A Fock sheaf for Givental quantization - Coates,T; Iritani,H (Kyoto Journal of Mathematics, 2018)
We give a global, intrinsic, and co-ordinate-free quantization formalism for Gromov{Witten invariants and their B-model counterparts, which simultaneously generalizes the quantizationformalisms described by Witten, Givental, and Aganagic{Bouchard{Klemm. Descendant potentialslive in a Fock sheaf, consisting of local functions on Givental's Lagrangian cone that satisfy the(3g2)-jet condition of Eguchi{Xiong; they also satisfy a certain anomaly equation, which gen-eralizes the Holomorphic Anomaly Equation of Bershadsky{Cecotti{Ooguri{Vafa. We interpretGivental's formula for the higher-genus potentials associated to a semisimple Frobenius manifold inthis setting, showing that, in the semisimple case, there is a canonical global section of the Focksheaf. This canonical section automatically has certain modularity properties. WhenXis a varietywith semisimple quantum cohomology, a theorem of Teleman implies that the canonical sectioncoincides with the geometric descendant potential de ned by Gromov{Witten invariants ofX. Weuse our formalism to prove a higher-genus version of Ruan's Crepant Transformation Conjecture forcompact toric orbifolds. When combined with our earlier joint work with Jiang, this shows that thetotal descendant potential for compact toric orbifoldXis a modular function for a certain group ofautoequivalences of the derived category ofX.