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 E ∨ → 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 K −1 X , 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).

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Let X be a smooth projective toric variety with k ample line bundles. Let Z be the zero locus of k generic sections. It is well-known that the ambient quantum D-module of Z is cyclic i.e., is defined by an ideal of differential operators. In this paper, we give an explicit construction of this ideal as a quotient ideal of a GKZ system associated to the toric data of X and the line bundles. This description can be seen as a "left cancellation procedure". We consider some examples where this description enables us to compute generators of this ideal, and thus to give a presentation of the ambient quantum D-module.

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