On the Cohomology of the Lie Superalgebra of Contact Vector Fields on S1|2

“…We recover, in this way, three of the four nontrivial 1-cocycles of k(1) with coefficients in F λ (see [1] for a classification). The fourth one,c 0 :…”

Section: The Associated 1-cocycles Of K(1)mentioning

confidence: 99%

“…Much in the same way, we define the Euclidean and affine 1-cocycles of K(N) for any N, with the help of the Cartan-like formulae (5.4), and (5.5). Using the results of Agrebaoui et al [1] on the cohomology of the Lie superalgebra of contact vector field on S 1|1 , we can claim that our three 1-cocycles on K(1) are, indeed, the generators of the three nontrivial cohomology spaces H 1 (K(1), F λ ), where λ = 0, 1 2 , 3 2 . The paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

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

Michel

Duval

2008

International Mathematics Research Notices

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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-

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“…We have analogous definition of weighted densities in super setting (see [2]) with dx replaced by α. …”

Section: The Space Of Polynomial Weighted Densities On K 1|1mentioning

confidence: 99%

“…The homomorphism condition gives, for the term L (2) λ,µ,λ ,µ in t λ,µ t λ ,µ , the following equation…”

Section: Integrability Conditionsmentioning

confidence: 99%

Cohomology of the Lie Superalgebra of Contact Vector Fields on 𝕂^1|1 and Deformations of the Superspace of Symbols

Basdouri¹,

Ammar²,

Fraj³

et al. 2021

JNMP

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Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra K(1) of contact vector fields on the (1,1)-dimensional real or complex superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but osp(1|2)-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal osp(1|2)-trivial deformations of the K(1)-module structure on the superspaces of symbols of differential operators. We prove that any generic formal osp(1|2)-trivial deformation of this K(1)-module is equivalent to a polynomial one of degree ≤ 4. This work is the simplest superization of a result by Bouarroudj [On sl (2)

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On the Cohomology of the Lie Superalgebra of Contact Vector Fields on S1|2

Cited by 29 publications

References 5 publications

Деформация вложений супералгебры Ли контактных векторных полей на $S^{1\mid 2}$ в супералгебру Ли псевдодифференциальных операторов на $S^{1\mid 2}$

Деформация вложений супералгебры Ли контактных векторных полей на $S^{1\mid 2}$ в супералгебру Ли псевдодифференциальных операторов на $S^{1\mid 2}$

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

Cohomology of the Lie Superalgebra of Contact Vector Fields on 𝕂^1|1 and Deformations of the Superspace of Symbols

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On the Cohomology of the Lie Superalgebra of Contact Vector Fields on S1|2

Cited by 29 publications

References 5 publications

Деформация вложений супералгебры Ли контактных векторных полей на $S^{1\mid 2}$ в супералгебру Ли псевдодифференциальных операторов на $S^{1\mid 2}$

Деформация вложений супералгебры Ли контактных векторных полей на $S^{1\mid 2}$ в супералгебру Ли псевдодифференциальных операторов на $S^{1\mid 2}$

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

Cohomology of the Lie Superalgebra of Contact Vector Fields on 𝕂1|1 and Deformations of the Superspace of Symbols

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Cohomology of the Lie Superalgebra of Contact Vector Fields on 𝕂^1|1 and Deformations of the Superspace of Symbols