[Paper Review] Narrow quantum D-modules and quantum Serre duality

Mark Shoemaker|arXiv (Cornell University)|Nov 5, 2018

Algebraic Geometry and Number Theory21 references4 citations

TL;DR

This paper introduces the narrow cohomology of non-compact manifolds or orbifolds, a subspace of cohomology that supports a well-defined, non-degenerate pairing despite non-compactness. It constructs a quantum D-module on this narrow cohomology, enabling a new formulation of quantum Serre duality as an isomorphism of quantum D-modules, including both the quantum connection and pairing, thereby resolving a long-standing issue in Gromov–Witten theory for non-compact targets.

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.

Motivation & Objective

• To define a well-behaved cohomology subspace—narrow cohomology—on non-compact manifolds or orbifolds where standard pairings fail.
• To construct a quantum D-module on this narrow cohomology, ensuring the quantum connection preserves the subspace and the pairing remains non-degenerate.
• To reframe quantum Serre duality as an isomorphism of quantum D-modules, including both the quantum connection and the pairing, rather than just a map on kernels.
• To resolve the issue that previous formulations of quantum Serre duality failed to preserve the pairing on non-compact targets like total spaces of vector bundles.

Proposed method

• Define narrow cohomology as the image of the forgetful map from compactly supported cohomology to ordinary cohomology: $ H^{*}_{ ext{nar}}( ilde{Y}) = ext{im}( heta: H^{*}_{ ext{c}}( ilde{Y}) \to H^{*}( ilde{Y}) ) $.
• Use the Poincaré duality pairing between $ H^{*}_{ ext{c}}( ilde{Y}) $ and $ H^{*}( ilde{Y}) $ to induce a non-degenerate pairing on $ H^{*}_{ ext{nar}}( ilde{Y}) $.
• Show that the quantum connection $ \nabla^{\tilde{Y}} $ preserves the narrow subspace $ H^{*}_{\text{nar}}(\tilde{Y}) $, ensuring the quantum D-module structure is well-defined on it.
• Define a restricted pairing $ S^{\tilde{Y},\text{nar}} $ on the narrow quantum D-module, which is flat with respect to the quantum connection.
• Prove that the map $ \pi^{*} \circ j_{*} $ induces an isomorphism of quantum D-modules between the narrow quantum D-module of $ \mathcal{Z} $ and the narrow quantum D-module of $ \tilde{Y} $, after a change of variables.
• Establish compatibility with integral structures via the functor $ (-1)^{\text{rk}(\mathcal{E})} \text{det}(\mathcal{E}) \otimes (\pi^{*} \circ j_{*})(-) $.

Experimental results

Research questions

• RQ1 Can a well-defined quantum D-module be constructed on a non-compact target $ \tilde{Y} $, despite the failure of the standard Poincaré pairing?
• RQ2 Can quantum Serre duality be upgraded from a map on kernels to an isomorphism of full quantum D-modules, including the pairing?
• RQ3 Is there a natural subspace of cohomology on non-compact $ \tilde{Y} $ that supports a non-degenerate pairing and is preserved by the quantum connection?
• RQ4 How does the narrow cohomology relate to the image of $ \pi^{*} \circ j_{*} $ in the kernel of $ \nabla^{\tilde{Y}} $?

Key findings

• The narrow cohomology $ H^{*}_{\text{nar}}(\tilde{Y}) $ is defined as the image of the forgetful map $ \theta: H^{*}_{\text{c}}(\tilde{Y}) \to H^{*}(\tilde{Y}) $, providing a finite-dimensional subspace with good duality properties.
• The induced pairing on $ H^{*}_{\text{nar}}(\tilde{Y}) $ is non-degenerate, resolving the issue of degeneracy in standard cohomology for non-compact spaces.
• The quantum connection $ \nabla^{\tilde{Y}} $ preserves the narrow subspace, so the quantum D-module structure restricts to $ H^{*}_{\text{nar}}(\tilde{Y}) $.
• A well-defined, flat pairing $ S^{\tilde{Y},\text{nar}} $ exists on the narrow quantum D-module, making it a full quantum D-module.
• The map $ \pi^{*} \circ j_{*} $ induces an isomorphism of quantum D-modules between $ \mathcal{Z} $ and $ \tilde{Y} $, after a change of variables, realizing quantum Serre duality as a D-module isomorphism.
• The construction is compatible with integral structures, extending Iritani’s integral structure to non-compact settings via the functor $ (-1)^{\text{rk}(\mathcal{E})} \text{det}(\mathcal{E}) \otimes (\pi^{*} \circ j_{*})(-) $.