Chiral fermions on a compact Riemann surface: A sheaf cohomology analysis

We prove that the correlation functions of a system of chiral fermions on a compact Riemann surface are determined by postulating their behaviour at coincident points and a principle of maximal analyticity. The proof proceeds by a reformulation as a problem of sheaf cohomology. Wick's theorem and the Fay identities are rigorous consequences of our analysis.