  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.
