N-point g-loop vertex for a fermionic theory with arbitrary spin

Abstract

We use the sewing procedure of the operator formalism to construct explicitly the N-point g-loop vertex V N; g for a free fermionic ( b, c)-system with conformal weight (λ, 1−λ). We show that this vertex has the structure we expect from geometrical arguments. We obtain also several geometrical objects, e.g. the holomorphic λ-differentials on an arbitrary Riemann surface, which turn out to be expressed as a Poincaré θ-series over all elements of the Schottky group. From V N; g we compute explicitly correlation functions for our system, finding agreement with the geometrical procedure.