Skip to main content
SpringerLink
Log in

String Structures and Trivialisations of a Pfaffian Line Bundle

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

Abstract

The present paper is a contribution to categorial index theory. Its main result is the calculation of the Pfaffian line bundle of a certain family of real Dirac operators as an object in the category of line bundles. Furthermore, it is shown how string structures give rise to trivialisations of that Pfaffian.

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. Atiyah M.F., Singer I.M.: The index of elliptic operators. I. Ann. Math. 87(2), 484–530 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bismut J.-M., Cheeger J.: η-invariants and their adiabatic limits. J. Amer. Math. Soc. 2(1), 33–70 (1989)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  4. Bunke U., Kreck M., Schick Th.: A geometric description of smooth cohomology. Ann. Math. Blaise Pascal 17(1), 1–16 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borthwick D.: The Pfaffian line bundle. Commun. Math. Phys. 149(3), 463–493 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. B. Booss. Topology and analysis. The Atiyah-Singer index formula and gauge-theoretic physics. Universitext, Berlin-New York: Springer-Verlag, 1977

  7. Brylinski, J.L.: Loop spaces, characteristic classes and geometric quantization. Volume 107 of Progress in Mathematics. Boston, MA: Birkhäuser Boston Inc., 1993

  8. Bunke U., Schick Th.: Smooth K-theory. Astérisque 328, 45–135 (2010)

    Google Scholar 

  9. Bunke U., Schick Th.: Real secondary index theory. Algebr. Geom. Topol. 8(2), 1093–1139 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bunke U., Schick Th.: Uniqueness of smooth extensions of generalized cohomology theories. J. Topol. 3(1), 110–156 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bunke, U., Schick, Th., Spitzweck, M., Thom, A.: Duality for topological abelian group stacks and T-duality. In: Cortinas, Guillermo (ed.) et al., K-theory and noncommutative geometry. Proceedings of the ICM 2006 satellite conference, Valladolid, Spain, 2006, Zürich: European Mathematical Society (EMS), 2008, pp. 227–347

  12. Bunke U.: Chern classes on differential K-theory. Pacific J. Math. 247(2), 313–322 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bunke, U.: Index theory, eta forms, and Deligne cohomology. Mem. Amer. Math. Soc. 198(928): vi+120, Providence, RI: Amer. Math. Soc., 2009

  14. Carey A.L., Johnson St., Murray M.K., Stevenson D., Wang B.L.: Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories. Commun. Math. Phys. 259(3), 577–613 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Cheeger, J., Simons, J.: Differential characters and geometric invariants. In: Geometry and topology (College Park, Md., 1983/84), Volume 1167 of Lecture Notes in Math., Berlin: Springer, 1985, pp. 50–80

  16. Deligne, P.: Le déterminant de la cohomologie. In: Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Volume 67 of Contemp. Math., Providence, RI: Amer. Math. Soc., 1987, pp. 93–177

  17. Franke, J.: Chern functors. In: Arithmetic algebraic geometry (Texel, 1989), Volume 89 of Progr. Math., Boston, MA: Birkhäuser Boston, 1991, pp. 75–152

  18. Freed D.S., Lott J.: An index theorem in differential K-theory. Geom. Topol. 14(2), 903–966 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Freed D.S., Moore G.W.: Setting the quantum integrand of M-theory. Commun. Math. Phys. 263(1), 89–132 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Freed, D.S.: On determinant line bundles. In: Mathematical aspects of string theory (San Diego, Calif., 1986), Singapore: World Sci. Publishing, 1987, pp. 189–238

  21. Freed, D.S.: K-theory in quantum field theory. In: Current developments in mathematics, 2001, Somerville, MA: Int. Press, 2002, pp. 41–87

  22. Heinloth, J.: Notes on differentiable stacks. In: Mathematisches Institut, Georg-August-Universität G öttingen: Seminars Winter Term 2004/2005, Göttingen: Universitätsdrucke Göttingen, 2005, pp. 1–32

  23. Hitchin, N.: Lectures on special Lagrangian submanifolds. In: Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), Volume 23 of AMS/IP Stud. Adv. Math., Providence, RI: Amer. Math. Soc., 2001, pp. 151–182

  24. Hopkins M.J., Singer I.M.: Quadratic functions in geometry, topology, and M-theory. J. Diff. Geom. 70(3), 329–452 (2005)

    MathSciNet  MATH  Google Scholar 

  25. Madsen I.: An integral Riemann-Roch theorem for surface bundles. Adv. Math. 225(6), 3229–3257 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Simons J., Sullivan D.: Axiomatic characterization of ordinary differential cohomology. J. Topol. 1(1), 45–56 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. St. Stolz, Teichner, P.: What is an elliptic object?. In: Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser. 308, Cambridge: Cambridge Univ. Press, 2004, pp. 247–343

  28. Stevenson D.: Bundle 2-gerbes. Proc. London Math. Soc. (3) 88(2), 405–435 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Waldorf K.: Multiplicative bundle gerbes with connection. Diff. Geom. Appl. 28(3), 313–340 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Waldorf K.: String connections and Chern-Cimons theory. Diff. Geom. Appl. 28(3), 313–340 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Waldorf K.: More morphisms between bundle gerbes. Theory Appl. Categ. 18(9), 240–273 (2007) (electronic)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. NWF I - Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Ulrich Bunke

Authors
  1. Ulrich Bunke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ulrich Bunke.

Additional information

Communicated by N.A. Nekrasov

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bunke, U. String Structures and Trivialisations of a Pfaffian Line Bundle. Commun. Math. Phys. 307, 675–712 (2011). https://doi.org/10.1007/s00220-011-1348-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-011-1348-0

Keywords

Search

Navigation