2007/10/31 by Jean-Philippe Michel, Christian Duval
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #hep-th #math-ph #math.DG #math.MP
paper · pdf · doi:10.1093/imrn/rnn054
published as International Mathematics Research Notices 2008 (2008) rnn054 · LaTeX, 50 pages. Minor improvements
openalex publication_date 2008/01/01 · arxiv created 2008/04/22 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/29
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 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 ℱλ(S1|1). The even cross-ratio yields a K(1) 1-cocycle with values in quadratic differentials, Q(S1|1), whose projection on corresponds to the super Schwarzian derivative arising in superconformal field theory. This leads to the classification of the cohomology spaces H1(K(1),ℱλ(S1|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(S1|2) of a K(2) 1-cocycle with values in Q(S1|2). The obstruction to obtain, for N ≥ 3, a projective cocycle is pointed out.