Gromov - Witten invariants and quantization of quadratic hamiltonians

Abstract

We describe a formalism based on quantization of quadratic hamiltonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures about Gromov-Witten invariants of compact symplectic manifolds and, more generally, Frobenius structures at higher genus. We state several results illustrating the formalism and its use. In particular, we establish Virasoro constraints for semisimple Frobenius structures and outline a proof of the Virasoro conjecture for Gromov -- Witten invariants of complex projective spaces and other Fano toric manifolds. Details will be published elsewhere.

To appear in Moscow Mathematical Journal. We correct erroneous formulation of Corollary 5.4

References in corpus (5)

• Quantum Cohomology and Virasoro Algebra
• Quantum Cohomology Rings of Toric Manifolds
• Virasoro constraints and the Chern classes of the Hodge bundle
• Relations between the correlators of the topological sigma-model coupled to gravity
• The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants

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• Twisted orbifold Gromov-Witten invariants
• Borodin-Okounkov formula, string equation and topological solutions of Drinfeld-Sokolov hierarchies
• The Eynard--Orantin recursion for the total ancestor potential
• Lecture notes on topological recursion and geometry
• Geometric interpretation of Zhou's explicit formula for the Witten-Kontsevich tau function
• Borcea-Voisin Mirror Symmetry for Landau-Ginzburg models
• Les Houches lecture notes on moduli spaces of Riemann surfaces