Divergence operators and odd Poisson brackets
[Opérateurs de divergence et crochets de Poisson impairs]
Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 419-456.

On définit les opérateurs de divergence sur les algèbres graduées et l’on montre qu’étant donné un crochet de Poisson impair sur l’algèbre, l’opérateur qui associe à un élément la divergence de la dérivation hamiltonienne qu’il définit est un générateur du crochet. C’est le “laplacien impair”, Δ de la quantification de Batalin-Vilkovisky. On étudie alors les générateurs des crochets de Poisson impairs sur les supervariétés, où l’opérateur de divergence peut être défini soit à l’aide d’un volume bérézinien, soit à l’aide d’une connexion graduée. Comme exemples, on trouve des générateurs du crochet de Schouten des multivecteurs sur une variété (la supervariété étant le fibré cotangent, où les coordonnées sur les fibres sont impaires), et ceux du crochet de Koszul-Schouten des formes différentielles sur une variété de Poisson (la supervariété étant le fibré tangent, avec des coordonnées impaires sur les fibres).

We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the “odd laplacian”, Δ, of Batalin-Vilkovisky quantization. We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections. Examples include generators of the Schouten bracket of multivectors on a manifold (the supermanifold being the cotangent bundle where the coordinates in the fibres are odd) and generators of the Koszul-Schouten bracket of forms on a Poisson manifold (the supermanifold being the tangent bundle, with odd coordinates on the fibres).

DOI : https://doi.org/10.5802/aif.1892
Classification : 17B70,  17B63,  58A50,  81S10,  53D17
Mots clés : algèbres de Lie graduées, algèbres de Gerstenhaber, algèbres de Batalin-Vilkovisky, crochet de Schouten, supervariété, volume bérézinien, connexion graduée, équation de Maurer-Cartan, "master equation" quantique
@article{AIF_2002__52_2_419_0,
     author = {Kosmann-Schwarzbach, Yvette and Monterde, Juan},
     title = {Divergence operators and odd Poisson brackets},
     journal = {Annales de l'Institut Fourier},
     pages = {419--456},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {52},
     number = {2},
     year = {2002},
     doi = {10.5802/aif.1892},
     zbl = {1054.53094},
     mrnumber = {1906481},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1892/}
}
Kosmann-Schwarzbach, Yvette; Monterde, Juan. Divergence operators and odd Poisson brackets. Annales de l'Institut Fourier, Tome 52 (2002) no. 2, pp. 419-456. doi : 10.5802/aif.1892. http://www.numdam.org/articles/10.5802/aif.1892/

[1] M. Alexandrov; M. Kontsevich; A. Schwarz; O. Zaboronsky The geometry of the master equation and topological quantum field theory, Int. J. Mod. Phys, Volume A12 (1997), pp. 1405-1429 | MR 1432574 | Zbl 01712530

[2] I. A. Batalin; G. A. Vilkovisky Gauge algebra and quantization, Phys. Lett., Volume B 102 (1981), pp. 27-31 | MR 616572

[3] I. A. Batalin; G. A. Vilkovisky Closure of the gauge algebra, generalized Lie equations and Feynman rules, Nuclear Physics, Volume B 234 (1984), pp. 106-124 | MR 736479

[4] F. A. Berezin Introduction to Superanalysis, D. Reidel, 1987

[5] J. V. Beltrán; J. Monterde Graded Poisson structures on the algebra of differential forms, Comment. Math. Helv., Volume 70 (1995), pp. 383-402 | Article | MR 1340100 | Zbl 0844.58025

[6] J. V. Beltrán; J. Monterde; O. A. Sánchez; - Valenzuela Graded Jacobi operators on the algebra of differential forms, Compositio Math., Volume 106 (1997), pp. 43-59 | Article | MR 1446150 | Zbl 0874.58017

[7] P. Deligne et al. Quantum Fields and Strings: A Course for Mathematicians, Volume vol. 1, part 1 (1999)

[8] B. De; Witt Supermanifolds, Cambridge Univ. Press, 1984

[9] A. Frölicher; A. Nijenhuis Theory of vector-valued differential forms, part I, Indag. Math, Volume 18 (1956), pp. 338-359 | MR 82554 | Zbl 0079.37502

[10] E. Getzler Batalin-Vilkovisky algebras and two-dimensional topological field theories, Commun. Math. Phys., Volume 159 (1994), pp. 265-285 | Article | MR 1256989 | Zbl 0807.17026

[11] H. Hata; B. Zwiebach Developing the covariant Batalin-Vilkovisky approach to string theory, Ann. Phys., Volume 229 (1994), pp. 177-216 | Article | MR 1257465 | Zbl 0784.53054

[12] D. Hernández; Ruipérez; J. Muñoz; Masqué Construction intrinsèque du faisceau de Berezin d'une variété graduée, Comptes Rendus Acad. Sci. Paris, Sér. I Math, Volume 301 (1985), pp. 915-918 | MR 829061 | Zbl 0592.58042

[13] D. Hernández; Ruipérez; J. Muñoz; Masqué Variational berezinian problems and their relationship with graded variational problems, Diff. Geometric Methods in Math. Phys. (Salamanca 1985) (Lect. Notes Math.), Volume 1251 (1987), pp. 137-149 | Zbl 0627.58020

[14] J. Huebschmann Poisson cohomology and quantization, J. für die reine und angew. Math., Volume 408 (1990), pp. 57-113 | Article | MR 1058984 | Zbl 0699.53037

[15] J. Huebschmann Lie-Rinehart algebras, Gerstenhaber algebras, and Batalin-Vilkovisky algebras, Ann. Inst. Fourier, Volume 48 (1998) no. 2, pp. 425-440 | Article | Numdam | MR 1625610 | Zbl 0973.17027

[16] J. Huebschmann Duality for Lie-Rinehart algebras and the modular class, J. für die reine und angew. Math., Volume 510 (1999), pp. 103-159 | Article | MR 1696093 | Zbl 1034.53083

[17] O. M. Khudaverdian Geometry of superspace with even and odd brackets, J. Math. Phys., Volume 32 (1991), pp. 1934-1937 | Article | MR 1112728 | Zbl 0737.58063

[18] O. M. Khudaverdian; Pyatov, P. N., Solodukhin, S. N., eds. Batalin-Vilkovisky formalism and odd symplectic geometry, Geometry and integrable models (Dubna 1994) (1996), pp. 144-181

[19] O. M. Khudaverdian; A. P. Nersessian On the geometry of the Batalin-Vilkovisky formalism, Mod. Phys. Lett., Volume A 8 (1993), pp. 2377-2385 | MR 1234886 | Zbl 1021.81948

[20] Y. Kosmann; - Schwarzbach From Poisson algebras to Gerstenhaber algebras, Ann. Inst. Fourier, Volume 46 (1996) no. 5, pp. 1243-1274 | Article | Numdam | MR 1427124 | Zbl 0858.17027

[21] Y. Kosmann; Schwarzbach Modular vector fields and Batalin-Vilkovisky algebras, Banach Center Publications, Volume 51 (2000), pp. 109-129 | MR 1764439 | Zbl 1018.17020

[22] Y. Kosmann; - Schwarzbach; F. Magri Poisson-Nijenhuis structures, Ann. Inst. Henri Poincaré, Volume A53 (1990), pp. 35-81 | Numdam | MR 1077465 | Zbl 0707.58048

[23] B. Kostant Graded manifolds, graded Lie theory and prequantization, Proc. Conf. Diff. Geom. Methods in Math. Phys. (Bonn 1975) (Lecture Notes Math.), Volume 570 (1977), pp. 177-306 | Zbl 0358.53024

[24] J.-L. Koszul Crochet de Schouten-Nijenhuis et cohomologie, Élie Cartan et les mathématiques d'aujourd'hui (Astérisque, hors série) (1985), pp. 257-271 | Zbl 0615.58029

[25] I. S. Krasil'shchik Schouten brackets and canonical algebras (Lecture Notes Math.), Volume 1334 (1988), pp. 79-110 | Zbl 0661.53059

[26] I. S. Krasil'shchik Supercanonical algebras and Schouten brackets, Mat. Zametki, Volume 49(1) (1991), pp. 70-76 | MR 1101552 | Zbl 0723.58020

[26] I. S. Krasil'shchik Supercanonical algebras and Schouten brackets, Mathematical Notes, Volume 49(1) (1991), pp. 50-54 | MR 1101552 | Zbl 0732.58016

[27] D. Leites Supermanifold Theory, Karelia Branch of the USSR Acad. of Sci., Petrozavodsk (in Russian). (1983)

[28] D. Leites Quantization and supermanifolds, The Schrödinger Equation, Supplément 3 in Berezin, Kluwer, 1991

[29] B. H. Lian; G. J. Zuckerman New perspectives on the BRST-algebraic structure of string theory, Commun. Math. Phys, Volume 154 (1993), pp. 613-646 | Article | MR 1224094 | Zbl 0780.17029

[30] Y. I. Manin Gauge Field Theory and Complex Geometry, Springer-Verlag, 1988 | MR 954833 | Zbl 0641.53001

[31] Y. I. Manin; I. B. Penkov The formalism of left and right connections on supermanifolds, Lectures on Supermanifolds, Geometrical Methods and Conformal Groups, Volume Doebner H.-D., Hennig, J. D.PalevT. D.eds. (1989), pp. 3-13 | Zbl 0824.58007

[32] J. Monterde; A. Montesinos Integral curves of derivations, Ann. Global Anal. Geom., Volume 6 (1988), pp. 177-189 | Article | MR 982764 | Zbl 0632.58017

[33] J. Monterde; O. A. Sánchez; Valenzuela The exterior derivative as a Killing vector field, Israel J. Math., Volume 93 (1996), pp. 157-170 | Article | MR 1380639 | Zbl 0853.58010

[34] I. B. Penkov 𝒟-modules on supermanifolds, Invent. Math., Volume 71 (1983), pp. 501-512 | MR 695902 | Zbl 0528.32012

[35] M. Rothstein Integration on noncompact supermanifolds, Trans. Amer. Math. Soc., Volume 299 (1987), pp. 387-396 | Article | MR 869418 | Zbl 0611.58014

[36] V. Schechtman Remarks on formal deformations and Batalin-Vilkovisky algebras (e-print, math.AG/9802006)

[37] A. Schwarz Geometry of Batalin-Vilkovisky quantization, Commun. Math. Phys., Volume 155 (1993), pp. 249-260 | Article | MR 1230027 | Zbl 0786.58017

[38] A. Schwarz Semi-classical approximation in Batalin-Vilkovisky formalism, Commun. Math. Phys., Volume 158 (1993), pp. 373-396 | Article | MR 1249600 | Zbl 0855.58005

[39] J. Stasheff; Sternheimer, D., Rawnsley, J., Gutt, S., eds. Deformation theory and the Batalin-Vilkovisky master equation, Deformation Theory and Symplectic Geometry (Ascona 1996) (1997), pp. 271-284

[40] I. Vaisman Lectures on the Geometry of Poisson Manifolds, Birkhäuser, 1994 | MR 1269545 | Zbl 0810.53019

[41] T. Voronov Geometric integration theory on supermanifolds, Sov. Sci. Rev. C Math, Volume 9 (1992), pp. 1-138 | MR 1202882 | Zbl 0839.58014

[42] A. Weinstein The modular automorphism group of a Poisson manifold, J. Geom. Phys., Volume 23 (1997), pp. 379-394 | Article | MR 1484598 | Zbl 0902.58013

[43] E. Witten A note on the antibracket formalism, Mod. Phys. Lett., Volume A5 (1990), pp. 487-494 | MR 1049114 | Zbl 1020.81931

[44] P. Xu Gerstenhaber algebras and BV-algebras in Poisson geometry, Commun. Math. Phys., Volume 200 (1999), pp. 545-560 | Article | MR 1675117 | Zbl 0941.17016