AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Quantum Field Theory: Batalin–Vilkovisky Formalism and Its Applications
About this Title
Pavel Mnev, Steklov Institute of Mathematics, St. Petersburg, Russia
Publication: University Lecture Series
Publication Year:
2019; Volume 72
ISBNs: 978-1-4704-5271-1 (print); 978-1-4704-5368-8 (online)
DOI: https://doi.org/10.1090/ulect/072
MathSciNet review: 3967709
MSC: Primary 81S40; Secondary 57R56, 81T13, 81T45
Table of Contents
Download chapters as PDF
Front/Back Matter
Chapters
- Introduction
- Classical Chern–Simons theory
- Feynman diagrams
- Batalin–Vilkovisky formalism
- Applications
- C. Albert, B. Bleile, and J. Fröhlich, Batalin-Vilkovisky integrals in finite dimensions, J. Math. Phys. 51 (2010), no. 1, 015213, 31. MR 2605846, DOI 10.1063/1.3278524
- M. Alexandrov, A. Schwarz, O. Zaboronsky, and M. Kontsevich, The geometry of the master equation and topological quantum field theory, Internat. J. Modern Phys. A 12 (1997), no. 7, 1405–1429. MR 1432574, DOI 10.1142/S0217751X97001031
- Michael Atiyah, Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 175–186 (1989). MR 1001453
- Scott Axelrod and I. M. Singer, Chern-Simons perturbation theory, Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, Vol. 1, 2 (New York, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 3–45. MR 1225107
- Scott Axelrod and I. M. Singer, Chern-Simons perturbation theory. II, J. Differential Geom. 39 (1994), no. 1, 173–213. MR 1258919
- John C. Baez and James Dolan, Higher-dimensional algebra and topological quantum field theory, J. Math. Phys. 36 (1995), no. 11, 6073–6105. MR 1355899, DOI 10.1063/1.531236
- I. A. Batalin and E. S. Fradkin, A generalized canonical formalism and quantization of reducible gauge theories, Phys. Lett. B 122 (1983), no. 2, 157–164. MR 697056, DOI 10.1016/0370-2693(83)90784-0
- I. A. Batalin, G. A. Vilkovisky, Relativistic S-matrix of dynamical systems with bosons and fermion constraints, Phys. Lett. B 69.3 (1977) 309–312.
- I. A. Batalin and G. A. Vilkovisky, Gauge algebra and quantization, Phys. Lett. B 102 (1981), no. 1, 27–31. MR 616572, DOI 10.1016/0370-2693(81)90205-7
- I. A. Batalin and G. A. Vilkovisky, Quantization of gauge theories with linearly dependent generators, Phys. Rev. D (3) 28 (1983), no. 10, 2567–2582. MR 726170, DOI 10.1103/PhysRevD.28.2567
- C. Becchi, A. Rouet, and R. Stora, Renormalization of gauge theories, Ann. Physics 98 (1976), no. 2, 287–321. MR 413861, DOI 10.1016/0003-4916(76)90156-1
- Alexander Beilinson and Vladimir Drinfeld, Chiral algebras, American Mathematical Society Colloquium Publications, vol. 51, American Mathematical Society, Providence, RI, 2004. MR 2058353, DOI 10.1090/coll/051
- F. A. Berezin, Vvedenie v algebru i analiz s antikommutiruyushchimi peremennymi, Moskov. Gos. Univ., Moscow, 1983 (Russian). With a preface by A. A. Kirillov. MR 732126
- F. A. Berezin and D. A. Leĭtes, Supermanifolds, Dokl. Akad. Nauk SSSR 224 (1975), no. 3, 505–508 (Russian). MR 0402795
- Damien Calaque, Tony Pantev, Bertrand Toën, Michel Vaquié, and Gabriele Vezzosi, Shifted Poisson structures and deformation quantization, J. Topol. 10 (2017), no. 2, 483–584. MR 3653319, DOI 10.1112/topo.12012
- Alberto S. Cattaneo, Paolo Cotta-Ramusino, Jürg Fröhlich, and Maurizio Martellini, Topological $BF$ theories in $3$ and $4$ dimensions, J. Math. Phys. 36 (1995), no. 11, 6137–6160. MR 1355902, DOI 10.1063/1.531238
- Alberto S. Cattaneo and Giovanni Felder, A path integral approach to the Kontsevich quantization formula, Comm. Math. Phys. 212 (2000), no. 3, 591–611. MR 1779159, DOI 10.1007/s002200000229
- Alberto S. Cattaneo, Giovanni Felder, and Lorenzo Tomassini, From local to global deformation quantization of Poisson manifolds, Duke Math. J. 115 (2002), no. 2, 329–352. MR 1944574, DOI 10.1215/S0012-7094-02-11524-5
- Alberto S. Cattaneo, Formality and star products, Poisson geometry, deformation quantisation and group representations, London Math. Soc. Lecture Note Ser., vol. 323, Cambridge Univ. Press, Cambridge, 2005, pp. 79–144. Lecture notes taken by D. Indelicato. MR 2166452, DOI 10.1017/CBO9780511734878.008
- Alberto S. Cattaneo and Pavel Mnëv, Remarks on Chern-Simons invariants, Comm. Math. Phys. 293 (2010), no. 3, 803–836. MR 2566163, DOI 10.1007/s00220-009-0959-1
- Alberto S. Cattaneo, Pavel Mnev, and Nicolai Reshetikhin, Classical BV theories on manifolds with boundary, Comm. Math. Phys. 332 (2014), no. 2, 535–603. MR 3257656, DOI 10.1007/s00220-014-2145-3
- A. S. Cattaneo, P. Mnev, N. Reshetikhin, Classical and quantum Lagrangian field theories with boundary, PoS (CORFU2011) 044.
- Alberto S. Cattaneo and Pavel Mnev, Wave relations, Comm. Math. Phys. 332 (2014), no. 3, 1083–1111. MR 3262621, DOI 10.1007/s00220-014-2130-x
- Alberto S. Cattaneo, Pavel Mnev, and Nicolai Reshetikhin, Perturbative quantum gauge theories on manifolds with boundary, Comm. Math. Phys. 357 (2018), no. 2, 631–730. MR 3767705, DOI 10.1007/s00220-017-3031-6
- A. S. Cattaneo, P. Mnev, N. Reshetikhin, Perturbative BV theories with Segal-like gluing, arXiv:1602.00741 [math-ph].
- A. S. Cattaneo, P. Mnev, N. Reshetikhin, A cellular topological quantum field theory, arXiv:1701.05874 [math.AT].
- Alberto S. Cattaneo and Carlo A. Rossi, Wilson surfaces and higher dimensional knot invariants, Comm. Math. Phys. 256 (2005), no. 3, 513–537. MR 2161270, DOI 10.1007/s00220-005-1339-0
- Alberto S. Cattaneo and Florian Schätz, Introduction to supergeometry, Rev. Math. Phys. 23 (2011), no. 6, 669–690. MR 2819233, DOI 10.1142/S0129055X11004400
- M. Chas, D. Sullivan, String topology, arXiv:math/9911159.
- Stefan Cordes, Gregory Moore, and Sanjaye Ramgoolam, Lectures on $2$D Yang-Mills theory, equivariant cohomology and topological field theories, Nuclear Phys. B Proc. Suppl. 41 (1995), 184–244. String theory, gauge theory and quantum gravity (Trieste, 1994). MR 1352757, DOI 10.1016/0920-5632(95)00434-B
- Kevin Costello, Renormalization and effective field theory, Mathematical Surveys and Monographs, vol. 170, American Mathematical Society, Providence, RI, 2011. MR 2778558, DOI 10.1090/surv/170
- Kevin Costello and Owen Gwilliam, Factorization algebras in quantum field theory. Vol. 1, New Mathematical Monographs, vol. 31, Cambridge University Press, Cambridge, 2017. MR 3586504, DOI 10.1017/9781316678626
- Pierre Deligne and John W. Morgan, Notes on supersymmetry (following Joseph Bernstein), Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) Amer. Math. Soc., Providence, RI, 1999, pp. 41–97. MR 1701597
- R. Dijkgraaf, A geometrical approach to two-dimensional conformal field theory, Ph.D. thesis, University of Utrecht (1989).
- Robbert Dijkgraaf and Edward Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), no. 2, 393–429. MR 1048699
- Johan L. Dupont, Curvature and characteristic classes, Lecture Notes in Mathematics, Vol. 640, Springer-Verlag, Berlin-New York, 1978. MR 0500997
- P. Etingof, Mathematical ideas and notions of quantum field theory, http://www-math.mit.edu/ etingof/lect.ps (2002).
- L. D. Faddeev, V. N. Popov, “Feynman diagrams for the Yang-Mills field,” Phys. Lett. B 25.1 (1967) 29–30.
- M. V. Fedoryuk, The method of steepest descent, Nauka, Moscow, 1977 (in Russian).
- Boris Fedosov, Deformation quantization and index theory, Mathematical Topics, vol. 9, Akademie Verlag, Berlin, 1996. MR 1376365
- Giovanni Felder and David Kazhdan, The classical master equation, Perspectives in representation theory, Contemp. Math., vol. 610, Amer. Math. Soc., Providence, RI, 2014, pp. 79–137. With an appendix by Tomer M. Schlank. MR 3220627, DOI 10.1090/conm/610/12124
- R. P. Feynman, Quantum theory of gravitation, Acta Phys. Polon. 24 (1963), 697–722. MR 165951
- Richard P. Feynman and Albert R. Hibbs, Quantum mechanics and path integrals, Emended edition, Dover Publications, Inc., Mineola, NY, 2010. Emended and with a preface by Daniel F. Styer. MR 2797644
- D. Fiorenza, An introduction to the Batalin-Vilkovisky formalism, arXiv:math/0402057.
- Jean M. L. Fisch and Marc Henneaux, Homological perturbation theory and the algebraic structure of the antifield-antibracket formalism for gauge theories, Comm. Math. Phys. 128 (1990), no. 3, 627–640. MR 1045888
- Klaus Fredenhagen and Katarzyna Rejzner, Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, Comm. Math. Phys. 317 (2013), no. 3, 697–725. MR 3009722, DOI 10.1007/s00220-012-1601-1
- Daniel S. Freed, Classical Chern-Simons theory. I, Adv. Math. 113 (1995), no. 2, 237–303. MR 1337109, DOI 10.1006/aima.1995.1039
- Daniel S. Freed, Classical Chern-Simons theory. II, Houston J. Math. 28 (2002), no. 2, 293–310. Special issue for S. S. Chern. MR 1898192
- E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys. 159 (1994), no. 2, 265–285. MR 1256989
- Ezra Getzler, Lie theory for nilpotent $L_\infty$-algebras, Ann. of Math. (2) 170 (2009), no. 1, 271–301. MR 2521116, DOI 10.4007/annals.2009.170.271
- Ezra Getzler, The Batalin-Vilkovisky cohomology of the spinning particle, J. High Energy Phys. 6 (2016), 017, front matter+16. MR 3538178, DOI 10.1007/JHEP06(2016)017
- J. Granåker, Unimodular L-infinity algebras, arXiv:0803.1763 [math.QA].
- V. K. A. M. Gugenheim and L. A. Lambe, Perturbation theory in differential homological algebra. I, Illinois J. Math. 33 (1989), no. 4, 566–582. MR 1007895
- Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, Mathematical Surveys, No. 14, American Mathematical Society, Providence, R.I., 1977. MR 0516965
- Marc Henneaux and Claudio Teitelboim, Quantization of gauge systems, Princeton University Press, Princeton, NJ, 1992. MR 1191617
- G. Hochschild, Bertram Kostant, and Alex Rosenberg, Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102 (1962), 383–408. MR 142598, DOI 10.1090/S0002-9947-1962-0142598-8
- Lars Hörmander, Linear partial differential operators, Springer-Verlag, Berlin-New York, 1976. MR 0404822
- Noriaki Ikeda, Two-dimensional gravity and nonlinear gauge theory, Ann. Physics 235 (1994), no. 2, 435–464. MR 1297824, DOI 10.1006/aphy.1994.1104
- Lisa C. Jeffrey, Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Comm. Math. Phys. 147 (1992), no. 3, 563–604. MR 1175494
- Hovhannes M. Khudaverdian, Semidensities on odd symplectic supermanifolds, Comm. Math. Phys. 247 (2004), no. 2, 353–390. MR 2063265, DOI 10.1007/s00220-004-1083-x
- Maxim Kontsevich, Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, Vol. II (Paris, 1992) Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 97–121. MR 1341841
- Maxim Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157–216. MR 2062626, DOI 10.1023/B:MATH.0000027508.00421.bf
- Maxim Kontsevich and Yan Soibelman, Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (Seoul, 2000) World Sci. Publ., River Edge, NJ, 2001, pp. 203–263. MR 1882331, DOI 10.1142/9789812799821_{0}007
- D. A. Leites, Theory of supermanifolds, KF Akad. Nauk SSSR, Petrozavodsk (1983) [in Russian].
- A. Losev, From Berezin integral to Batalin-Vilkovisky formalism: a mathematical physicist’s point of view, in “Felix Berezin: the life and death of the mastermind of supermathematics,” World Scientific (2007).
- Andrey S. Losev, Pavel Mnev, and Donald R. Youmans, Two-dimensional abelian $BF$ theory in Lorenz gauge as a twisted $\mathcal {N}=(2,2)$ superconformal field theory, J. Geom. Phys. 131 (2018), 122–137. MR 3815232, DOI 10.1016/j.geomphys.2018.05.009
- Jacob Lurie, On the classification of topological field theories, Current developments in mathematics, 2008, Int. Press, Somerville, MA, 2009, pp. 129–280. MR 2555928
- Yu. I. Manin, Kalibrovochnye polya i kompleksnaya geometriya, “Nauka”, Moscow, 1984 (Russian). MR 787979
- Yuri I. Manin, Topics in noncommutative geometry, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1991. MR 1095783, DOI 10.1515/9781400862511
- A. A. Migdal, Recursion equations in gauge field theories, Soviet Journal of Experimental and Theoretical Physics 42 (1976) 413.
- P. Mnëv, Notes on simplicial BF theory, Mosc. Math. J. 9 (2009), no. 2, 371–410, back matter (English, with English and Russian summaries). MR 2568441, DOI 10.17323/1609-4514-2009-9-2-371-410
- P. Mnev, Discrete $BF$ theory, Ph.D. dissertation; arXiv:0809.1160 [hep-th].
- P. Mnev, N. Reshetikhin, Faddeev-Popov theorem for BRST integrals, unpublished discussions, 2009.
- Pavel Mnev, A construction of observables for AKSZ sigma models, Lett. Math. Phys. 105 (2015), no. 12, 1735–1783. MR 3420597, DOI 10.1007/s11005-015-0788-4
- P. Mnev, Finite-dimensional BV integrals, unpublished draft, 2013.
- Tony Pantev, Bertrand Toën, Michel Vaquié, and Gabriele Vezzosi, Shifted symplectic structures, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 271–328. MR 3090262, DOI 10.1007/s10240-013-0054-1
- Michael E. Peskin and Daniel V. Schroeder, An introduction to quantum field theory, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1995. Edited and with a foreword by David Pines. MR 1402248
- G. Ponzano, T. Regge, Semiclassical limit of Racah coefficients, in “Spectroscopic and group theoretical methods in physics,” ed. F. Bloch, North-Holland Publ. Co., Amsterdam (1968) 1–58.
- N. Reshetikhin, Lectures on quantization of gauge systems, in “New paths towards quantum gravity,” Springer, Berlin, Heidelberg (2010) 125–190.
- N. Reshetikhin and V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597. MR 1091619, DOI 10.1007/BF01239527
- Vladimir S. Retakh, Lie-Massey brackets and $n$-homotopically multiplicative maps of differential graded Lie algebras, J. Pure Appl. Algebra 89 (1993), no. 1-2, 217–229. MR 1239561, DOI 10.1016/0022-4049(93)90095-B
- Dmitry Roytenberg, AKSZ-BV formalism and Courant algebroid-induced topological field theories, Lett. Math. Phys. 79 (2007), no. 2, 143–159. MR 2301393, DOI 10.1007/s11005-006-0134-y
- Florian Schätz, BFV-complex and higher homotopy structures, Comm. Math. Phys. 286 (2009), no. 2, 399–443. MR 2472031, DOI 10.1007/s00220-008-0705-0
- Peter Schaller and Thomas Strobl, Poisson structure induced (topological) field theories, Modern Phys. Lett. A 9 (1994), no. 33, 3129–3136. MR 1303989, DOI 10.1142/S0217732394002951
- A. S. Schwarz, The partition function of degenerate quadratic functional and Ray-Singer invariants, Lett. Math. Phys. 2 (1977/78), no. 3, 247–252. MR 676337, DOI 10.1007/BF00406412
- A. S. Schwarz, Topological quantum field theories, hep-th/0011260.
- Albert Schwarz, Geometry of Batalin-Vilkovisky quantization, Comm. Math. Phys. 155 (1993), no. 2, 249–260. MR 1230027
- G. B. Segal, The definition of conformal field theory, Differential geometrical methods in theoretical physics (Como, 1987) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 250, Kluwer Acad. Publ., Dordrecht, 1988, pp. 165–171. MR 981378
- Pavol Ševera, On the origin of the BV operator on odd symplectic supermanifolds, Lett. Math. Phys. 78 (2006), no. 1, 55–59. MR 2271128, DOI 10.1007/s11005-006-0097-z
- Jim Stasheff, 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 1640453, DOI 10.1090/conm/219/03076
- Frank Thuillier, Deligne-Beilinson cohomology in $U(1)$ Chern-Simons theories, Mathematical aspects of quantum field theories, Math. Phys. Stud., Springer, Cham, 2015, pp. 233–271. MR 3330244
- V. G. Turaev and O. Ya. Viro, State sum invariants of $3$-manifolds and quantum $6j$-symbols, Topology 31 (1992), no. 4, 865–902. MR 1191386, DOI 10.1016/0040-9383(92)90015-A
- I. V. Tyutin, Gauge invariance in field theory and statistical physics in operator formalism, Lebedev Physics Institute preprint 39 (1975).
- A. Yu. Vaĭntrob, Lie algebroids and homological vector fields, Uspekhi Mat. Nauk 52 (1997), no. 2(314), 161–162 (Russian); English transl., Russian Math. Surveys 52 (1997), no. 2, 428–429. MR 1480150, DOI 10.1070/RM1997v052n02ABEH001802
- Steven Weinberg, The quantum theory of fields. Vol. II, Cambridge University Press, Cambridge, 1996. Modern applications. MR 1411911, DOI 10.1017/CBO9781139644174
- Alan Weinstein, Symplectic categories, Port. Math. 67 (2010), no. 2, 261–278. MR 2662868, DOI 10.4171/PM/1866
- Hassler Whitney, Geometric integration theory, Princeton University Press, Princeton, N. J., 1957. MR 0087148
- K. G. Wilson, J. Kogut, The renormalization group and the $\epsilon$ expansion, Physics Reports 12.2 (1974) 75–199.
- Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772
- Edward Witten, A note on the antibracket formalism, Modern Phys. Lett. A 5 (1990), no. 7, 487–494. MR 1049114, DOI 10.1142/S0217732390000561
- Edward Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), no. 1, 153–209. MR 1133264
- E. Witten, “Notes on supermanifolds and integration,” arXiv:1209.2199.
- S. Zelditch, Method of stationary phase, https://pcmi.ias.edu/files/zelditchStationary\%20phase2.pdf.
- J. Zinn-Justin, Trends in elementary particle theory, Lecture Notes in Physics 37, eds. H. Rollnik and K. Dietz, Springer Berlin (1975).