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 our previous coauthored work with Mavlyutov, but they admit natural flat connections with respect to the appropriate parameters. This solves a longstanding questionof relating the Gelfand-Kapranov-Zelevinsky (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 our coauthored work with Mavlyutov, and we prove a statement on intersection cohomology of dual cones that may be of independent interest.
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