Quantum Serre Theorem as a Duality between Quantum D-Modules

Abstract

We give an interpretation of quantum Serre theorem 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 Ev → X, and (3) the quantum D-module of a submanifold Z ⊂ X cut out by a regular section of E. When E is the anticanonical line bundle KX-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).