GROMOV - WITTEN INVARIANTS AND QUANTIZATION OF QUADRATIC HAMILTONIANS

By Alexander Givental. 2001

500 citations recorded by the source corpus.

We describea formalism based on quantizationof quadratichamil- tonians 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 Gro- mov - Witten invariants of complex projective spaces and other Fano toric manifolds. Details will be published elsewhere.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/874c5d03fc20a5ce75e878eb67852792e2a94b68

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Citations in the corpus

Matrix models, topological strings, and supersymmetric gauge theories (2002, background): “18 invariants (see for example [ 31 ,32]). If we manage to formulate a matrix model dual of”
Simple singularities and integrable hierarchies (2003, methodology): “2 in [9] specialized to Frobenius structures of weightedhomogeneous singularities.”
An integral structure in quantum cohomology and mirror symmetry for toric orbifolds (2009, background): “Givental’s symplectic space [39, 25] is the loop space on H∗ orb(X ) with a loop parameter z.”
Relations in the tautological ring of the moduli space of curves (2013, methodology): “Using the classification of semisimple CohFTs by Givental-Teleman [14, 36], we derive an explicit formula in the tautological ring ofMg,n for Witten’s 3-spin class and use the vanishing property to establish the relations Rg,n.”
The Extended Toda Hierarchy (2003, background): “The crucial point in proving the validity of this conjecture in full genera is the Givental’s result on the Virasoro constraints for CP 1 [18, 19].”
Relations on Mbar_{g,n} via 3-spin structures (2013, unknown): “Using the classiﬁcation of semisimple CohFTs by Givental-Teleman [6, 20], we derive an explicit formula in the tautological ring of M g,n for Witten’s 3-spin class and use the vanishing property to establish the relation set P .”
The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants (2002, methodology): “[6] T. Coates and A. Givental, Quantum Riemann-Roch, Lefschetz and Serre. math.”
Computation of Open Gromov–Witten Invariants for Toric Calabi–Yau 3-Folds by Topological Recursion, a Proof of the BKMP Conjecture (2012, background): “Also, we mention that this theorem can be seen as a formulation of Givental’s relations [28, 29], although we shall not pursue in that direction.”