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Witten’s conjecture and the Virasoro conjecture for genus up to two. (English) Zbl 1114.14034
Jarvis, Tyler J. (ed.) et al., Gromov-Witten theory of spin curves and orbifolds. AMS special session, San Francisco, CA, USA, May 3–4, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3534-3/pbk). Contemporary Mathematics 403, 31-42 (2006).
Summary: This is an expository paper based on the results in [Y.-P. Lee, Witten’s conjecture, Virasoro conjecture, and invariance of tautological equations, preprint, arXiv:math/0311100] and [Y.-P. Lee and A. Givental, in preparation]. The main goal is to show one may prove the following two conjectures for genus up to two:
(1) E. Witten’s conjecture on the relations between higher spin curves and the Gelfand-Dickey hierarchy [in: Topological methods in modern mathematics. Proc. symp. hon. John Milnor’s State Univ. New York, Stony Brook, USA, 1991. 235–269 (1993; Zbl 0812.14017)];
(2) the Virasoro conjecture for conformal semisimple Frobenius manifolds [T. Eguchi, K. Hori and Ch. Sh. Xion, Phys. Lett. B402, 71–80 (1997; Zbl 0933.81050)].
For the entire collection see [Zbl 1091.14002].

MSC:
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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