Skip to main content

On the Chern Character in Higher Twisted K-Theory and Spherical T-Duality

Abstract

In this paper, we construct for higher twists that arise from cohomotopy classes, the Chern character in higher twisted K-theory, that maps into higher twisted cohomology. We show that it gives rise to an isomorphism between higher twisted K-theory and higher twisted cohomology over the reals. Finally we compute spherical T-duality in higher twisted K-theory and higher twisted cohomology in very general cases.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Arlettaz, D.: The order of the differentials in the Atiyah–Hirzebruch spectral sequence. K-theory 6(4), 347–361 (1992)

  2. 2.

    Atiyah, M.F.: K-Theory. Benjamin, New York (1967)

    MATH  Google Scholar 

  3. 3.

    Atiyah, M.F., Hirzebruch, F.: Vector bundles and homogeneous spaces. Proc. Symp. Pure Math. 3, 7–38 (1961)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Atiyah, M.F., Segal, G.: Twisted K-Theory and Cohomology. Inspired by S. S. Chern. Nankai Tracts Math, vol. 11. World Sci. Publ., Hackensack (2006)

    Google Scholar 

  5. 5.

    Atiyah, M.F., Segal, G.: Twisted K-theory. Ukr. Mat. Visn. 1, 287–330 (2004)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Blackadar, B.: K-Theory for Operator Algebras (Mathematical Sciences Research Institute Publications 5). Springer, New York (1986)

    Google Scholar 

  7. 7.

    Borsuk, K.: Sur les groupes des classes de transformations continues. C. R. Acad. Sci. Paris 202, 1400–1403 (1936)

    MATH  Google Scholar 

  8. 8.

    Borsuk K.: Theory of retracts. Państwowe Wydawn 44. Naukowe (1967)

  9. 9.

    Bouwknegt, P., Mathai, V.: D-branes, B-fields and twisted K-theory. J. High Energy Phys. 03, 007 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Bouwknegt, P., Carey, A., Mathai, V., Murray, M., Stevenson, D.: Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys. 228, 17–49 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  11. 11.

    Bouwknegt, P., Evslin, J., Mathai, V.: T-duality: topology change from H-flux. Commun. Math. Phys. 249, 383–415 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    Bouwknegt, P., Evslin, J., Mathai, V.: Topology and H-flux of T-dual manifolds. Phys. Rev. Lett. 92(18), 3 (2004)

  13. 13.

    Bouwknegt, P., Evslin, J., Mathai, V.: Spherical T-duality. Commun. Math. Phys. 337, 909–954 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Bouwknegt, P., Evslin, J., Mathai, V.: Spherical T-duality II: an infinity of spherical T-duals for non-principal \(SU(2)\)-bundles. J. Geom. Phys. 92, 46–54 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    Bouwknegt, P., Evslin, J., Mathai, V.: Spherical T-duality and the spherical Fourier–Mukai transform. J. Geom. Phys. 133, 303–314 (2018)

    ADS  MathSciNet  Article  Google Scholar 

  16. 16.

    Brook, D.: Higher Twisted K-Theory. MPhil thesis, University of Adelaide (2020)

  17. 17.

    Bunke, U., Schick, T.: On the topology of T-duality. Rev. Math. Phys. 17, 77–112 (2005)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Cavalcanti, G., Gualtieri, M.: Generalized complex geometry and T-duality. In: A Celebration of the Mathematical Legacy of Raoul Bott (CRM Proceedings & Lecture Notes), pp. 341–366. American Mathematical Society (2010)

  19. 19.

    Christensen, J.D., Sinnamon, G., Wu, E.: The D topology for diffeological spaces. Pac. J. Math. 272, 87–110 (2014)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Cuntz, J.: K-theory for certain \(C^{*}\)-algebras. Ann. Math. 113, 181–197 (1981)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Dadarlat, M., Pennig, U.: A Dixmier–Douady theory for strongly self-absorbing \(C^*\)-algebras. J. Reine Angew. Math. 718, 153–181 (2016)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Freed, Daniel S., Hopkins, Michael J., Teleman, Constantin: Loop groups and twisted K-theory I. J. Topol. 4(4), 737–798 (2011)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Gomez Guerra, J.: Models of Twisted K-Theory. Ph.D. Thesis, University of Michigan (2008)

  24. 24.

    Griffiths, P., Morgan, J.: Rational Homotopy Theory and Differential Forms. Progress in Mathematics, vol. 16, 2nd edn. Birkhäuser, Berlin (2013)

    Book  Google Scholar 

  25. 25.

    Garmendia, A., Villatoro, J.: Integration of Singular Foliations via Paths. arXiv:1912.02148 [math.DG] (2019)

  26. 26.

    Hector, G., Macías-Virgós, E., Sanmartín-Carbón, E.: De Rham cohomology of diffeological spaces and foliations. Indag. Math. 21, 212–220 (2011)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Hori, K.: D-branes, T-duality, and index theory. Adv. Theor. Math. Phys. 3, 281–342 (1999)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Husemoller, D.: Fibre Bundles. Springer, New York (1994)

    Book  Google Scholar 

  29. 29.

    Iglesias-Zemmour, P., Karshon, Y., Zadka, M.: Orbifolds as diffeologies. Trans. Am. Math. Soc. 362, 2811–2831 (2010)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Iglesias-Zemmour, P.: Diffeology. Mathematical Surveys and Monographs, vol. 185. American Mathematical Society, Providence (2013)

    MATH  Google Scholar 

  31. 31.

    Kuribayashi, K.: Simplicial cochain algebras for diffeological spaces. arXiv:1902.10937v5 [math.AT] (2019)

  32. 32.

    Li, W., Liu, W., Wang, H.: On a spectral sequence for twisted cohomologies. Chin. Ann. Math. 35B, 633–658 (2014)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Lind, J.A., Sati, H., Westerland, C.: Twisted iterated algebraic K-theory and topological T-duality for sphere bundles. Ann. K-Theory 5, 1–42 (2020)

    MathSciNet  Article  Google Scholar 

  34. 34.

    MacDonald, L.E.: Hierarchies of holonomy groupoids for foliated bundles. arXiv:2004.13929 [math.DG] (2020)

  35. 35.

    MacDonald, L.E.: The holonomy groupoids of singularly foliated bundles. arXiv:2006.14271 [math.DG] (2020)

  36. 36.

    Madsen, I., Snaith, V., Tornehave, J.: Infinite loop maps in geometric topology. Math. Proc. Camb. Philos. Soc. 81, 399–430 (1977)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Mathai, V., Rosenberg, J.: T-duality for torus bundles with H-fluxes via noncommutative topology. Commun. Math. Phys. 253, 705–721 (2005)

    ADS  MathSciNet  Article  Google Scholar 

  38. 38.

    Mathai, V., Stevenson, D.: Chern character in twisted K-theory: equivariant and holomorphic cases. Commun. Math. Phys. 236, 161–186 (2003)

    ADS  MathSciNet  Article  Google Scholar 

  39. 39.

    Mathai, V., Wu, S.: Analytic torsion for twisted de Rham complexes. J. Differ. Geom. 88, 297–332 (2011)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Minasian, R., Moore, G.: K-theory and Ramond–Ramond charge. J. High Energy Phys. (1997). https://doi.org/10.1088/1126-6708/1997/11/002

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Pennig, U.: A non-commutative model for higher twisted K-theory. J. Topol. 9(1), 27–50 (2016)

  42. 42.

    Polchinski, J.: String Theory. Vol. I. An Introduction to the Bosonic String. Reprint of the Edition (Cambridge Monographs on Mathematical Physics). Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  43. 43.

    Polchinski, J.: String Theory. Vol. II. Superstring Theory and Beyond. Reprint of 2003 Edition (Cambridge Monographs on Mathematical Physics). Cambridge University Press, Cambridge (2005)

    Google Scholar 

  44. 44.

    Raeburn, I., Rosenberg, J.: Crossed products of continuous-trace \(C^*\)-algebras by smooth actions. Trans. Am. Math. Soc. 305, 1–45 (1988)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Segal, G.: Categories and cohomology theories. Topology 13, 293–312 (1974)

    MathSciNet  Article  Google Scholar 

  46. 46.

    Soriau, J.-M.: Groupes différentiels. In: García, P.L., Pérez-Rendón, A., Souriau, J.M. (eds.) Differential Geometrical Methods in Mathematical Physics. Lecture Notes in Mathematics, vol. 836. Springer, Berlin (1980)

    Google Scholar 

  47. 47.

    Teleman, C.: K-theory and the moduli space of bundles on a surface and deformations of the Verlinde algebra. In: Ulrike, T. (ed.) Topology,Geometry and Quantum Field Theory, volume 308 of London Mathematical Society Lecture Note Series, pp. 358–378. Cambridge University Press, Cambridge (2004)

  48. 48.

    Walschap, G.: The Euler class as a cohomology generator. Ill. J. Math. 46, 165–169 (2002)

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, vol. 94. Springer, Berlin (1971)

    Google Scholar 

  50. 50.

    Wegge-Olsen, N.E.: \(K\)-Theory and \(C^{*}\)-Algebras: A Friendly Approach. Oxford Science Publications, Oxford (1993)

    MATH  Google Scholar 

  51. 51.

    Westerland, C.: Topological T-duality is twisted Atiyah duality, preprint. arXiv:1503.00210

  52. 52.

    Witten, E.: D-branes and K-theory. J. High Energy Phys. (1998). https://doi.org/10.1088/1126-6708/1998/12/019

    Article  MATH  Google Scholar 

Download references

Acknowledgements

HS and VM were partially supported by funding from the Australian Research Council, through the Australian Laureate Fellowship FL170100020. LM and VM were partially supported by funding from the Australian Research Council, through the Discovery Project Grant DP200100729. HS wishes to acknowledge Jarah Evslin for useful discussions about spectral sequences and their differentials.

All three authors would like to thank the two anonymous referees for their comments on the paper, which have helped to improve the exposition.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Varghese Mathai.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by S. Gukov

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Macdonald, L., Mathai, V. & Saratchandran, H. On the Chern Character in Higher Twisted K-Theory and Spherical T-Duality. Commun. Math. Phys. 385, 331–368 (2021). https://doi.org/10.1007/s00220-021-04096-w

Download citation