| Title:
|
Gerstenhaber and Batalin-Vilkovisky algebras; algebraic, geometric, and physical aspects (English) |
| Author:
|
Roger, Claude |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
45 |
| Issue:
|
4 |
| Year:
|
2009 |
| Pages:
|
301-324 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We shall give a survey of classical examples, together with algebraic methods to deal with those structures: graded algebra, cohomologies, cohomology operations. The corresponding geometric structures will be described(e.g., Lie algebroids), with particular emphasis on supergeometry, odd supersymplectic structures and their classification. Finally, we shall explain how BV-structures appear in Quantum Field Theory, as a version of functional integral quantization. (English) |
| Keyword:
|
supergeometry |
| Keyword:
|
odd symplectic manifolds |
| Keyword:
|
functional integral quantization |
| Keyword:
|
Graded Lie Algebras |
| Keyword:
|
Hochschild cohomology |
| MSC:
|
16-02 |
| MSC:
|
16E40 |
| MSC:
|
16E45 |
| MSC:
|
17B56 |
| MSC:
|
17B70 |
| MSC:
|
53D17 |
| MSC:
|
53D55 |
| MSC:
|
58A50 |
| MSC:
|
58D29 |
| MSC:
|
81T70 |
| idZBL:
|
Zbl 1212.58004 |
| idMR:
|
MR2591684 |
| . |
| Date available:
|
2009-12-22T07:53:31Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/137462 |
| . |
| Reference:
|
[1] Akman, F.: Multibraces on the Hochschild space.J. Pure Appl. Algebra 167 (2002), no. 2-3, 129–163. MR MR1874538 (2002k:17001) Zbl 1008.17001, MR 1874538, 10.1016/S0022-4049(01)00026-3 |
| Reference:
|
[2] Batalin, I. A., Vilkovisky, G. A.: Gauge algebra and quantization.Phys. Lett. B 102 (1981), no. 1, 27–31. MR MR616572 (82j:81047) MR 0616572, 10.1016/0370-2693(81)90205-7 |
| Reference:
|
[3] Batalin, I. A., Vilkovisky, G. A.: Quantization of gauge theories with linearly dependent generators.Phys. Rev. D (3) 28 (1983), no. 10, 2567–2582. MR MR726170 (85i:81068a) MR 0726170, 10.1103/PhysRevD.28.2567 |
| Reference:
|
[4] Bonechi, F., Zabzine, M.: Poisson Sigma model on the sphere.Comm. Math. Phys. 285 (2009), no. 3, 1033–1063. Zbl 1247.53092, MR 2470915, 10.1007/s00220-008-0615-1 |
| Reference:
|
[5] Bouwknegt, P., McCarthy, J., Pilch, K.: The ${\mathcal{W}}_3$ algebra.Lecture Notes in Physics. New Series m: Monographs, vol. 42, Springer-Verlag, Berlin, 1996. MR MR1423803 (97m:17029) MR 1423803 |
| Reference:
|
[6] Brylinski, J.-L., : A differential complex for Poisson manifolds.J. Differential Geom. 28 (1988), no. 1, 93–114. MR MR950556 (89m:58006) Zbl 0634.58029, MR 0950556 |
| Reference:
|
[7] Cahen, M., Gutt, S., De Wilde, M.: Local cohomology of the algebra of $C^{\infty }$ functions on a connected manifold.Lett. Math. Phys. 4 (1980), no. 3, 157–167. MR MR583079 (81j:58046) MR 0583079, 10.1007/BF00316669 |
| Reference:
|
[8] Calvez-Carillo, I., Tonks, A., Vallette, B.: Homotopy Batalin-Vilkovisky algebras.Preprint, 2008. MR MR2062626 (2005i:53122) |
| Reference:
|
[9] Cartan, H., Eilenberg, S.: Homological Algebra.Princeton University Press, Princeton, NJ, 1956. Zbl 0075.24305, MR 0077480 |
| Reference:
|
[10] Cattaneo, A.: On the BV-formalism.Preprint, Zürich Universität, 2005. MR MR2062626 (2005i:53122) |
| Reference:
|
[11] Connes, A.: Noncommutative geometry.Academic Press Inc., San Diego, CA, 1994. MR MR1303779 (95j:46063) Zbl 0818.46076, MR 1303779 |
| Reference:
|
[12] De Wilde, M. and Lecomte, P., : An homotopy formula for the Hochschild cohomology.Compositio Math. 96 (1995), no. 1, 99–109. MR MR1323727 (96f:16012) MR 1323727 |
| Reference:
|
[13]
: Quantum fields and strings: a course for mathematicians. Vol. 1, 2.American Mathematical Society, Providence, RI, 1999, Material from the Special Year on Quantum Field Theory held at the Institute for Advanced Study, Princeton, NJ, 1996–1997. MR MR1701618 (2000e:81010) Zbl 0984.00503, MR 1701618 |
| Reference:
|
[14] Gerstenhaber, M.: The cohomology structure of an associative ring.Ann. of Math. (2) 78 (1963), 267–288. MR MR0161898 (28 #5102) Zbl 0131.27302, MR 0161898 |
| Reference:
|
[15] Gerstenhaber, M., Schack, S. D.: Algebraic cohomology and deformation theory.Deformation theory of algebras and structures and applications (Il Ciocco, 1986), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 247, Kluwer Acad. Publ., Dordrecht, 1988, pp. 11–264. MR MR981619 (90c:16016) Zbl 0676.16022, MR 0981619 |
| Reference:
|
[16] Grothendieck, A.: Éléments de géométrie algébrique. I. Le langage des schémas.Inst. Hautes Études Sci. Publ. Math. (1960), no. 4, 228. MR MR0217083 (36 #177a) Zbl 0118.36206, MR 0163908 |
| Reference:
|
[17] Hochschild, G., Kostant, B., Rosenberg, A.: Differential forms on regular affine algebras.Trans. Amer. Math. Soc. 102 (1962), 383–408. MR MR0142598 (26 #167) Zbl 0102.27701, MR 0142598, 10.1090/S0002-9947-1962-0142598-8 |
| Reference:
|
[18] Huebschmann, J.: Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras.Ann. Inst. Fourier (Grenoble) 48 (1998), no. 2, 425–440. MR MR1625610 (99b:17021) Zbl 0973.17027, MR 1625610, 10.5802/aif.1624 |
| Reference:
|
[19] Khudaverdian, H. M.: Semidensities on odd symplectic supermanifolds.Comm. Math. Phys. 247 (2004), no. 2, 353–390. MR MR2063265 (2005e:58004) Zbl 1057.58002, MR 2063265, 10.1007/s00220-004-1083-x |
| Reference:
|
[20] Khudaverdian, H. M., Voronov, Th. Th.: Differential forms and odd symplectic geometry.2006, http://www.citebase.org/abstract?id=oai:arXiv.org:math/0606560. MR 2462360 |
| Reference:
|
[21] Kontsevich, M.: Deformation quantization of Poisson manifolds.Lett. Math. Phys. 66 (2003), no. 3, 157–216. MR MR2062626 (2005i:53122) Zbl 1058.53065, MR 2062626, 10.1023/B:MATH.0000027508.00421.bf |
| Reference:
|
[22] Kosmann-Schwarzbach, Y.: Exact Gerstenhaber algebras and Lie bialgebroids.Acta Appl. Math. 41 (1995), no. 1-3, 153–165, Geometric and algebraic structures in differential equations. MR MR1362125 (97i:17021) Zbl 0837.17014, MR 1362125, 10.1007/BF00996111 |
| Reference:
|
[23] Lichnerowicz, A.: Les variétés de Poisson et leurs algèbres de Lie associées.J. Differential Geom. 12 (1977), no. 2, 253–300. MR MR0501133 (58 #18565) Zbl 0405.53024, MR 0501133 |
| Reference:
|
[24] Mac Lane, S., : Homology.Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1975 edition. MR MR1344215 (96d:18001) Zbl 0818.18001, MR 1344215 |
| Reference:
|
[25] Markl, M., Shnider, S., Stasheff, J.: Operads in algebra, topology and physics.Mathematical Surveys and Monographs, vol. 96, Amer. Math. Soc., Providence, RI, 2002. MR MR1898414 (2003f:18011) Zbl 1017.18001, MR 1898414 |
| Reference:
|
[26] Nijenhuis, A.: Geometric aspects of formal differential operations on tensors fields.Proc. Internat. Congress Math. 1958, Cambridge Univ. Press, New York, 1960, pp. 463–469. MR MR0170293 (30 #531) MR 0170293 |
| Reference:
|
[27] Peetre, J.: Une caractérisation abstraite des opérateurs différentiels.Math. Scand. 7 (1959), 211–218. Zbl 0089.32502, MR 0112146 |
| Reference:
|
[28] Penkava, M., Schwarz, A.: On some algebraic structures arising in string theory.Perspectives in mathematical physics, Conf. Proc. Lecture Notes Math. Phys., III, Int. Press, Cambridge, MA, 1994, pp. 219–227. MR MR1314668 (96b:81121) Zbl 0871.17021, MR 1314668 |
| Reference:
|
[29] Poletaeva, E.: Analogues of Riemann tensors for the odd metric on supermanifolds.Acta Appl. Math. 31 (1993), no. 2, 137–169. MR MR1223168 (94d:58166) Zbl 0795.53025, MR 1223168, 10.1007/BF00990540 |
| Reference:
|
[30] Schreiber, U.: On the BV-formalism.Preprint, The n-category café, Department of Physics University of Texas at Austin (2006). MR MR2062626 (2005i:53122) |
| Reference:
|
[31] Schwarz, A.: Geometry of Batalin-Vilkovisky quantization.Comm. Math. Phys. 155 (1993), no. 2, 249–260. MR MR1230027 (95f:81095) Zbl 0786.58017, MR 1230027, 10.1007/BF02097392 |
| Reference:
|
[32]
: Felix Berezin: the life and death of the mastermind of supermathematics.World Scientific, 2007, Singapore, 2007, Andrei Losev: from Berezin integral to Batalin-Vilkovisky formalism: a mathematical physicist's point of view. MR 2406259 |
| Reference:
|
[33] Stasheff, J.: The (secret?) homological algebra of the Batalin-Vilkovisky approach.Secondary calculus and cohomological physics (Moscow, 1997), Contemp. Math., vol. 219, Amer. Math. Soc., Providence, RI, 1998, pp. 195–210. MR MR1640453 (2000f:18011) Zbl 0969.17012, MR 1640453 |
| Reference:
|
[34] Tougeron, J.-C., : Idéaux de fonctions différentiables.Springer Verlag, Berlin, Heidelberg, New York, 1972. Zbl 0251.58001, MR 0440598 |
| Reference:
|
[35] Voronov, A. A., Gerstenhaber, M.: Higher-order operations on the Hochschild complex.Funktsional. Anal. i Prilozhen. 29 (1995), no. 1, 1–6, 96. MR MR1328534 (96g:18006) MR 1328534, 10.1007/BF01077036 |
| Reference:
|
[36] Waldmann, S.: Poisson-Geometrie und Deformationsquantisierung. Eine Einführung.Springer-Verlag, Berlin-Heidelberg, 2007. Zbl 1139.53001 |
| Reference:
|
[37] Xu, Ping, : Gerstenhaber algebras and BV-algebras in Poisson geometry.Comm. Math. Phys. 200 (1999), no. 3, 545–560. MR MR1675117 (2000b:17025) MR 1675117 |
| . |
