| Title:
|
Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications (English) |
| Author:
|
Qiu, Jian |
| Author:
|
Zabzine, Maxim |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
47 |
| Issue:
|
5 |
| Year:
|
2011 |
| Pages:
|
415-471 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
These notes are intended to provide a self-contained introduction to the basic ideas of finite dimensional Batalin-Vilkovisky (BV) formalism and its applications. A brief exposition of super- and graded geometries is also given. The BV–formalism is introduced through an odd Fourier transform and the algebraic aspects of integration theory are stressed. As a main application we consider the perturbation theory for certain finite dimensional integrals within BV-formalism. As an illustration we present a proof of the isomorphism between the graph complex and the Chevalley-Eilenberg complex of formal Hamiltonian vectors fields. We briefly discuss how these ideas can be extended to the infinite dimensional setting. These notes should be accessible to both physicists and mathematicians. (English) |
| Keyword:
|
Batalin-Vilkovisky formalism |
| Keyword:
|
graded symplectic geometry |
| Keyword:
|
graph homology |
| Keyword:
|
perturbation theory |
| MSC:
|
16E45 |
| MSC:
|
58A50 |
| MSC:
|
97K30 |
| idZBL:
|
Zbl 1265.58003 |
| idMR:
|
MR2876945 |
| . |
| Date available:
|
2011-12-16T15:30:57Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141789 |
| . |
| Reference:
|
[1] Axelrod, S., Singer, I. M.: Chern–Simons perturbation theory II.J. Differential Geom. 39 (1994), 173–213. Zbl 0889.53053, MR 1258919 |
| Reference:
|
[2] Bar–Natan, D.: On the Vassiliev Knot Invariants.Topology 34 (1995), 423–472. MR 1318886, 10.1016/0040-9383(95)93237-2 |
| Reference:
|
[3] Batalin, I. A., Vilkovisky, G. A.: Gauge algebra and quantization.Phys. Lett. B 102 (1981), 27–31. MR 0616572, 10.1016/0370-2693(81)90205-7 |
| Reference:
|
[4] Batalin, I. A., Vilkovisky, G. A.: Quantization of gauge theories with linearly dependent generators.Phys. Rev. D 28 (1983), 2567–2582, [Erratum-ibid. D 30 (1984) 508]. MR 0726170, 10.1103/PhysRevD.28.2567 |
| Reference:
|
[5] Carmeli, C., Caston, L., Fioresi, R.: Mathematical foundation of supersymmetry.EMS Ser. Lect. Math., 2011, with an appendix I. Dimitrov. MR 2840967 |
| Reference:
|
[6] Cattaneo, A. S., Felder, G.: A path integral approach to the Kontsevich quantization formula.Comm. Math. Phys. 212 (2000), 591–611, [arXiv:math/9902090]. Zbl 1038.53088, MR 1779159, 10.1007/s002200000229 |
| Reference:
|
[7] Cattaneo, A. S., Fiorenza, D., Longoni, R.: Graded Poisson Algebras.Encyclopedia of Mathematical Physics (Françoise, J.-P., Naber, G. L., Tsou, S. T., eds.), vol. 2, Oxford, Elsevier, 2006, pp. 560–567. |
| Reference:
|
[8] Cattaneo, A. S., Mnëv, P.: Remarks on Chern–Simons invariants.Comm. Math. Phys. 293 (2010), 803–836, [arXiv:0811.2045 [math.QA]]. Zbl 1246.58018, MR 2566163, 10.1007/s00220-009-0959-1 |
| Reference:
|
[9] Conant, J., Vogtmann, K.: On a theorem of Kontsevich.Algebr. Geom. Topol. 3 (2003), 1167–1224, [arXiv:math/0208169]. Zbl 1063.18007, MR 2026331, 10.2140/agt.2003.3.1167 |
| Reference:
|
[10] Deligne, P., Morgan, J. W.: Notes on supersymmetry (following Joseph Bernstein). Quantum fields and strings: a course for mathematicians.Amer. Math. Soc., Providence, RI 1, 2 (1999), 41–97, Vol. 1, 2 (Princeton, NJ, 1996/1997). MR 1701597 |
| Reference:
|
[11] Getzler, E.: Batalin–Vilkovisky algebras and two–dimensional topological field theories.Comm. Math. Phys. 159 (1994), 265–285. Zbl 0807.17026, MR 1256989, 10.1007/BF02102639 |
| Reference:
|
[12] Hamilton, A.: A super–analogue of Kontsevich’s theorem on graph homology.[arXiv:math/0510390v1]. Zbl 1173.17020 |
| Reference:
|
[13] Hamilton, A., Lazarev, A.: Graph cohomology classes in the Batalin–Vilkovisky formalism.J. Geom. Phys. 59 (2009), 555–575. Zbl 1204.53070, MR 2518988, 10.1016/j.geomphys.2009.01.007 |
| Reference:
|
[14] Hochschild, G., Serre, J–P.: Cohomology of Lie algebras.Ann. of Math. (2) 57 (3) (1953), 591–603. Zbl 0053.01402, MR 0054581, 10.2307/1969740 |
| Reference:
|
[15] Kontsevich, M.: Formal (non)–commutative symplectic geometry.The Gelfand Mathematical Seminars, 1990 – 1992, Birkhäuser, 1993, pp. 173–187. Zbl 0821.58018, MR 1247289 |
| Reference:
|
[16] Kontsevich, M.: Feynman diagrams and low–dimensional topology.First European Congress of Mathematics, 1992, Paris, Progress in Mathematics 120, vol. II, Birkhäuser, 1994, pp. 97–121. Zbl 0872.57001, MR 1341841 |
| Reference:
|
[17] Polyak, M.: Feynman diagrams for pedestrians and mathematicians.Graphs and patterns in mathematics and theoretical physics, vol. 73, Proc. Sympos. Pure Math., 2005, [arXiv:math/0406251], pp. 15–42. Zbl 1080.81047, MR 2131010 |
| Reference:
|
[18] Qiu, J., Zabzine, M.: Knot invariants and new weight systems from general 3D TFTs.arXiv:1006.1240 [hep-th]. |
| Reference:
|
[19] Qiu, J., Zabzine, M.: Odd Chern–Simons theory, Lie algebra cohomology and characteristic classes.Comm. Math. Phys. 300 (2010), 789–833, arXiv:0912.1243 [hep-th]. Zbl 1214.81262, MR 2736963, 10.1007/s00220-010-1102-z |
| Reference:
|
[20] Roytenberg, D.: On the structure of graded symplectic supermanifolds and Courant algebroids.Quantization, Poisson brackets and beyond (Manchester, 2001), Contemp. Math., 2002, arXiv:math/0203110, pp. 169–185. Zbl 1036.53057, MR 1958835 |
| Reference:
|
[21] Sawon, J.: Rozansky–Witten invariants of hyperkähler manifolds.Ph.D. thesis, Oxford, 1999. Zbl 0946.32014, MR 1708931 |
| Reference:
|
[22] Sawon, J.: Perturbative expansion of Chern–Simons theory.Geom. Topol. Monogr. 8 (2006), 145–166, arXiv:math/0504495. Zbl 1108.81034, MR 2402824, 10.2140/gtm.2006.8.145 |
| Reference:
|
[23] Schwarz, A. S.: Geometry of Batalin–Vilkovisky quantization.Comm. Math. Phys. 155 (1993), 249–260, arXiv:hep-th/9205088. Zbl 0786.58017, MR 1230027, 10.1007/BF02097392 |
| Reference:
|
[24] Schwarz, A. S.: Quantum observables, Lie algebra homology and TQFT.Lett. Math. Phys. 49 (2) (1999), 115–122, arXiv:hep-th/9904168. Zbl 1029.81064, MR 1728307, 10.1023/A:1007684424728 |
| Reference:
|
[25] Varadarajan, V. S.: Supersymmetry for mathematicians: an introduction.Courant Lecture Notes in Mathematics, 11 ed., AMS, New York, 2004. Zbl 1142.58009, MR 2069561 |
| . |
