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Communications in Mathematical Physics

, Volume 372, Issue 1, pp 213–260 | Cite as

Globalization for Perturbative Quantization of Nonlinear Split AKSZ Sigma Models on Manifolds with Boundary

  • Alberto S. CattaneoEmail author
  • Nima Moshayedi
  • Konstantin Wernli
Article
  • 36 Downloads

Abstract

We describe a covariant framework to construct a globalized version for the perturbative quantization of nonlinear split AKSZ Sigma Models on manifolds with and without boundary, and show that it captures the change of the quantum state as one changes the constant map around which one perturbs. This is done by using concepts of formal geometry. Moreover, we show that the globalized quantum state can be interpreted as a closed section with respect to an operator that squares to zero. This condition is a generalization of the modified Quantum Master Equation as in the BV-BFV formalism, which we call the modified “differential” Quantum Master Equation.

Notes

Acknowledgements

We thank I. Contreras for helpful comments. Moreover, we want to thank the referee for very important and helpful comments.

References

  1. 1.
    Alexandrov, M., Schwarz, A., Zaboronsky, O., Kontsevich, M.: The geometry of the master equation and topological quantum field theory. Int. J. Mod. Phys. A 12(7), 1405–1429 (1997)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Axelrod, S., Singer, I. M.: Chern–Simons perturbation theory. In: Differential Geometric Methods in Theoretical Physics, Proceedings, New York, vol. 1, pp. 3–45 (1991)Google Scholar
  3. 3.
    Axelrod, S., Singer, I.M.: Chern-Simons perturbation theory II. J. Differ. Geom. 39(1), 173–213 (1994)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Batalin, I.A., Fradkin, E.S.: Operator quantization and abelization of dynamical systems subject to first-class constraints. La Rivista Del Nuovo Cimento Series 3 9(10), 1–48 (1986)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Batalin, I., Fradkin, E.: A generalized canonical formalism and quantization of reducible gauge theories. Phys. Lett. B 122(2), 157–164 (1983)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Batalin, I., Vilkovisky, G.: Gauge algebra and quantization. Phys. Lett. B 102(1), 27–31 (1981)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Batalin, I., Vilkovisky, G.: Relativistic S-matrix of dynamical systems with boson and fermion constraints. Phys. Lett. B 69(3), 309–312 (1977)ADSCrossRefGoogle Scholar
  8. 8.
    Bonechi, F., Cattaneo, A.S., Mnev, P.: The Poisson sigma model on closed surfaces. J. High Energy Phys. 1, 99 (2012)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bott, R.: Some aspects of invariant theory in differential geometry. In: Differential Operators on Manifolds. Springer, Berlin, pp. 49–145 (2010)Google Scholar
  10. 10.
    Bott, R., Cattaneo, A.S.: Integral invariants of 3-manifolds. J. Differ. Geom. 48(1), 91–133 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Brunner, I., Herbst, M., Lerche, W., Scheuner, B.: Landau–Ginzburg realization of open string TFT. In: JHEP 2006 (2006)Google Scholar
  12. 12.
    Cattaneo, A.S., Contreras, I.: Groupoids and Poisson sigma models with boundary. In: Geometric, Algebraic and Topological Methods for Quantum Field Theory. World Scientific, Singapore (2013)Google Scholar
  13. 13.
    Cattaneo, A.S., Contreras, I.: Relational symplectic groupoids. Lett. Math. Phys. 105(5), 723–767 (2015)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Cattaneo, A.S., Felder, G.: A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys. 212, 591–611 (2000)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Cattaneo, A.S., Felder, G.: On the AKSZ formulation of the Poisson sigma model. Lett. Math. Phys. 56(2), 163–179 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cattaneo, A.S., Felder, G.: On the globalization of Kontsevich’s star product and the perturbative Poisson Sigma Model. Progr. Theor. Phys. Suppl. 144, 38–53 (2001)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Cattaneo, A.S., Felder, G., Tomassini, L.: From local to global deformation quantization of Poisson manifolds. Duke Math J. 115(2), 329–352 (2002)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Cattaneo, A.S., Mnev, P., Reshetikhin, N.: Classical and quantum Lagrangian field theories with boundary. In: PoS CORFU2011, p. 44 (2011)Google Scholar
  19. 19.
    Cattaneo, A.S., Mnev, P., Reshetikhin, N.: Classical BV theories on manifolds with boundary. Commun. Math. Phys. 332(2), 535–603 (2014)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Cattaneo, A.S., Mnev, P., Reshetikhin, N.: Perturbative quantum gauge theories on manifolds with boundary. Commun. Math. Phys. 357(2), 631–730 (2017)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Cattaneo, A.S., Mnev, P., Wernli, K.: Split Chern–Simons theory in the BV-BFV formalism. In: Quantization, Geometry and Noncommutative Structures in Mathematics and Physics. Springer, Belin, pp. 293–324 (2017)Google Scholar
  22. 22.
    Cattaneo, A.S., Moshayedi, N.: Introduction to the BV-BFV formalism. (2019). arXiv:1905.08047
  23. 23.
    Cattaneo, A.S., Moshayedi, N., Wernli, K.: On the Globalization of the Poisson sigma model in the BV-BFV formalism (2018). arXiv:1808.01832v1 [math-ph]
  24. 24.
    Cattaneo, A.S., Moshayedi, N., Wernli, K.: Relational symplectic groupoid quantization for constant Poisson structures. Lett. Math. Phys. 107(9), 1649–1688 (2017)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    Contreras, I.: Relational symplectic groupoids and Poisson sigma models with boundary. Ph.D. thesis. Universitäat Zürich (2013)Google Scholar
  26. 26.
    Costello, K.: A geometric construction of the Witten genus, II (2011). arXiv:1112.0816v2
  27. 27.
    Costello, K.: Renormalization and effective field theory, vol. 170. In: Mathematical Surveys and Monographs. American Mathematical Society (AMS) (2011)Google Scholar
  28. 28.
    Dolgushev, V.: Covariant and equivariant formality theorems. Adv. Math. 191(1), 147–177 (2005)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Feynman, R.P.: Mathematical formulation of the quantum theory of electromagnetic interaction. Phys. Rev. 80(3), 440–457 (1950)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Feynman, R.P.: Space-time approach to quantum electrodynamics. Phys. Rev. 76(6), 769–789 (1949)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Fradkin, E.S., Vilkovisky, G.A.: Quantization of relativistic systems with constraints: equivalence of canonical and covariant formalisms in quantum theory of gravitational field. In: CERN Preprint CERN-TH-2332 (1977)Google Scholar
  32. 32.
    Fradkin, E., Vilkovisky, G.: Quantization of relativistic systems with constraints. Phys. Lett. B 55(2), 224–226 (1975)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Fulton, W., MacPherson, R.: A compactification of configuration spaces. Ann. of Math. (2) 139(1), 183–225 (1994)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Gelfand, I.M., Kazhdan, D.A.: Some problems of the differential geometry and the calculation of cohomologies of Lie algebras of vector fields. Dokl. Akad. Nauk Ser. Fiz. 200, 269–272 (1971)Google Scholar
  35. 35.
    Glimm, J., Jaffe, A.: Quantum Physics. Springer, New York (1987)CrossRefGoogle Scholar
  36. 36.
    Grady, R., Li, Q., Li, S.: Batalin–Vilkovisky quantization and the algebraic index. Adv. Math. 317(7), 575–639 (2017)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Gwilliam, O., Grady, R.: One-dimensional Chern–Simons theory and the \(\hat{A}\) genus. Algebr. Geom. Topol. 14(4), 2299–2377 (2014)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton University Press, Princeton (1994)zbMATHGoogle Scholar
  39. 39.
    Ikeda, N.: Two-dimensional gravity and nonlinear gauge theory. Ann. Phys. 235(2), 435–464 (1994)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Iraso, R., Mnev, P.: Two-dimensional Yang–Mills theory on surfaces with corners in Batalin–Vilkovisky formalism (2018). arXiv:1806.04172v1 [math-ph]
  41. 41.
    Kapustin, A., Li, Y.: D-branes in Landau–Ginzburg models and algebraic geometry. In: JHEP 2003 (2004)Google Scholar
  42. 42.
    Khudaverdian, H.M.: Semidensities on odd symplectic supermanifolds. Commun. Math. Phys. 247(2), 353–390 (2004)ADSMathSciNetCrossRefGoogle Scholar
  43. 43.
    Kontsevich, M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157–216 (2003)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Lazaroiu, C.I.: On the boundary coupling of topological Landau–Ginzburg models. In: JHEP 2005 (2005)Google Scholar
  45. 45.
    Liao, H.-Y., Stiénon, M.: Formal exponential map for graded manifolds. Int. Math. Res. Not. 2019(3), 700–730 (2019)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Mnev, P.: Discrete BF theory (2008). arXiv:0809.1160
  47. 47.
    Mnev, P.: Lectures on Batalin–Vilkovisky formalism and its applications in topological quantum field theory (2017). arXiv:1707.08096 [math-ph]
  48. 48.
    Polyak, M.: Feynman diagrams for pedestrians and mathematicians. Proc. Symp. Pure Math. 73, 15–42 (2005)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Reshetikhin, N.: Lectures on quantization of gauge systems. In: New Paths Towards Quantum Gravity. Springer, Berlin, pp. 125–190 (2010)Google Scholar
  50. 50.
    Schaller, P., Strobl, T.: Introduction to Poisson sigma models. In: Grosse, H., Pittner, L. (eds.) Low-Dimensional Models in Statistical Physics and Quantum Field Theory, pp. 321–333. Springer, Berlin (1995)zbMATHGoogle Scholar
  51. 51.
    Schaller, P., Strobl, T.: Poisson structure induced (topological) field theories. Mod. Phys. Lett. A 09(33), 3129–3136 (1994)ADSMathSciNetCrossRefGoogle Scholar
  52. 52.
    Schwarz, A.: Geometry of Batalin–Vilkovisky quantization. Commun. Math. Phys. 155(2), 249–260 (1993)ADSMathSciNetCrossRefGoogle Scholar
  53. 53.
    Ševera, P.: On the origin of the BV operator on odd symplectic supermanifolds. Lett. Math. Phys. 78(1), 55–59 (2006)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121(3), 351–399 (1989)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZurichSwitzerland

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