Quantum Serre theorem as a duality between quantum D-modules

TL;DR

This paper interprets quantum Serre duality as a duality between twisted quantum D-modules, revealing deep relationships among quantum D-modules of a variety, its bundle, and submanifolds, with applications to mirror symmetry.

Contribution

It provides a new interpretation of quantum Serre duality as a duality of twisted quantum D-modules, including a non-equivariant limit and connections to Fourier-Laplace transforms.

Findings

Identifies the quantum D-module of a variety twisted by a convex bundle with that of the total space of its dual.

Establishes a relationship between quantum D-modules of a submanifold and the ambient space.

Shows the duality pairing corresponds to Dubrovin's second metric in special cases.

Abstract

We give an interpretation of quantum Serre of Coates and Givental as a duality of twisted quantum D-modules. This interpretation admits a non-equivariant limit, and we obtain a precise relationship among (1) the quantum D-module of X twisted by a convex vector bundle E and the Euler class, (2) the quantum D-module of the total space of the dual bundle E^\vee over X, and (3) the quantum D-module of a submanifold Z\subset X cut out by a regular section of E. When E is the anticanonical line bundle K_X^{-1}, we identify these twisted quantum D-modules with second structure connections with different parameters, which arise as Fourier-Laplace transforms of the quantum D-module of X. In this case, we show that the duality pairing is identified with Dubrovin's second metric (intersection form).