# Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications

Archivum Mathematicum (2011)

- Volume: 047, Issue: 5, page 415-471
- ISSN: 0044-8753

## Access Full Article

top## Abstract

top## How to cite

topQiu, Jian, and Zabzine, Maxim. "Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications." Archivum Mathematicum 047.5 (2011): 415-471. <http://eudml.org/doc/246633>.

@article{Qiu2011,

abstract = {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.},

author = {Qiu, Jian, Zabzine, Maxim},

journal = {Archivum Mathematicum},

keywords = {Batalin-Vilkovisky formalism; graded symplectic geometry; graph homology; perturbation theory; Batalin-Vilkovisky formalism; graded symplectic geometry; graph homology; perturbation theory},

language = {eng},

number = {5},

pages = {415-471},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications},

url = {http://eudml.org/doc/246633},

volume = {047},

year = {2011},

}

TY - JOUR

AU - Qiu, Jian

AU - Zabzine, Maxim

TI - Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications

JO - Archivum Mathematicum

PY - 2011

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 047

IS - 5

SP - 415

EP - 471

AB - 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.

LA - eng

KW - Batalin-Vilkovisky formalism; graded symplectic geometry; graph homology; perturbation theory; Batalin-Vilkovisky formalism; graded symplectic geometry; graph homology; perturbation theory

UR - http://eudml.org/doc/246633

ER -

## References

top- Axelrod, S., Singer, I. M., Chern–Simons perturbation theory II, J. Differential Geom. 39 (1994), 173–213. (1994) Zbl0889.53053MR1258919
- Bar–Natan, D., 10.1016/0040-9383(95)93237-2, Topology 34 (1995), 423–472. (1995) MR1318886DOI10.1016/0040-9383(95)93237-2
- Batalin, I. A., Vilkovisky, G. A., 10.1016/0370-2693(81)90205-7, Phys. Lett. B 102 (1981), 27–31. (1981) MR0616572DOI10.1016/0370-2693(81)90205-7
- Batalin, I. A., Vilkovisky, G. A., 10.1103/PhysRevD.28.2567, Phys. Rev. D 28 (1983), 2567–2582, [Erratum-ibid. D 30 (1984) 508]. (1983) MR0726170DOI10.1103/PhysRevD.28.2567
- Carmeli, C., Caston, L., Fioresi, R., Mathematical foundation of supersymmetry, EMS Ser. Lect. Math., 2011, with an appendix I. Dimitrov. (2011) MR2840967
- Cattaneo, A. S., Felder, G., 10.1007/s002200000229, Comm. Math. Phys. 212 (2000), 591–611, [arXiv:math/9902090]. (2000) Zbl1038.53088MR1779159DOI10.1007/s002200000229
- 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. (2006)
- Cattaneo, A. S., Mnëv, P., 10.1007/s00220-009-0959-1, Comm. Math. Phys. 293 (2010), 803–836, [arXiv:0811.2045 [math.QA]]. (2010) Zbl1246.58018MR2566163DOI10.1007/s00220-009-0959-1
- Conant, J., Vogtmann, K., 10.2140/agt.2003.3.1167, Algebr. Geom. Topol. 3 (2003), 1167–1224, [arXiv:math/0208169]. (2003) Zbl1063.18007MR2026331DOI10.2140/agt.2003.3.1167
- 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). (1999) MR1701597
- Getzler, E., 10.1007/BF02102639, Comm. Math. Phys. 159 (1994), 265–285. (1994) Zbl0807.17026MR1256989DOI10.1007/BF02102639
- Hamilton, A., A super–analogue of Kontsevich’s theorem on graph homology, [arXiv:math/0510390v1]. Zbl1173.17020
- Hamilton, A., Lazarev, A., 10.1016/j.geomphys.2009.01.007, J. Geom. Phys. 59 (2009), 555–575. (2009) Zbl1204.53070MR2518988DOI10.1016/j.geomphys.2009.01.007
- Hochschild, G., Serre, J–P., 10.2307/1969740, Ann. of Math. (2) 57 (3) (1953), 591–603. (1953) Zbl0053.01402MR0054581DOI10.2307/1969740
- Kontsevich, M., Formal (non)–commutative symplectic geometry, The Gelfand Mathematical Seminars, 1990 – 1992, Birkhäuser, 1993, pp. 173–187. (1993) Zbl0821.58018MR1247289
- 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. (1994) Zbl0872.57001MR1341841
- 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. (2005) Zbl1080.81047MR2131010
- Qiu, J., Zabzine, M., Knot invariants and new weight systems from general 3D TFTs, arXiv:1006.1240 [hep-th].
- Qiu, J., Zabzine, M., 10.1007/s00220-010-1102-z, Comm. Math. Phys. 300 (2010), 789–833, arXiv:0912.1243 [hep-th]. (2010) Zbl1214.81262MR2736963DOI10.1007/s00220-010-1102-z
- 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. (2002) Zbl1036.53057MR1958835
- Sawon, J., Rozansky–Witten invariants of hyperkähler manifolds, Ph.D. thesis, Oxford, 1999. (1999) Zbl0946.32014MR1708931
- Sawon, J., 10.2140/gtm.2006.8.145, Geom. Topol. Monogr. 8 (2006), 145–166, arXiv:math/0504495. (2006) Zbl1108.81034MR2402824DOI10.2140/gtm.2006.8.145
- Schwarz, A. S., 10.1007/BF02097392, Comm. Math. Phys. 155 (1993), 249–260, arXiv:hep-th/9205088. (1993) Zbl0786.58017MR1230027DOI10.1007/BF02097392
- Schwarz, A. S., 10.1023/A:1007684424728, Lett. Math. Phys. 49 (2) (1999), 115–122, arXiv:hep-th/9904168. (1999) Zbl1029.81064MR1728307DOI10.1023/A:1007684424728
- Varadarajan, V. S., Supersymmetry for mathematicians: an introduction, Courant Lecture Notes in Mathematics, 11 ed., AMS, New York, 2004. (2004) Zbl1142.58009MR2069561

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.