Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T23:32:00.165Z Has data issue: false hasContentIssue false
  • English
  • Français

Thom isomorphism and push-forward map in twisted K-theory

Published online by Cambridge University Press:  30 November 2007

Alan L. Carey  and
Bai-Ling Wang
Alan L. Carey
Affiliation:
acarey@maths.anu.edu.au Mathematical Sciences Institute, Australian National University, Canberra ACT 0200, Australia
Bai-Ling Wang
Affiliation:
wangb@maths.anu.edu.au Department of Mathematics, Australian National University, Canberra ACT 0200, Australia
Get access

Abstract

We establish the Thom isomorphism in twisted K-theory for any real vector bundle and develop the push-forward map in twisted K-theory for any differentiable map f : X → Y (not necessarily K-oriented). We also obtain the wrong way functoriality property for the push-forward map in twisted K-theory. For D-branes satisfying Freed-Witten's anomaly cancellation condition in a manifold with a non-trivial B-field, we associate a canonical element in the twisted K-group to get the so-called D-brane charges.

Keywords

Twisted K-theoryThom isomorphismpush-forward map and D-brane charges
Type
Research Article
Information
Journal of K-Theory , Volume 1 , Issue 2 , April 2008 , pp. 357 - 393
Copyright
Copyright © ISOPP 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Atiyah, M., Hirzebruch, F. Vector bundles and homogeneous spaces, Proceedings of Symposium in Pure Mathematics, Vol. 3, 7–38. Am. Math. Soc. 1961CrossRefGoogle Scholar
2. Atiyah, M., Segal, G. Twisted K-theory, preprintGoogle Scholar
3. Bouwknegt, P., Evslin, J., Mathai, V. T-Duality: Topology Change from H-flux. Commun. Math. Phys. 249 (2004) 383–415CrossRefGoogle Scholar
4. Berline, N., Getzler, E., Vergne, M. Heat kernels and Dirac operators, Springer-Verlag. Berlin, 1992CrossRefGoogle Scholar
5. Bismut, J.M. Local index theory, eta invariants and holomorphic torsion: a survey. Surveys in differential geometry, Vol. III, 1–76, Int. Press, Boston, MA, 1998Google Scholar
6. Bouwknegt, P., Dawson, P., Ridout, D. D-branes on group manifolds and fusion rings, JHEP 0212 (2002) 065Google Scholar
7. Bouwknegt, P., Carey, A., Mathai, V., Murray, M., Stevenson, D. Twisted K-theory and K-theory of bundle gerbes. Comm. Math. Phys. Vol. 228 no. 1, 17–45, 2002CrossRefGoogle Scholar
8. Bouwknegt, P., Mathai, V. D-branes, B-fields and twisted K-theory, J. High Energy Phys. 03 (2000) 007, hep-th/0002023 D-Branes, RR-Fields and Duality on Noncommutative Manifolds hep-th/0607020Google Scholar
9. Carey, A., Johnson, S., Murray, M., Stevenson, D., Wang, B.L. Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories, Comm. Math. Phys. Vol. 259 No. 3 577–613, 2005CrossRefGoogle Scholar
10. Carey, A., Wang, B.L. On the relationship of gerbes to the odd families index theorem, preprint, math.DG/0407243Google Scholar
11. Carey, A., Wang, B.L. Fusion of Symmetric D-branes and Verlinde ring, preprint, math.ph/0505040Google Scholar
12. Connes, A., Skandalis, G. The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci. 20 (1984), no. 6, 1139–1183CrossRefGoogle Scholar
13. Donavan, P., Karoubi, M. Graded Brauer groups and K-theory with lcoal coefficients, Publ. Math. de IHES, Vol 38, 5–25, 1970CrossRefGoogle Scholar
14. Freed, D., Hopkins, M.J. and Teleman, C. Twisted K-theory and Loop Group Representations, math.AT/0312155, preprintGoogle Scholar
15. Freed, D., Witten, E. Anomalies in String Theory with D-Branes, hep-th/9907189Google Scholar
16. Felder, G., Frohlich, J., Fuchs, J., Schweigert, C. The geometry of WZW branes. J. Geom. Phys. 34 (2000), no. 2, 162–190CrossRefGoogle Scholar
17. Gaberdiel, M., Gannon, T., Roggenkamp, D. The D-branes of SU(n), JHEP 0407 (2004) 015CrossRefGoogle Scholar
18. Gawedzki, K. and Reis, N. WZW branes and gerbes. Rev. Math. Phys. 14 (2002), no. 12, 1281–1334CrossRefGoogle Scholar
19. Kapustin, A. D-branes in a topologically non-trivial B-field, Adv. Theor. Math. Phys. 4 (2001) 127, hep-th/9909089CrossRefGoogle Scholar
20. Karoubi, M. K-theory, an introduction. Grundlehren der math. Wiss. Nr 226. Springer Verlag (1978)Google Scholar
21. Lawson, H. B., Michelsohn, M-L Spin Geometry. Princeton University Press, 1989Google Scholar
22. Maldacena, J., Seiberg, N., Moore, G. Geometrical interpretation of D-branes in gauged WZW models. J. High Energy Phys. 2001, no. 7. hep-th/0105038CrossRefGoogle Scholar
23. Mathai, V., Melrose, R., Singer, I. The index of projective families of elliptic operators, Geom. Topol. 9(2005) 341–373CrossRefGoogle Scholar
24. Mathai, V., Stevenson, D. Chern character in twisted K-theory: equivariant and holomorphic cases, Commun. Math. Phys. 228 (2002) 17–49Google Scholar
25. Gerbes, J. Mickelsson, (twisted) K-theory, and the supersymmetric WZW model. Infinite dimensional groups and manifolds, 93–107, IRMA Lect. Math. Theor. Phys., 5, de Gruyter, Berlin, 2004Google Scholar
26. Murray, M. K. Bundle gerbes, J. London Math. Soc. (2) 54 (1996), no. 2, 403–416CrossRefGoogle Scholar
27. Murray, M. k., Singer, M. A. Singer Gerbes, Clifford modules and the index theorem. Ann. Global Anal. Geom. 26 (2004), no. 4, 355–367CrossRefGoogle Scholar
28. Parker, E. The Brauer group of Graded continuous trace C*-algebras. Trans. of Amer. Math. Soc., Vol 308, No. 1, 1988, 115–132Google Scholar
29. Parker, E. Graded continuous trace C*-algebras and duality. Operator algebras and topology (Craiova, 1989), 130–145, Pitman Res. Notes Math. Ser., 270Google Scholar
30. Pressley, A., Segal, G. Loop groups, Oxford University Press, Oxford, 1988Google Scholar
31. Rosenberg, J. Continuous-trace algebras from the bundle-theoretic point of view. Jour. of Australian Math. Soc. Vol A 47, 368–381, 1989CrossRefGoogle Scholar
32. Schafer-Nameki, S. K-theoretical boundary rings in N = 2 coset models. Nucl. Phys. B 706 (2005) 531. hep-th/0408060CrossRefGoogle Scholar
33. Segal, G. Equivariant K-theory Publications Math. de IHES, Vol. 34, 129–151, 1968CrossRefGoogle Scholar
34. Tu, J., Xu, P., Laurent-Gengoux, C. Twisted K-theory of differentiable stacks, Ann. Sci. Ecole Norm. Sup. (4) Vol 37, 841–910, 2004CrossRefGoogle Scholar
35. Witten, E. D-branes and K-theory J. High Energy Phys. 1998, no. 12, Paper 19CrossRefGoogle Scholar
36. Witten, E. Overview of K-theory applied to strings, Int. J. Mod. Phys. A16 (2001) 693, hep-th/0007175CrossRefGoogle Scholar