×

On the projective geometry of the supercircle: a unified construction of the super cross-ratio and Schwarzian derivative. (English) Zbl 1145.53005

Summary: We consider the standard contact structure on the supercircle \(S^{1|1}\) and the supergroups \(E(1|1)\), \(\text{Aff}(1|1)\) and \(\operatorname{Sp}O(2|1)\) of contactomorphisms, defining the Euclidean, affine, and projective geometries, 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, 1-cocycles of the group of contactomorphisms \(K(1)\) with values in tensor densities \({\mathcal F}_\lambda(S^{1|1})\). The even cross-ratio yields a \(K(1)\) 1-cocycle with values in quadratic differentials, \({\mathcal Q}(S^{1|1})\), whose projection on \({\mathcal F}_{\frac32}(S^{1|1})\) corresponds to the super Schwarzian derivative arising in superconformal field theory. This leads to the classification of the cohomology spaces \(H^1(K(1),{\mathcal F}_\lambda(S^{1|1}))\).
The construction is extended to the case of \(S^{1|N}\). 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 \({\mathcal F}_1(S^{1|2})\) of a \(K(2)\) 1-cocycle with values in \({\mathcal Q}(S^{1|2})\). The obstruction to obtain, for \(N\geq 3\), a projective cocycle is pointed out.

MSC:

53A20 Projective differential geometry
58A50 Supermanifolds and graded manifolds
20J06 Cohomology of groups
PDFBibTeX XMLCite
Full Text: DOI arXiv