic equations (HQEs) that governs the Gromov–Witten invariants of the Fano orbifold projective curve a1,a2,a31. The vertex operators in our construction are given in terms of the K–theory of a1,a2,a31 via Iritani’s Γ–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 1 to all Fano orbifold curves.

Keywords
Gromov–Witten theory, Fano orbifold curves, ADE-Toda hierarchies
Mathematical Subject Classification 2010
Primary: 14N35, 17B69
References
Publication
Received: 12 December 2014
Accepted: 5 November 2015
Published: 15 September 2016
Proposed: Yasha Eliashberg
Seconded: Gang Tian, Jim Bryan
Authors
Todor Milanov
Kavli IPMU
University of Tokyo (WPI)
5-1-5 Kashiwanoha
Kashiwa 2778583
Japan
Yefeng Shen
Department of Mathematics
Stanford University
450 Serra Mall, Building 380
Stanford, CA 94305
United States
Hsian-Hua Tseng
Department of Mathematics
Ohio State University
100 Math Tower
231 West 18th Avenue
Columbus, OH 43210
United States