General Genus Zhu Recursion for Vertex Operator Algebras

We describe Zhu recursion for a vertex operator algebra (VOA) on a general genus Riemann surface in the Schottky uniformization where \(n\)-point correlation functions are written as linear combinations of \((n-1)\)-point functions with universal coefficients. These coefficients are identified with specific geometric structures on the Riemann surface. We apply Zhu recursion to the Heisenberg VOA and determine all its correlation functions. For a general VOA, Zhu recursion with respect to the Virasoro vector is shown to lead to conformal Ward identities expressed in terms of derivatives with respect to the surface moduli. We derive linear partial differential equations for the Heisenberg VOA partition function and various structures such as the bidifferential of the second kind, holomorphic 1-forms and the period matrix. We also compute the genus \(g\) partition function for an even lattice VOA.