On the projective geometry of the supercircle: a unified construction of the super cross-ratio and Schwarzian derivative

We consider the standard contact structure on the supercircle, S1|1, and the supergroups E(1|1), Aff(1|1) and SpO(2|1) of contactomorphisms, defining the Euclidean, affine and projective geometry respectively. Using the new notion of p|q-transitivity, we construct in synthetic fashion even and odd invariants characterizing each geometry, and obtain an even and an odd super cross-ratios. Starting from the even invariants, we derive, using a superized Cartan formula, one-cocycles of the group of contactomorphisms, K(1), with values in tensor densities Fλ(S 1|1). The even cross-ratio yields a K(1) one-cocycle with values in quadratic differentials, Q(S1|1), whose projection on F 3 2 (S1|1) corresponds to the super Schwarzian derivative arising in superconformal field theory. This leads to the classification of the cohomology spaces H(K(1),Fλ(S 1|1)). The construction is extended to the case of S1|N . All previous invariants admit a prolongation for N > 1, as well as the associated Euclidean and affine cocycles. The super Schwarzian derivative is obtained from the even cross-ratio, for N = 2, as a projection to F1(S 1|2) of a K(2) one-cocycle with values in Q(S1|2). The obstruction to obtain, for N ≥ 3, a projective cocycle is pointed out.