Witten's conjecture, Virasoro conjecture, and invariance of tautological equations

Abstract

The main goal of this paper is to prove the following two conjectures for genus up to two: 1. Witten's conjecture on the relations between higher spin curves and Gelfand--Dickey hierarchy. 2. Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds. The main technique used in the proof is the invariance of tautological equations under loop group action.

References in corpus (6)

• Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants
• Gromov-Witten theory, Hurwitz numbers, and Matrix models, I
• Quantum Cohomology and Virasoro Algebra
• Relations between the correlators of the topological sigma-model coupled to gravity
• A descendent relation in genus 2
• Genus-2 Gromov-Witten invariants for manifolds with semisimple quantum cohomology

Cited by in corpus (8)

• Tautological relations and the r-spin Witten conjecture
• BCOV theory via Givental group action on cohomological field theories
• An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions
• Givental graphs and inversion symmetry
• BCOV's Feynman rule of quintic $3$-folds
• Invariance of tautological equations I: conjectures and applications
• $N=1$ formal genus $0$ Gromov-Witten theories and Givental's formalism
• Invariance of tautological equations II: Gromov--Witten theory