Modular classes of Q-manifolds: a review and some applications

Bruce, Andrew James

Archivum mathematicum, Tome 53 (2017) no. 4, pp. 203-219

Abstract (VO)

A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. The notion of the modular class of a Q-manifold – which is viewed as the obstruction to the existence of a Q-invariant Berezin volume – is not well know. We review the basic ideas and then apply this technology to various examples, including $L_{\infty}$-algebroids and higher Poisson manifolds.

Détail

DOI : 10.5817/AM2017-4-203

Classification : 17B66, 53D17, 57R20, 58A50
Keywords: Q-manifolds; modular classes; characteristic classes; higher Poisson manifolds; $L_{\infty }$-algebroids

Bibliographie

[1] Batalin, I.A., Vilkovisky, G.A.: Gauge algebra and quantization. Phys. Lett. B 102 (1) (1981), 27–31. | DOI | MR

[2] Batalin, I.A., Vilkovisky, G.A.: Quantization of gauge theories with linearly dependent generators. Phys. Rev. D (3) 28 (10) (1983), 2567–2582. | DOI | MR

[3] Batalin, I.A., Vilkovisky, G.A.: Closure of the gauge algebra, generalized Lie equations and Feynman rules. Nuclear Phys. B 234 (1) (1984), 106–124. | MR

[4] Bonavolontà, G., Poncin, N.: On the category of Lie n-algebroids. J. Geom. Phys. 73 (2013), 70–90, arXiv:1207.3590. | DOI | MR | Zbl

[5] Braun, C., Lazarev, A.: Unimodular homotopy algebras and Chern-Simons theory. J. Pure Appl. Algebra 219 (11 (2015), 5158–5194, arXiv:1309.3219. | DOI | MR

[6] Bruce, A.J.: From $L_{\infty }$-algebroids to higher Schouten/Poisson structures. Rep. Math. Phys. 67 (2) (2011), 157–177, arXiv:1007.1389. | DOI | MR | Zbl

[7] Bruce, A.J., Grabowska, K., Grabowski, J.: Linear duals of graded bundles and higher analogues of (Lie) algebroids. J. Geom. Phys. 101 (2016), 71–99, arXiv:1409.0439. | DOI | MR | Zbl

[8] Carmeli, C., Caston, L., Fioresi, R.: Mathematical foundations of supersymmetry. EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2011, xiv+287pp., ISBN: 978-3-03719-097-5. | MR | Zbl

[9] Damianou, P.A., Fernandes, R.L.: Integrable hierarchies and the modular class. Ann. Inst. Fourier (Grenoble) 58 (1) (2008), 107–137, arXiv:math/0607784. | DOI | MR | Zbl

[10] Evens, S., Lu, J.H., Weinstein, A.: Transverse measures, the modular class and a cohomology pairing for Lie algebroids. Quart. J. Math. Oxford Ser. (2) 50 (1999), 417–436, arXiv:dg-ga/9610008. | DOI | MR | Zbl

[11] Fernandes, R.L.: Lie algebroids, holonomy and characteristic classes. Adv. Math. 170 (1) (2002), 119–179, arXiv:math/0007132. | DOI | MR | Zbl

[12] Grabowski, J.: Modular classes of skew algebroid relations. Transform. Groups 17 (4) (2011), 989–1010, arXiv:1108.2366. | DOI | MR

[13] Grabowski, J.: Modular classes revisited. J. Geom. Methods Mod. Phys 11 (9) (2014), 11pp., arXiv:1311.3962. | MR | Zbl

[14] Grabowski, J., Marmo, G., Michor, P.W.: Homology and modular classes of Lie algebroids. Ann. Inst. Fourier (Grenoble) 56 (1) (2006), 69–83, arXiv:math/0310072. | DOI | MR | Zbl

[15] Granåker, J.: Unimodular L-infinity algebras. preprint (2008), arXiv:0803.1763.

[16] Khudaverdian, H.M.: Laplacians in odd symplectic geometry. Quantization, Poisson brackets and beyond, Contemp. Math., vol. 315, Amer. Math. Soc., Providence, RI, 2002. | DOI | MR | Zbl

[17] Khudaverdian, H.M., Voronov, Th.Th.: On odd Laplace operators. Lett. Math. Phys. 62 (2) (2002), 127–142, arXiv:math/0205202. | DOI | MR | Zbl

[18] Khudaverdian, H.M., Voronov, Th.Th.: Higher Poisson brackets and differential forms. Geometric methods in physics, AIP Conf. Proc., 1079, Amer. Inst. Phys., Melville, NY, 2008, arXiv:0808.3406, pp. 203–215. | MR | Zbl

[19] Kosmann-Schwarzbach, Y.: Poisson manifolds, Lie algebroids, modular classes: a survey. SIGMA (2008), paper 005, 30pp., arXiv:0710.3098. | MR | Zbl

[20] Koszul, J.: Crochet de Schouten-Nijenhuis et cohomologie,The mathematical heritage of Élie Cartan (Lyon, 1984). Astérisque, Numéro Hors Série (1985), 257–271. | MR

[21] Kotov, A., Strobl, T.: Characteristic classes associated to Q-bundles. Int. J. Geom. Methods Mod. Phys. 12 (1) (2015), 26 pp., 1550006 arXiv:0711.4106. | DOI | MR | Zbl

[22] Lyakhovich, S.L., Mosman, E.A., Sharapov, A.A.: Characteristic classes of Q-manifolds: classification and applications. J. Geom. Phys. 60 (5) (2010), 729–759, arXiv:0906.0466. | DOI | MR | Zbl

[23] Lyakhovich, S.L., Sharapov, A.A.: Characteristic classes of gauge systems. Nuclear Phys. B 703 (3) (2004), 419–453, arXiv:0906.0466. | DOI | MR | Zbl

[24] Mackenzie, K.C.H.: Double Lie algebroids and second-order geometry, I. Adv. Math. 94 (2) (1992), 180–239. | DOI | MR | Zbl

[25] Mackenzie, K.C.H.: Double Lie algebroids and second-order geometry, II. Adv. Math. 154 (1) (2000), 46–75. | DOI | MR | Zbl

[26] Manin, Y.I.: Gauge field theory and complex geometry. Fundamental Principles of Mathematical Sciences, vol. 289, Springer-Verlag, Berlin, 2nd ed., 1997, xii+346 pp. ISBN: 3-540-61378-1. | MR | Zbl

[27] Mehta, R.A.: Q-algebroids and their cohomology. J. Symplectic Geom. 7 (3) (2009), 263–293, arXiv:math/0703234. | DOI | MR | Zbl

[28] Roytenberg, D.: On the structure of graded symplectic supermanifolds and Courant algebroids. Quantization, Poisson brackets and beyond, Contemp. Math., vol. 315, Amer. Math. Soc., Providence, RI, 2002, arXiv:math/0203110, pp. 169–185. | MR | Zbl

[29] Roytenberg, D.: The modular class of a differential graded manifold, talk presented at the International Workshop on Gauge Theories, Supersymmetry and Mathematical Physics. Lyon, France, 2010, 6-10 April 2010.

[30] Shander, V.N.: Orientations of supermanifolds. Funct. Anal. Appl. 22 (1) (1988), 80–82. | DOI | MR | Zbl

[31] Sheng, Y., Zhu, C.: Higher extensions of Lie algebroids. Commun. Contemp. Math. 0 (2013), 1650034, arXiv:1103.5920. | MR

[32] Vaĭntrob, A.Yu.: Lie algebroids and homological vector fields. Russ. Math. Surv. 52 (1997), 428–429. | DOI | MR | Zbl

[33] Varadarajan, V.S.: Supersymmetry for mathematicians: an introduction. Courant Lecture Notes in Mathematics, 11. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2004, viii+300 pp. ISBN: 0-8218-3574-2. | MR | Zbl

[34] Voronov, Th.: Q-manifolds and Mackenzie theory: an overview. preprint (2007), arXiv:0709.4232. | MR

[35] Voronov, Th.: Higher derived brackets and homotopy algebras. J. Pure Appl. Algebra 202 (1–3) (2005), 133–153, arXiv:math/0304038. | DOI | MR | Zbl

[36] Voronov, Th.: Q-manifolds and Mackenzie theory. Comm. Math. Phys. 315 (2012), 279–310. | DOI | MR | Zbl

[37] Weinstein, A.: The modular automorphism group of a Poisson manifold. J. Geom. Phys. 23 (1997), 379–394. | DOI | MR | Zbl