Advertisement

String Structures and Trivialisations of a Pfaffian Line Bundle

  • Ulrich BunkeEmail author
Article
First Online:

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.

Keywords

Vector Bundle Line Bundle Dirac Operator Bundle Versus Spin Bundle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

References

  1. AS68.
    Atiyah M.F., Singer I.M.: The index of elliptic operators. I. Ann. Math. 87(2), 484–530 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  2. BC89.
    Bismut J.-M., Cheeger J.: η-invariants and their adiabatic limits. J. Amer. Math. Soc. 2(1), 33–70 (1989)MathSciNetzbMATHGoogle Scholar
  3. BGV92.
    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. BKS09.
    Bunke U., Kreck M., Schick Th.: A geometric description of smooth cohomology. Ann. Math. Blaise Pascal 17(1), 1–16 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. Bor92.
    Borthwick D.: The Pfaffian line bundle. Commun. Math. Phys. 149(3), 463–493 (1992)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. Bos77.
    B. Booss. Topology and analysis. The Atiyah-Singer index formula and gauge-theoretic physics. Universitext, Berlin-New York: Springer-Verlag, 1977Google Scholar
  7. Bry93.
    Brylinski, J.L.: Loop spaces, characteristic classes and geometric quantization. Volume 107 of Progress in Mathematics. Boston, MA: Birkhäuser Boston Inc., 1993Google Scholar
  8. BS07.
    Bunke U., Schick Th.: Smooth K-theory. Astérisque 328, 45–135 (2010)Google Scholar
  9. BS08.
    Bunke U., Schick Th.: Real secondary index theory. Algebr. Geom. Topol. 8(2), 1093–1139 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  10. BS09.
    Bunke U., Schick Th.: Uniqueness of smooth extensions of generalized cohomology theories. J. Topol. 3(1), 110–156 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  11. BSST08.
    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. 227–347Google Scholar
  12. Bun.
    Bunke U.: Chern classes on differential K-theory. Pacific J. Math. 247(2), 313–322 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. Bun09.
    Bunke, U.: Index theory, eta forms, and Deligne cohomology. Mem. Amer. Math. Soc. 198(928): vi+120, Providence, RI: Amer. Math. Soc., 2009Google Scholar
  14. CJM+05.
    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)MathSciNetADSzbMATHCrossRefGoogle Scholar
  15. CS85.
    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. 50–80Google Scholar
  16. Del87.
    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. 93–177Google Scholar
  17. Fra91.
    Franke, J.: Chern functors. In: Arithmetic algebraic geometry (Texel, 1989), Volume 89 of Progr. Math., Boston, MA: Birkhäuser Boston, 1991, pp. 75–152Google Scholar
  18. FL09.
    Freed D.S., Lott J.: An index theorem in differential K-theory. Geom. Topol. 14(2), 903–966 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  19. FM06.
    Freed D.S., Moore G.W.: Setting the quantum integrand of M-theory. Commun. Math. Phys. 263(1), 89–132 (2006)MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. Fre03.
    Freed, D.S.: On determinant line bundles. In: Mathematical aspects of string theory (San Diego, Calif., 1986), Singapore: World Sci. Publishing, 1987, pp. 189–238Google Scholar
  21. Fre02.
    Freed, D.S.: K-theory in quantum field theory. In: Current developments in mathematics, 2001, Somerville, MA: Int. Press, 2002, pp. 41–87Google Scholar
  22. Hei05.
    Heinloth, J.: Notes on differentiable stacks. In: Mathematisches Institut, Georg-August-Universität G öttingen: Seminars Winter Term 2004/2005, Göttingen: Universitätsdrucke Göttingen, 2005, pp. 1–32Google Scholar
  23. Hit01.
    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. 151–182Google Scholar
  24. HS05.
    Hopkins M.J., Singer I.M.: Quadratic functions in geometry, topology, and M-theory. J. Diff. Geom. 70(3), 329–452 (2005)MathSciNetzbMATHGoogle Scholar
  25. Mad09.
    Madsen I.: An integral Riemann-Roch theorem for surface bundles. Adv. Math. 225(6), 3229–3257 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  26. SS08.
    Simons J., Sullivan D.: Axiomatic characterization of ordinary differential cohomology. J. Topol. 1(1), 45–56 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  27. ST04.
    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. 247–343Google Scholar
  28. Ste04.
    Stevenson D.: Bundle 2-gerbes. Proc. London Math. Soc. (3) 88(2), 405–435 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  29. Wala.
    Waldorf K.: Multiplicative bundle gerbes with connection. Diff. Geom. Appl. 28(3), 313–340 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  30. Walb.
    Waldorf K.: String connections and Chern-Cimons theory. Diff. Geom. Appl. 28(3), 313–340 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  31. Wal07.
    Waldorf K.: More morphisms between bundle gerbes. Theory Appl. Categ. 18(9), 240–273 (2007) (electronic)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.NWF I - MathematikUniversität RegensburgRegensburgGermany

Personalised recommendations

Cite article

Buy options