On Integrable Structure behind the Generalized WDVV Equations

Abstract

In the theory of quantum cohomologies the WDVV equations imply integrability of the system $(I\partial_\mu - zC_\mu)\psi = 0$. However, in generic situation -- of which an example is provided by the Seiberg-Witten theory -- there is no distinguished direction (like $t^0$) in the moduli space, and such equations for $\psi$ appear inconsistent. Instead they are substituted by $(C_\mu\partial_\nu - C_\nu\partial_\mu)\psi^{(\mu)} \sim (F_\mu\partial_\nu - F_\nu\partial_\mu)\psi^{(\mu)} = 0$, where matrices $(F_\mu)_{\alpha\beta} = \partial_\alpha \partial_\beta \partial_\mu F$.