Gordan-Rankin-Cohen operators on superstrings

Abstract

We distinguish two classifications of bidifferential operators: between (A) spaces of modular forms and (B) spaces of weighted densities. (A) The invariant under the projective action of $\text{SL}(2;\mathbb{Z})$ binary differential operators between spaces of modular forms of integer or half-integer weight on the 1-dimensional manifold were found by Gordan (called transvectants), rediscovered and classified by Rankin and Cohen (called brackets), and, in still another context, by Janson and Peetre. The invariant under the algebraic supergroup $\text{OSp}(1|2; \mathbb{Z})$ super modular forms of integer and half-integer weight on $(1|1)$-dimensional superstrings with contact structure were introduced, bidifferential operators between them classified and further studied by Gieres-Theisen, Cohen-Manin-Zagier, and Gargoubi-Ovsienko. (B) For any complex weights, we classify the analogs of Gordan-Rankin-Cohen (briefly: GRC) binary differential operators between spaces of weighted densities invariant under $\mathfrak{pgl}(2)$. For any complex weights, we classify the analogs of GRC-operators between spaces of weighted densities invariant under the Lie superalgebra $\mathfrak{osp}(1|2)$. In the case of $(1|1)$-dimensional superstring without any additional structure, we also classify the analogs of GRC-operators between spaces of any weighted densities invariant under the Lie superalgebra $\mathfrak{pgl}(1|2)$.