On stringy cohomology spaces

TL;DR

This paper introduces modified stringy cohomology spaces with flat connections, linking GKZ hypergeometric systems to stringy cohomology, and constructs products using vertex algebra techniques, addressing longstanding questions.

Contribution

It defines new stringy cohomology spaces with flat connections and establishes their relation to GKZ systems, also fixing a gap in previous work and constructing products via vertex algebras.

Findings

Spaces have the same dimension as previous definitions.

Spaces admit natural flat connections.

Constructed products using vertex algebra techniques.

Abstract

We modify the definition of the families of $A$ and $B$ stringy cohomology spaces associated to a pair of dual reflexive Gorenstein cones. The new spaces have the same dimension as the ones defined in the joint paper with Mavlyutov \cite{BM}, but they admit natural flat connections with respect to the appropriate parameters. This solves a longstanding question of relating GKZ hypergeometric system to stringy cohomology. We construct products on these spaces by vertex algebra techniques. In the process, we fix a minor gap in \cite{BM} and prove a statement on intersection cohomology of dual cones that may be of independent interest.