Problems of fields on super Riemann surfaces

Abstract

Superconformal fields of type k on a compact super Riemann surface are defined and studied. It is shown that if there are non-trivial fermionic moduli, the space of such fields can behave in a highly anomalous way; with a module structure which depends strongly on the moduli. (This will only happen for -1 ≤ k ≤ 2, but this includes the case of superscalar fields.) The proofs are explicit for genus 1, the actual spaces of fields being computed; and implicit for higher genus, via a spectral sequence. The implications for superstring theory are indicated.