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Pré-Publication, Document De Travail Année : 2007

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

Résumé

We consider the standard contact structure on the supercircle, $\rS^{1|1}$, and the supergroups $\rE(1|1)$, $\Aff(1|1)$ 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 $\cF_\l(\rS^{1|1})$. The even cross-ratio yields a $K(1)$ one-cocycle with values in quadratic differentials, $\cQ(\rS^{1|1})$, whose projection on $\cF_\frac{3}{2}(\rS^{1|1})$ corresponds to the super Schwarzian derivative arising in superconformal field theory. This leads to the classification of the cohomology spaces $H^1(K(1),\cF_\l(\rS^{1|1}))$. The construction is extended to the case of $\rS^{1|N}$. 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 $\cF_1(\rS^{1|2})$ of a $K(2)$ one-cocycle with values in~$\cQ(\rS^{1|2})$. The obstruction to obtain, for $N\geq 3$, a projective cocycle is pointed out.
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Dates et versions

hal-00177542 , version 1 (08-10-2007)
hal-00177542 , version 2 (31-10-2007)
hal-00177542 , version 3 (22-04-2008)

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Jean-Philippe Michel, Christian Duval. On the projective geometry of the supercircle: a unified construction of the super cross-ratio and Schwarzian derivative. 2007. ⟨hal-00177542v1⟩
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