Skip to main content
SpringerLink
Log in

A Variant of K-Theory and Topological T-Duality for Real Circle Bundles

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

Abstract

For a space with involutive action, there is a variant of K-theory. Motivated by T-duality in type II orbifold string theory, we establish that a twisted version of the variant enjoys a topological T-duality for Real circle bundles, i.e., circle bundles with real structure.

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.: K-theory and reality. Quart. J. Math. Oxford Ser. (2) 17, 367–386 (1966)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Atiyah, M.F., Hopkins, M.: A variant of K-theory: K ±. In: Topology, Geometry and Quantum Field Theory, pp. 5–17. London Mathematical Society Lecture Note Series, vol. 308. Cambridge University Press, Cambridge (2004)

  3. Baraglia, D.: Conformal Courant algebroids and orientifold T-duality. Int. J. Geom. Methods Mod. Phys. 10(2), 1250084 (2013)

  4. Baraglia, D.: Topological T-duality for general circle bundles. Pure Appl. Math. Q. (to appear) arXiv:1105.0290

  5. Bouwknegt P., Evslin J., Mathai V.: T-duality: topology change from H-flux. Comm. Math. Phys. 249(2), 383–415 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Bouwknegt, P., Evslin, J., Mathai, V.: Topology and H-flux of T-dual manifolds. Phys. Rev. Lett. 92(18), 181601 (2013)

  7. Braun, V., Stefanski, B. jr.: Orientifolds and K-theory.arXiv:hep-th/0206158

  8. Brown, K.S. Cohomology of groups. Corrected reprint of the 1982 original. In: Graduate Texts in Mathematics, vol. 87. Springer, New York (1994)

  9. Brylinski, J.-L.: Gerbes on complex reductive Lie groups. arXiv:math/0002158

  10. Bunke U., Schick T.: On the topology of T-duality. Rev. Math. Phys. 17(1), 77–112 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Freed D.S., Hopkins M.J., Teleman C.: Loop groups and twisted K-theory I. J. Topol. 4(4), 737–798 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Freed, D.S., Hopkins, M.J., Teleman, C.: Consistent orientation of moduli spaces. In: The Many Facets of Geometry, pp. 395–419. Oxford University Press, Oxford (2010)

  13. Gaberdiel M.R., Stefański B. Jr.: Dirichlet branes on orbifolds. Nuclear Phys. B 578(1-2), 58–84 (2000)

  14. Gawȩdzki K., Suszek R.R., Waldorf K.: Bundle gerbes for orientifold sigma models. Adv. Theor. Math. Phys. 15(3), 621–687 (2011)

    Article  MathSciNet  Google Scholar 

  15. Grothendieck A.: Sur quelques points d’algèbre homologique. Tôhoku Math. J. (2) 9, 119–221 (1957)

    MATH  MathSciNet  Google Scholar 

  16. Dabholkar, A.: Lectures on orientifolds and duality.arXiv:hep-th/9804208

  17. Deligne P.: Théorie de Hodge. III. Inst. Hautes Études Sci. Publ. Math. No. 44, 5–77 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  18. Distler, J., Freed, D.S., Moore, G.W.: Orientifold précis. In: Mathematical Foundations of Quantum Field Theory and Perturbative String Theory, pp. 159–172. Proceedings of Symposos Pure Mathematics, vol. 83. American Mathematical Society, Providence, RI (2011)

  19. Donovan P., Karoubi M.: Graded Brauer groups and K-theory with local coefficients. Inst. Hautes Études Sci. Publ. Math. No. 38, 5–25 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  20. Doran, C., Mendez-Diez, S., Rosenberg, J.: T-duality For Orientifolds and Twisted KR-theory. Lett. Math. Phys. doi:10.1007/s11005-014-0715-0

  21. Hattori A., Yoshida T.: Lifting compact group actions in fiber bundles. Japan. J. Math. (N.S.) 2(1), 13–25 (1976)

    MATH  MathSciNet  Google Scholar 

  22. Hausmann J.-C., Holm T., Puppe V.: Conjugation spaces. Algebr. Geom. Topol. 5, 923–964 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hori K.: D-branes, T-duality, and index theory. Adv. Theor. Math. Phys. 3(2), 281–342 (1999)

    MATH  MathSciNet  Google Scholar 

  24. Kahn B.: Construction de classes de Chern équivariantes pour un fibré vectoriel réel. Comm. Algebra 15(4), 695–711 (1987)

    MATH  MathSciNet  Google Scholar 

  25. Lawson, H.B. Jr., Michelsohn, M.-L.: Spin geometry. In: Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton, NJ (1989)

  26. Mathai, V., Rosenberg, J.: T-duality for circle bundles via noncommutative gometry. Adv. Theor. Math. Phys. (to appear) arXiv:1306.4198

  27. Mathai V., Wu S.: Topology and flux of T-dual manifolds with circle actions. Comm. Math. Phys. 316(1), 279–286 (2012)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. May, J.P.: Equivariant homotopy and cohomology theory. With contributions by M. Cole, G. Comezaña, S. Costenoble, A.D. Elmendorf, J.P.C. Greenlees, L.G. Lewis, Jr., R.J. Piacenza, G. Triantafillou, S. Waner. In: CBMS Regional Conference Series in Mathematics, vol. 91. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI (1996)

  29. May, J.P.: Simplicial objects in algebraic topology. Reprint of the 1967 original. In: Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL (1992)

  30. Mendez-Diez S., Rosenberg J.: K-theoretic matching of brane charges in S- and U-duality. Adv. Theor. Math. Phys. 16(6), 1591–1618 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  31. Moutuou, E.M.: Twisted groupoid KR-Theory, Ph.D. thesis, Université de Lorraine-Metz, and Universität Paderborn (2012)

  32. Pitsch W., Scherer J.: Conjugation spaces and equivariant Chern classes. Bull. Belg. Math. Soc. Simon Stevin 20(1), 77–90 (2013)

    MATH  MathSciNet  Google Scholar 

  33. Rosenberg J.: Continuous-trace algebras from the bundle theoretic point of view. J. Aust. Math. Soc. Ser. A 47(3), 368–381 (1989)

    Article  MATH  Google Scholar 

  34. Rosenberg J.: The Künneth Theorem in equivariant K-theory for actions of a cyclic group of order 2. Algebr. Geom. Topol. 13(2), 1225–1241 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  35. Schreiber U., Schweigert C., Waldorf K.: Unoriented WZW models and holonomy of bundle gerbes. Comm. Math. Phys. 274(1), 31–64 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  36. Segal G.: Equivariant K-theory. Inst. Hautes Études Sci. Publ. Math. No. 34, 129–151 (1968)

    Article  MATH  Google Scholar 

  37. Sen A.: Duality and orbifolds. Nuclear Phys. B 474(2), 361–375 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  38. Spanier, E. H.: Algebraic Topology. Corrected reprint. Springer, New York-Berlin (1981)

  39. Vafa, C.: Lectures on Strings and Dualities. arXiv:hep-th/9702201

  40. Witten, E.: D-branes and K-theory. J. High Energy Phys. 12, Paper 19 (electronic) (1998)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Shinshu University, 3-1-1 Asahi, Matsumoto, Nagano, 390-8621, Japan

    Kiyonori Gomi

Authors
  1. Kiyonori Gomi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kiyonori Gomi.

Additional information

Communicated by H. Ooguri

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gomi, K. A Variant of K-Theory and Topological T-Duality for Real Circle Bundles. Commun. Math. Phys. 334, 923–975 (2015). https://doi.org/10.1007/s00220-014-2153-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-2153-3

Keywords

Search

Navigation