Czechoslovak Mathematical Journal, Vol. 71, No. 4, pp. 1235-1248, 2021
Projectively equivariant quantization and symbol on supercircle $S^{1|3}$
Taher Bichr
Received March 30, 2019. Published online September 20, 2021.
Abstract: Let $\mathcal{D}_{\lambda,\mu} $ be the space of linear differential operators on weighted densities from $\mathcal{F}_{\lambda}$ to $\mathcal{F}_{\mu}$ as module over the orthosymplectic Lie superalgebra $\mathfrak{osp}(3|2)$, where $\mathcal{F}_{\lambda} $, $l\in\nobreak\mathbb{C}$ is the space of tensor densities of degree $\lambda$ on the supercircle $S^{1|3}$. We prove the existence and uniqueness of projectively equivariant quantization map from the space of symbols to the space of differential operators. An explicite expression of this map is also given.
Keywords: differential operator; density; equivariant quantization and orthosymplectic algebra
References: [1] C. Duval, P. Lecomte, V. Ovsienko: Conformally equivariant quantization: Existence and uniqueness. Ann. Inst. Fourier 49 (1999), 1999-2029. DOI 10.5802/aif.1744 | MR 1738073 | Zbl 0932.53048
[2] P. Grozman, D. Leites, I. Shchepochkina: Invariant operators on supermanifolds and standard models. Multiple Facets of Quantization and Supersymmetry. World Scientific, River Edge (2002), 508-555. DOI 10.1142/9789812777065_0031 | MR 1964915 | Zbl 1037.58003
[3] P. B. A. Lecomte: Classification projective des espaces d'opérateurs différentiels agissant sur les densités. C. R. Acad. Sci. Paris., Sér. I, Math. 328 (1999), 287-290. (In French.) DOI 10.1016/S0764-4442(99)80211-0 | MR 1675939 | Zbl 1017.17022
[4] P. B. A. Lecomte: Towards projectively equivariant quantization. Prog. Theor. Phys., Suppl. 144 (2001), 125-132. DOI 10.1143/PTPS.144.125 | MR 2023850 | Zbl 1012.53069
[5] P. B. A. Lecomte, V. Y. Ovsienko: Projectively equivariant symbol calculus. Lett. Math. Phys. 49 (1999), 173-196. DOI 10.1023/A:1007662702470 | MR 1743456 | Zbl 0989.17015
[6] D. A. Leites, Y. Kochetkov, A. Weintrob: New invariant differential operators on supermanifolds and pseudo-(co)homology. General Topology and Applications. Lecture Notes in Pure and Applied Mathematics 134. Marcel Dekker, New York (1991), 217-238. MR 1142806 | Zbl 0773.58004
[7] T. Leuther, P. Mathonet, F. Radoux: One $osp(p+1,q+1|2r)$-equivariant quantizations. J. Geom. Phys. 62 (2012), 87-99. DOI 10.1016/j.geomphys.2011.09.003 | MR 2854196 | Zbl 1238.17016
[8] P. Mathonet, F. Radoux: Projectively equivariant quantizations over the superspace $R^{p|q}$. Lett. Math. Phys. 98 (2011), 311-331. DOI 10.1007/s11005-011-0474-0 | MR 2852986 | Zbl 1279.53083
[9] N. Mellouli: Projectively equivariant quantization and symbol calculus in dimension $1|2$. Available at https://arxiv.org/abs/1106.5246v1 (2011), 9 pages.
[10] V. Y. Ovsienko, O. D. Ovsienko, Y. V. Chekanov: Classification of contact-projective structures on the supercircles. Russ. Math. Surv. 44 (1989), 212-213. DOI 10.1070/RM1989v044n03ABEH002135 | MR 1024056 | Zbl 0727.58006
[11] I. Shchepochkina: How to realize a Lie algebra by vector fields. Theor. Math. Phys. 147 (2006), 821-838. DOI 10.1007/s11232-006-0078-5 | MR 2254724 | Zbl 1177.17015
Affiliations: Taher Bichr, Département de Mathématiques, Faculté des sciences de Sfax, Route de la Soukra km 4, 3000 Sfax BP 1171, Tunisia, e-mail: taher-bechr@hotmail.fr
