Mathematical Physics
[Submitted on 8 Oct 2007 (v1), last revised 22 Apr 2008 (this version, v3)]
Title:On the projective geometry of the supercircle: a unified construction of the super cross-ratio and Schwarzian derivative
View PDFAbstract: We consider the standard contact structure on the supercircle, S^{1|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 F_\lambda(S^{1|1}). The even cross-ratio yields a K(1) one-cocycle with values in quadratic differentials, Q(S^{1|1}), whose projection on F_{3/2}(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),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 F_1(S^{1|2}) of a K(2) one-cocycle with values in Q(S^{1|2}). The obstruction to obtain, for N\geq 3, a projective cocycle is pointed out.
Submission history
From: Christian Duval [view email] [via CCSD proxy][v1] Mon, 8 Oct 2007 13:12:41 UTC (37 KB)
[v2] Wed, 31 Oct 2007 19:27:58 UTC (37 KB)
[v3] Tue, 22 Apr 2008 11:02:41 UTC (38 KB)
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