Operator formalism and tau function for supersymmetric ghosts in higher genus

Abstract

We derive the relation and establish the consistency between two different approaches to the operator bosonization of (super)ghosts on Riemann surfaces, the global operator formalism and the tau function technique. We solve an apparent puzzle between the free field representation provided by vertex operators acting on the tau function, and the non-free superghost operator insertions (as dictated by global effects on the Riemann surface). A version of the global operator formalism is proposed which renders all insertions within one fixed coordinate patch free. The bridge between the two bosonization approaches is then provided by an “operator valued algebro-geometric super tau function”. This can be explicitly derived from the modified operator formalism and in turn yields the ordinary tau function, showing at the same time an “equivariance” under the action of the vertex operators and fusions with operator insertions.