Comptes Rendus
no. 7-8
Homological Algebra/Algebraic Geometry
Noncommutative Batalin–Vilkovisky geometry and matrix integrals
[Géométrie de Batalin–Vilkovisky non-commutative et intégrales matricielles]

Présenté par : Maxime Kontsevitch

Serguei Barannikov1
1 École normale superieure, 45, rue d'Ulm, 75230 Paris, France
Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 359-362.

J'associe un nouveau type d'intégrales matricielles super-symétriques avec une solution arbitraire de l'équation noncommutative de Batalin–Vilkovisky. Le cas le plus simple est une extension super-symétrique du modèle de Kontsevich de la gravité 2-dimensionnelle.

I study the new type of supersymmetric matrix models associated with any solution to the quantum master equation of the noncommutative Batalin–Vilkovisky geometry. The asymptotic expansion of the matrix integrals gives homology classes in the Kontsevich compactification of the moduli spaces, which I associated with the solutions to the quantum master equation in my previous paper. I associate with the Bernstein–Leites matrix superalgebra equipped with an odd differentiation, whose square is nonzero, the family of cohomology classes of the compactification. This family is the generating function for the products of the tautological classes. The simplest example of my matrix integrals in the case of dimension zero is a supersymmetric extension of the Kontsevich model of 2-dimensional gravity.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.02.002
Serguei Barannikov 1

1 École normale superieure, 45, rue d'Ulm, 75230 Paris, France 
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Serguei Barannikov. Noncommutative Batalin–Vilkovisky geometry and matrix integrals. Comptes Rendus. Mathématique, Volume 348 (2010) no. 7-8, pp. 359-362. doi : 10.1016/j.crma.2010.02.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.02.002/

[1] S. Barannikov Modular operads and non-commutative Batalin–Vilkovisky geometry, Internat. Math. Res. Notices, Volume 2007 (2007) Article ID: rnm075, Preprint MPIM (Bonn) 2006-48

[2] S. Barannikov, Supersymmetric matrix integrals and σ-model, Preprint hal-00443592, 2009

[3] S. Barannikov, Supersymmetry and cohomology of graph complexes, Preprint hal-00429963, 2009

[4] J.N. Bernstein; D.A. Leites The superalgebra Q(n), the odd trace, and the odd determinant, Dokl. Bolg. Akad. Nauk, Volume 35 (1982) no. 3, pp. 285-286

[5] E. Getzler; M. Kapranov Modular operads, Compositio Math., Volume 110 (1998) no. 1, pp. 65-126

[6] Quantum Fields and Strings: A Course for Mathematicians, vols. 1, 2 (P. Deligne et al., eds.), AMS, Providence, RI, 1999

[7] M. Kontsevich Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys., Volume 147 (1992) no. 1, pp. 1-23

[8] M. Kontsevich Feynman diagrams and low-dimensional topology, Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 97-121

Cité par Sources :

☆ Preprint NI06043 (25/09/2006), Isaac Newton Institute for Mathematical Sciences. Preprint HAL, the CNRS electronic archive, 00102085 (28/09/2006).

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