1-Cocycles on the group of contactomorphisms on the supercircle S ^1|3 generalizing the Schwarzian derivative

Abstract

The relative cohomology H_diff ¹(K(1|3), osp(2, 3);D _γ,µ(S ^1|3)) of the contact Lie superalgebra K(1|3) with coefficients in the space of differential operators D _γ,µ(S ^1|3) acting on tensor densities on S ^1|3, is calculated in N.Ben Fraj, I. Laraied, S. Omri (2013) and the generating 1-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative 1-cocycle s(X _f) = D₁D₂D₃(_f)α₃ ^1/2, X ^f ∈ K(1|3) which is invariant with respect to the conformal subsuperalgebra osp(2, 3) of K(1|3).

In this work we study the supergroup case. We give an explicit construction of 1-cocycles of the group of contactomorphisms K(1|3) on the supercircle S1|3 generating the relative cohomology H_diff ¹(K(1|3), PC(2, 3); D _γ,µ(S ^1|3) with coefficients in D _γ,µ(S ^1|3). We show that they possess properties similar to those of the super-Schwarzian derivative 1-cocycle S ₃(Φ) = E _Φ ^-1 (D₁(D₂),D₃)α₃ ^1/2, Φ ∈ K(1|3) introduced by Radul which is invariant with respect to the conformal group PC(2, 3) of K(1|3). These cocycles are expressed in terms of S ³(Φ) and possess its properties.