Kyoto Journal of Mathematics
- Kyoto J. Math.
- Volume 58, Number 4 (2018), 695-864.
A Fock sheaf for Givental quantization
We give a global, intrinsic, and coordinate-free quantization formalism for Gromov–Witten invariants and their B-model counterparts, which simultaneously generalizes the quantization formalisms described by Witten, Givental, and Aganagic–Bouchard–Klemm. Descendant potentials live in a Fock sheaf, consisting of local functions on Givental’s Lagrangian cone that satisfy the -jet condition of Eguchi–Xiong; they also satisfy a certain anomaly equation, which generalizes the holomorphic anomaly equation of Bershadsky–Cecotti–Ooguri–Vafa. We interpret Givental’s formula for the higher-genus potentials associated to a semisimple Frobenius manifold in this setting, showing that, in the semisimple case, there is a canonical global section of the Fock sheaf. This canonical section automatically has certain modularity properties. When is a variety with semisimple quantum cohomology, a theorem of Teleman implies that the canonical section coincides with the geometric descendant potential defined by Gromov–Witten invariants of . We use our formalism to prove a higher-genus version of Ruan’s crepant transformation conjecture for compact toric orbifolds. When combined with our earlier joint work with Jiang, this shows that the total descendant potential for a compact toric orbifold is a modular function for a certain group of autoequivalences of the derived category of .
Kyoto J. Math., Volume 58, Number 4 (2018), 695-864.
Received: 5 January 2015
Revised: 17 November 2015
Accepted: 12 December 2016
First available in Project Euclid: 27 July 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds [See also 14N35] 53D50: Geometric quantization
Coates, Tom; Iritani, Hiroshi. A Fock sheaf for Givental quantization. Kyoto J. Math. 58 (2018), no. 4, 695--864. doi:10.1215/21562261-2017-0036. https://projecteuclid.org/euclid.kjm/1532656825