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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn054, 47 pages, doi:10.1093/imrn/rnn054 published on June 13, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

On the Projective Geometry of the Supercircle: A Unified Construction of the Super Cross-Ratio and Schwarzian Derivative

Jean-Philippe Michel and Christian Duval

Centre de Physique Théorique, CNRS, Luminy, Case 907 F-13288 Marseille Cedex 9, France

Correspondence: Correspondence to be sent to: duval{at}cpt.univ-mrs.fr

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 F{lambda}(S1|1). The even cross-ratio yields a K(1) 1-cocycle with values in quadratic differentials, Q(S1|1), whose projection on Formula corresponds to the super Schwarzian derivative arising in superconformal field theory. This leads to the classification of the cohomology spaces H1(K(1),F{lambda}(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.


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