Rankin–Cohen brackets on supermodular forms

Abstract

Let ℂ1|1 denote the supercomplex plane with local coordinate (z,θ) where θ2=0, and with the canonical supersymmetric (SUSY) structure given by the maximal non-integrable structure distribution of rank 0|1 generated by the odd vector field D=∂∂θ+θ∂∂z. It admits a super action of the super Lie group OSP(1|2,R). This action induces a one parameter action of OSP(1|2,R) on the super-ring, R of C∞-superfunctions on ℂ1|1. A supermodular form is a superfunction F∈R invariant by the action of OSP(1|2,R) on the super-ring R (see Cohen et al., 1997).In this paper, we construct a whole family of operators transforming a given pair of supermodular forms into another supermodular form and apply these operators to define a deformed associative product.