Communications in Mathematical Physics

, Volume 357, Issue 2, pp 631–730 | Cite as

Perturbative Quantum Gauge Theories on Manifolds with Boundary

  • Alberto S. Cattaneo
  • Pavel Mnev
  • Nicolai ReshetikhinEmail author


This paper introduces a general perturbative quantization scheme for gauge theories on manifolds with boundary, compatible with cutting and gluing, in the cohomological symplectic (BV–BFV) formalism. Explicit examples, like abelian BF theory and its perturbations, including nontopological ones, are presented.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albert, C., Bleile, B., Fröhlich, J.: Batalin–Vilkovisky integrals in finite dimensions. J. Math. Phys. 51(1), 015213 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alekseev A., Mnev P.: One-dimensional Chern–Simons theory. Commun. Math. Phys. 307, 185–227 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alexandrov M., Kontsevich M., Schwarz A., Zaboronsky O.: The geometry of the master equation and topological quantum field theory. Int. J. Mod. Phys. A12, 1405–1430 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Anselmi D.: A general field-covariant formulation of quantum field theory. Eur. Phys. J. C 73(3), 1–19 (2013)Google Scholar
  5. 5.
    Atiyah M.: Topological quantum field theories. Inst. Hautes Etudes Sci. Publ. Math. 68, 175–186 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Axelrod S., Della Pietra S., Witten E.: Geometric quantization of Chern Simons gauge theory. Representations 34, 39 (1991)zbMATHGoogle Scholar
  7. 7.
    Axelrod, S., Singer, I.M.: Chern–Simons perturbation theory. In: Catto, S., Rocha, A. (eds.) Proceedings of the XXth DGM Conference, pp. 3–45. World Scientific, Singapore (1992)Google Scholar
  8. 8.
    Axelrod S., Singer I.M.: Chern–Simons perturbation theory. II. J. Differ. Geom. 39, 173–213 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bar-Natan D.: On the Vassiliev knot invariants. Topology 34, 423–472 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Batalin I.A., Vilkovisky G.A.: Gauge algebra and quantization. Phys. Lett. B 102, 27 (1981)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Batalin I.A., Fradkin E.S.: A generalized canonical formalism and quantization of reducible gauge theories. Phys. Lett. B 122(2), 157–164 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Batalin I.A., Vilkovisky G.A.: Relativistic S-matrix of dynamical systems with boson and fermion costraints. Phys. Lett. B 69(3), 309–312 (1977)ADSCrossRefGoogle Scholar
  13. 13.
    Bates, S., Weinstein, A.: Lectures on the geometry of quantization. Berkeley Mathematics Lecture Notes, vol. 8 (1997)Google Scholar
  14. 14.
    Bonechi F., Cattaneo A.S., Mnev P.: The Poisson sigma model on closed surfaces. JHEP 2012(99), 1–27 (2012)zbMATHGoogle Scholar
  15. 15.
    Bott R., Cattaneo A.S.: Integral invariants of 3-manifolds. J. Differ. Geom. 48, 91–133 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cappell S., DeTurck D., Gluck H., Miller E.Y.: Cohomology of harmonic forms on Riemannian manifolds with boundary. Forum Math. 18, 923–931 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cattaneo, A.S.: Configuration space integrals and invariants for 3-manifolds and knots. In: Nencka, H. (ed.) Low Dimensional Topology, Cont. Math., vol. 233, pp. 153–165 (1999)Google Scholar
  18. 18.
    Cattaneo A.S., Contreras I.: Relational symplectic groupoids. Lett. Math. Phys. 105, 723–767 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cattaneo A.S., Felder G.: A path integral approach to the Kontsevich deformation quantization formula. Commun. Math. Phys. 212(3), 591–611 (2000)ADSCrossRefzbMATHGoogle Scholar
  20. 20.
    Cattaneo A.S., Felder G.: Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model. Lett. Math. Phys. 69, 157–175 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cattaneo A.S., Felder G.: Effective Batalin–Vilkovisky theories, equivariant configuration spaces and cyclic chains. Prog. Math. 287, 111–137 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Cattaneo A.S., Mnev P.: Remarks on Chern–Simons invariants. Commun. Math. Phys. 293, 803–836 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Cattaneo, A.S., Mnev, P., Reshetikhin, N.: Classical BV theories on manifolds with boundary. Commun. Math. Phys. 332(2), 535–603 (2014). arXiv:1201.0290
  24. 24.
    Cattaneo, A.S., Mnev, P., Reshetikhin, N.: Classical and quantum Lagrangian field theories with boundary. In: Proceedings of the Corfu Summer Institute 2011 School and Workshops on Elementary Particle Physics and Gravity. PoS CORFU2011 (2011) 044. arXiv:1207.0239
  25. 25.
    Cattaneo, A.S., Mnev, P., Reshetikhin, N.: Cellular BV–BFV–BF theory (in preparation)Google Scholar
  26. 26.
    Cattaneo, A.S., Mnev, P., Wernli, K.: Split Chern–Simons theory in the BV–BFV formalism. In: Cardona, A. et al. (eds.) Quantization, Geometly and Noncommutative structures in Mathematics and Physics. Mathematical Physics Studies. Springer (2017)Google Scholar
  27. 27.
    Cattaneo A.S., Rossi C.A.: Higher-dimensional BF theories in the Batalin–Vilkovisky formalism: the BV action and generalized Wilson loops. Commun. Math. Phys. 221, 591–657 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Cheeger J.: Analytic torsion and the heat equation. Ann. Math. 109((2), 259–322 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Collins J.C.: Renormalization. Cambridge University Press, Cambridge (1986)Google Scholar
  30. 30.
    Costello, K.: Renormalization and Effective Field Theory, vol. 170. AMS (2011)Google Scholar
  31. 31.
    Felder, G., Kazhdan, D.: The classical master equation. In: Perspectives in Representation Theory. Cont. Math. arXiv:1212.1631
  32. 32.
    Friedrichs K.O.: Differential forms on Riemannian manifolds. Commun. Pure Appl. Math. 8, 551–590 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Fröhlich J., King C.: The Chern–Simons theory and knot polynomials. Commun. Math. Phys. 126(1), 167–199 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Fulton W., MacPherson R.: A compactification of configuration spaces. Ann. Math. 139, 183–225 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Gelfand I., Kazhdan D.: Some problems of differential geometry and the calculation of the cohomology of Lie algebras of vector fields. Sov. Math. Dokl. 12, 1367–1370 (1971)zbMATHGoogle Scholar
  36. 36.
    Ibort, A., Spivak, A.: Covariant Hamiltonian field theories on manifolds with boundary: Yang–Mills theories. arXiv:1506.00338
  37. 37.
    Khudaverdian H.: Semidensities on odd symplectic supermanifolds. Commun. Math. Phys. 247(2), 353–390 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Kontsevich, M.: Feynman Diagrams and Low-Dimensional Topology. First European Congress of Mathematics, Paris 1992, Volume II, Progress in Mathematics, vol. 120. Birkhäuser, p. 120 (1994)Google Scholar
  39. 39.
    Kontsevich M.: Vassiliev’s knot invariants. Adv. Sov. Math. 16, 137–150 (1993)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Kontsevich M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157–216 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Lück W.: Analytic and topological torsion for manifolds with boundary and symmetry. J. Differ. Geom. 37(2), 263–322 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Merkulov S., Vallette B.: Deformation theory of representations of prop(erad)s I. J. Reine Angew. Math. 634, 51–106 (2009)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Merkulov S., Vallette B.: Deformation theory of representations of prop(erad)s II. J. Reine Angew. Math. 636, 123–174 (2009)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Milnor J.: A duality theorem for Reidemeister torsion. Ann. Math. 76, 137–147 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Mnev, P.: Discrete BF theory. arXiv:0809.1160
  46. 46.
    Morrey C.B. Jr: A variational method in the theory of harmonic integrals, II. Am. J. Math. 78, 137–170 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Ray D.B., Singer I.M.: R-torsion and the Laplacian on Riemannian manifolds. Adv. Math. 7, 145–210 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Schätz F.: BFV-complex and higher homotopy structures. Commun. Math. Phys. 286, 399–443 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Schlegel, V.: Gluing manifolds in the Cahiers topos. arXiv:1503.07408 [math.DG]
  50. 50.
    Schwarz A.S.: The partition function of degenerate quadratic functionals and Ray–Singer invariants. Lett. Math. Phys. 2, 247–252 (1978)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Schwarz A.S.: Geometry of Batalin–Vilkovisky quantization. Commun. Math. Phys. 155(2), 249–260 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Schwarz, A.S.: Topological quantum field theories. arXiv:hep-th/0011260
  53. 53.
    Segal, G.: The definition of conformal field theory. In: Differential Geometrical Methods in Theoretical Physics. Springer, Netherlands, pp. 165–171 (1988)Google Scholar
  54. 54.
    Ševera P.: On the origin of the BV operator on odd symplectic supermanifolds. Lett. Math. Phys. 78, 55–59 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Stasheff J.: Homological reduction of constrained Poisson algebras. J. Differ. Geom. 45, 221–240 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Stolz, S., Teichner, P.: Supersymmetric Euclidean field theories and generalized cohomology, a survey. preprint (2008)Google Scholar
  57. 57.
    Turaev V.: Introduction to Combinatorial Torsions. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  58. 58.
    Vishik S.M.: Generalized Ray–Singer conjecture. I. A manifold with a smooth boundary. Commun. Math. Phys. 167(1), 1–102 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Witten E.: On quantum gauge theories in two dimensions. Commun. Math. Phys. 141(1), 153–209 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Wu S.: Topological quantum field theories on manifolds with a boundary. Commun. Math. Phys. 136, 157–168 (1991)ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institut für Mathematik, Universität ZürichZurichSwitzerland
  2. 2.Max Planck Institute for MathematicsBonnGermany
  3. 3.St. Petersburg Department of V. A. Steklov Institute of Mathematics of the Russian Academy of SciencesSaint PetersburgRussian Federation
  4. 4.University of Notre DameNotre DameUSA
  5. 5.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  6. 6.Physics DepartmentSt. Petersburg UniversitySt. PetersburgRussia
  7. 7.KdV Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands

Personalised recommendations