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

Abstract

On the projective geometry of the supercircle: a unified construction of the super cross-ratio and Schwarzian derivative J.-P. Michel ‡ C. Duval § Centre de Physique Théorique, CNRS, Luminy, Case 907 F-13288 Marseille Cedex 9 (France) ¶ Abstract 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 λ (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 spacesThe 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 ≥ 3, a projective cocycle is pointed out. Keywords IntroductionThe cross-ratio is the fundamental object of projective geometry; it is a projective invariant of the circle S 1 (or, rather, of RP 1 ). The main objective of this article is to propose and justify from a group theoretical analysis a super-analogue of the cross-ratio in the case of the supercircle S 1|N , and to deduce then, from the Cartan formula (1.2), the associated Schwarzian derivative for N = 1, 2.It is well-known that the circle, S 1 , admits three different geometries, namely the Euclidean, affine and projective geometries, as highlighted by Ghys [14]. They are defined by the groups (R, +), Aff(1, R) and PGL(2, R), or equivalently by their characteristic invariants, the distance, the distance ratio, and the cross-ratio. From these invariants we can obtain, using Cartan-like formulae, three 1-cocycles of Diff + (S 1 ) with coefficients in some tensorial density modules F λ (S 1 ) with λ ∈ R; see [9]. They are the generators of the three nontrivial cohomology spaces H 1 (Diff + (S 1 ), F λ ), with λ = 0, 1, 2, as proved in [12].The purpose of this article is to extend these results to the supercircle, S 1|N , en- Quite independently, and from a more mathematical point of view, Manin [23] introduced the even and odd cross-ratios, for N = 1, 2, by resorting to linear super-

Supertransvectants and Symplectic Geometry

“…where ad − bc − αβ = 1, e 2 + 2γδ = 1, αe = aδ − cγ and βe = bδ − dγ (cf. [2,16]). We denote by F λ the superspace of functions C ∞ C (S 1|1 ) equipped with the following OSp(1|2)action…”

“…which defines the contact structure on S 1|1 since it spanns the kernel of the contact 1-form α = dx + ξ dξ, see [13,3,16] (Manin [15] calls this vector field the canonical SUSY-structure) 1 . It is characterized by the relations for the Lie superbrackets…”

Schwarzian mechanics via nonlinear realizations

“…2 A conventional way of introducing a super-Schwarzian derivative is to compute a (finite) superconformal transformation of the super stress-energy tensor underlying a 2d N -extended conformal field theory, in which it shows up as the anomalous term [2]- [5]. Alternatively, one can study the cocycles describing central extensions of infinite dimensional Lie superalgebras (see, e.g., [6]). Because for N ≥ 5 the construction of the central term operator is problematic [7], the N = 1, 2, 3, 4 instances mentioned above seem to exhaust all available options.…”

“…A variant in [5] carries an external vector index and is linked to a superconformal field theory which involves non-primary fields. Mathematicians report an obstruction in constructing nontrivial cocycles for N > 3 (see, e.g., the discussion in [6]). A proposal in [12] seems to lack the invariance under finite SU (1, 1|2) transformations.…”

Second-Order Conformally Equivariant Quantization in Dimension 1|2

“…This space is not isomorphic to the corresponding space of symbols, as a K(2)-module. The obstructions to the existence of such an isomorphism are given by (the infinitesimal version of) the Schwarzian derivative, see [11] and references therein. We thus restrict the module structure on D 2 λµ S 1|2 to the orthosymplectic Lie superalgebra osp (2|2) naturally embedded to K (2).…”