Skip to main content
SpringerLink
Log in

Geometric Quantization and the Metric Dependence of the Self-Dual Field Theory

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We investigate the metric dependence of the partition function of the self-dual p-form gauge field on an arbitrary Riemannian manifold. Using geometric quantization of the space of middle-dimensional forms, we derive a projectively flat connection on its space of polarizations. This connection governs metric dependence of the partition function of the self-dual field. We show that the dependence is essentially given by the Cheeger half-torsion of the underlying manifold. We compute the local gravitational anomaly and show how our derivation relates to the classical computation based on index theory. As an application, we show that the one-loop determinant of the (2, 0) multiplet on a Calabi-Yau threefold coincides with the square root of the one-loop determinant of the B-model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Witten E.: Five-brane effective action in M-theory. J. Geom. Phys. 22, 103–133 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Woodhouse, N.M.J.: Geometric quantization. New York: Oxford Mathematical Monographs The Clarendon Press Oxford University Press, Second ed., 1992

  3. Axelrod S., Della Pietra S., Witten E.: Geometric quantization of Chern-Simons gauge theory. J. Diff. Geom. 33, 787–902 (1991)

    MathSciNet  MATH  Google Scholar 

  4. Ray D.B., Singer I.M.: R-torsion and the Laplacian on Riemannian manifolds. Adv. in Math. 7, 145–210 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ray D.B., Singer I.M.: Analytic torsion for complex manifolds. Ann. Math. 98(1), 154–177 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ray, D.B., Singer, I.M.: Analytic torsion. In: Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), Providence, RI: Amer. Math. Soc., 1973, pp. 167–181

  7. Branson, T.: Q-curvature and spectral invariants. In: Proceedings of the 24th Winter School “Geometry and Physics”, Slovák, J., Čadek, M., eds., Palermo: Circolo Matematico di Palermo, 2005, pp. 11–55

  8. Bershadsky M., Cecotti S., Ooguri H., Vafa C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311–428 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Fang, H., Lu, Z., Yoshikawa, K.-I.: Analytic torsion for Calabi-Yau threefolds. J.Diff.Geom. 80, 175 (2008)

    Google Scholar 

  10. Dijkgraaf, R., Verlinde, E.P., Vonk, M.: On the partition sum of the NS five-brane. http://arXiv.org/abs/hep-th/0205281v1, 2002

  11. Alexandrov S., Persson D., Pioline B.: Fivebrane instantons, topological wave functions and hypermultiplet moduli spaces. JHEP 03, 111 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  12. Alvarez-Gaume L., Witten E.: Gravitational Anomalies. Nucl. Phys. B 234, 269 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  13. Witten E.: Global gravitational anomalies. Commun. Math. Phys. 100, 197 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. Henningson M.: Global anomalies in M-theory. Nucl. Phys. B 515, 233–245 (1998)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Hopkins M.J., Singer I.M.: Quadratic functions in geometry, topology, and M-theory. J. Diff. Geom. 70, 329 (2005)

    MathSciNet  MATH  Google Scholar 

  16. Segal G.: Unitary representations of some infinite dimensional groups. Commun. Math. Phys. 80, 301–342 (1981)

    Article  ADS  MATH  Google Scholar 

  17. Belov, D., Moore, G.W.: Holographic action for the self-dual field. http://arXiv.org/abs/hep-th/0605038v1, 2006

  18. Cheeger, J., Simons, J.: Differential characters and geometric invariants. In: Geometry and Topology, Vol. 1167 of Lecture Notes in Mathematics, Berlin / Heidelberg: Springer, 1985, pp. 50–80

  19. Freed D.S., Moore G.W., Segal G.: Heisenberg groups and noncommutative fluxes. Ann. Phys. 322, 236–285 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Monnier, S.: The anomaly bundle of the self-dual field theory. http://arXiv.org/abs/1109.2904v1 [hep-th], 2011

  21. Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators. Vol. 298 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag, 1992

  22. Rosenberg, S.: The Laplacian on a Riemannian manifold. London Mathematical Society student texts. Cambridge: Cambridge University Press, 1997

  23. Rubei E.: Lazzeri’s Jacobian of oriented compact Riemannian manifolds. Ark. Mat. 38(2), 381–397 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Birkenhake, C., Lange, H.: Complex tori. Vol. 177 of Progress in Mathematics. Boston, MA: Birkhäuser Boston Inc., 1999

  25. Birkenhake, C., Lange, H.: Complex abelian varieties. Vol. 302 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Second ed. Berlin: Springer-Verlag, 2004

  26. Putman, A.: The Picard group of the moduli space of curves with level structures. Duke Math. J. 161(4), 623–674 (2012)

    Google Scholar 

  27. Sato M.: The abelianization of a symmetric mapping class group. Math. Proc. Cambridge Phil. Soc. 147, 369 (2009)

    Article  ADS  MATH  Google Scholar 

  28. Monnier, S.: On the half-torsion. In progress

  29. Bismut J.-M., Lott J.: Flat vector bundles, direct images and higher real analytic torsion. J. Amer. Math. Soc. 8(2), 291–363 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Quillen D.: Determinants of Cauchy-Riemann operators on a Riemann surface. Funct. Anal. Appl. 19, 31–34 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  31. Freed, D.S.: On determinant line bundles. In: Mathematical aspects of string theory, S. Yau, ed., Vol. 1 of Advanced series in mathematical physics. Singapore: World Scientific, 1986, pp. 189–238

  32. Atiyah M.: The logarithm of the Dedekind eta-function. Mathematische Ann. 278, 335–380 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  33. Alvarez-Gaume L., Ginsparg P.H.: The Structure of Gauge and Gravitational Anomalies. Ann. Phys. 161, 423 (1985)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  34. Bismut J.-M., Freed D.S.: The analysis of elliptic families. I. Metrics and connections on determinant bundles. Commun. Math. Phys. 106(1), 159–176 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Bismut J.-M., Freed D.S.: The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem. Commun. Math. Phys. 107(1), 103–163 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  36. Schwarz, J.H.: The M theory five-brane. http://arXiv.org/abs/hep-th/9706197, 1997

  37. Pestun V., Witten E.: The Hitchin functionals and the topological B-model at one loop. Lett. Math. Phys. 74, 21–51 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  38. Becker, K., Becker, M., Schwarz J.H.: String Theory and M-Theory: A Modern Introduction. Cambridge: Cambridge University Press, 2007

  39. Witten E.: On background independent open string field theory. Phys. Rev. D 46, 5467–5473 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  40. Gunaydin M., Neitzke A., Pioline B.: Topological wave functions and heat equations. JHEP 12, 070 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  41. Aganagic M., Bouchard V., Klemm A.: Topological Strings and (Almost) Modular Forms. Commun. Math. Phys. 277, 771–819 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. Schwarz A., Tang X.: Quantization and holomorphic anomaly. JHEP 03, 062 (2007)

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Physique Théorique de l’École Normale Supérieure, CNRS UMR 8549, 24 rue Lhomond, 75231, Paris Cedex 05, France

    Samuel Monnier

Authors
  1. Samuel Monnier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Samuel Monnier.

Additional information

Communicated by N. A. Nekrasov

Rights and permissions

Reprints and permissions

About this article

Cite this article

Monnier, S. Geometric Quantization and the Metric Dependence of the Self-Dual Field Theory. Commun. Math. Phys. 314, 305–328 (2012). https://doi.org/10.1007/s00220-012-1525-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-012-1525-9

Keywords

Search

Navigation