Narrow quantum D-modules and quantum Serre duality

Abstract

Given Y a non-compact manifold or orbifold, we define a natural subspace of the cohomology of Y called the narrow cohomology. We show that despite Y being non-compact, there is a well-defined and non-degenerate pairing on this subspace. The narrow cohomology proves useful for the study of genus zero Gromov-Witten theory. When Y is a smooth complex variety or Deligne-Mumford stack, one can define a quantum D-module on the narrow cohomology of Y. This yields a new formulation of quantum Serre duality.

Extremal transitions via quantum Serre duality

“…In this scenario, we will introduce an analogous notion of quantum D-module, namely, narrow quantum D-module. See [39] for details of this construction.…”

“…By [39,Proposition 4.6], the connection ∇ Y and the fundamental solution L Y (t, z)z − Gr z ρ(Y ) preserve the narrow cohomology. We denote the restrictions to H * CR,nar (Y ) by ∇ Y,nar and L Y,nar (t, z)z − Gr z ρ(Y ) respectively.…”

“…In particular we have the following relation between ∇ Y and ∇ Y,c . Proposition 2.14 (Proposition 4.4 of [39]). The connections ∇ Y and ∇ Y,c are dual with respect to the pairing S Y (−, −):…”

“…The correspondence was formulated as a relationship between quantum D-modules in [26]. The formulation below using narrow quantum D-modules was given by the second author in [39].…”

“…Z and T ). This was reformulated by the second author in [39] as an isomorphism between the ambient quantum D-module of Z and the narrow quantum D-module of T (resp. Z and T ).…”