Communications in Mathematical Physics

, Volume 249, Issue 2, pp 383–415 | Cite as

T-Duality: Topology Change from H-Flux

  • Peter Bouwknegt
  • Jarah Evslin
  • Varghese MathaiEmail author


T-duality acts on circle bundles by exchanging the first Chern class with the fiberwise integral of the H-flux, as we motivate using E 8 and also using S-duality. We present known and new examples including NS5-branes, nilmanifolds, lens spaces, both circle bundles over Open image in new window P n , and the AdS 5 ×S 5 to AdS 5 × Open image in new window P 2 ×S 1 with background H-flux of Duff, Lü and Pope. When T-duality leads to M-theory on a non-spin manifold the gravitino partition function continues to exist due to the background flux, however the known quantization condition for G 4 receives a correction. In a more general context, we use correspondence spaces to implement isomorphisms on the twisted K-theories and twisted cohomology theories and to study the corresponding Grothendieck-Riemann-Roch theorem. Interestingly, in the case of decomposable twists, both twisted theories admit fusion products and so are naturally rings.


Manifold Partition Function Quantization Condition General Context Chern Class 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Buscher, T.: A symmetry of the string background field equations. Phys. Lett. B194, 59–62 (1987); Buscher, T.: Path integral derivation of quantum duality in nonlinear sigma models. Phys. Lett. B201, 466–472 (1988)Google Scholar
  2. 2.
    Roček, M., Verlinde, E.: Duality, quotients, and currents. Nucl. Phys. 373, 630–646 (1992)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Álvarez, E., Álvarez-Gaumé, L., Lozano, Y.: An introduction to T-duality in string theory. Nucl. Phys. Proc. Suppl. 41, 1–20 (1995)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bergshoeff, E., Hull, C.M., Ortin, T.: Dualty in the type-II superstring effective action. Nucl. Phys. B451, 547–578 (1995)Google Scholar
  5. 5.
    Álvarez, E., Álvarez-Gaumé, L., Barbón, J.L.F., Lozano, Y.: Some global aspects of duality in string theory. Nucl. Phys. B415, 71–100 (1994)Google Scholar
  6. 6.
    Duff, M.J. , Lü, H., Pope, C.N.: AdS5× S5 untwisted. Nucl. Phys. B532, 181–209 (1998)Google Scholar
  7. 7.
    Gurrieri, S., Louis, J., Micu, A., Waldram, D.: Mirror symmetry in generalized Calabi-Yau compactifications. Nucl. Phys. B654, 61–113 (2003)Google Scholar
  8. 8.
    Kachru, S., Schulz, M., Tripathy, P., Trivedi, S.: New supersymmetric string compactifications. J. High Energy Phys. 03, 061 (2003)CrossRefGoogle Scholar
  9. 9.
    Hori, K.: D-branes, T-duality, and index theory. Adv. Theor. Math. Phys. 3, 281–342 (1999)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Gukov, S.: K-theory, reality, and orientifolds. Commun. Math. Phys. 210, 621–639 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Sharpe, E.R.: D-branes, derived categories, and Grothendieck groups. Nucl. Phys. B561, 433–450 (1999)Google Scholar
  12. 12.
    Olsen, K., Szabo, R.J.: Constructing D-Branes from K-Theory. Adv. Theor. Math. Phys. 3, 889–1025 (1999)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Moore, G., Saulina, N.: T-duality, and the K-theoretic partition function of TypeIIA superstring theory. Nud. Phys. B670, 27–89 (2003)Google Scholar
  14. 14.
    Strominger, A., Yau, S.-T., Zaslow, E.: Mirror symmetry is T-duality. Nucl. Phys. B479, 243–259 (1996)Google Scholar
  15. 15.
    Bott, R., Tu, L.: Differential forms in algebraic topology. Graduate Texts in Mathematics 82, New York: Springer Verlag, 1982Google Scholar
  16. 16.
    Brylinski, J.-L.: Loop spaces, characteristic classes and geometric quantization. Prog. Math. 107, Boston: Birkhäuser Boston, 1993Google Scholar
  17. 17.
    Mathai, V., Melrose, R.B., Singer, I.M.: The index of projective families of elliptic operators., 2002
  18. 18.
    Mathai, V., Melrose, R.B., Singer, I.M.: Work in progressGoogle Scholar
  19. 19.
    Minasian, R., Moore, G.: K-theory and Ramond-Ramond charge. J. High Energy Phys. 11, 002 (1997)Google Scholar
  20. 20.
    Witten, E.: D-Branes and K-Theory. J. High Energy Phys. 12, 019 (1998)zbMATHGoogle Scholar
  21. 21.
    Moore, G., Witten, E.: Self duality, Ramond-Ramond fields, and K-theory. J. High Energy Phys. 05, 032 (2000)zbMATHGoogle Scholar
  22. 22.
    Bouwknegt, P., Carey, A., Mathai, V., Murray, M., Stevenson, D.: Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys. 228, 17–45 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Mathai, V., Stevenson, D.: Chern Character in Twisted K-Theory: Equivariant and Holomorphic Cases. Commun. Math. Phys. 236, 161–186 (2003)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Raeburn, I., Rosenberg, J.: Crossed products of continuous-trace C*-algebras by smooth actions. Trans. Am. Math. Soc. 305, 1–45 (1988)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Connes, A.: An analogue of the Thom isomorphism for crossed products of a C* algebra by an action of Open image in new window. Adv. Math. 39, 31–55 (1981)Google Scholar
  26. 26.
    Álvarez-Gaumé, L., Ginsparg, P.: The Structure of Gauge and Gravitational Anomalies. Ann. Phys. 161, 423 (1985) Erratum-ibid. 171, 233 (1986)MathSciNetGoogle Scholar
  27. 27.
    Witten, E.: On Flux Quantization in M-Theory and the Effective Action. J. Geom. Phys. 22, 1–13 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Evslin, J.: Twisted K-Theory from Monodromies. J. High Energy Phys. 05, 030 (2003)CrossRefGoogle Scholar
  29. 29.
    Diaconescu, E., Moore, G., Witten, E.: E8 Gauge Theory, and a Derivation of K-Theory from M-Theory. Adv. Theor. Math. 6, 1031–1134 (2003)Google Scholar
  30. 30.
    Gomez, C., Manjarin, J. J.: Dyons, K-theory and M-theory., 2001
  31. 31.
    Adams, A., Evslin, J.: The Loop Group of E8 and K-Theory from 11d. J. High Energy Phys. 02, 029 (2003)CrossRefGoogle Scholar
  32. 32.
    Morrison, D. R.: Half K3 surfaces. Talk at Strings 2002, Cambridge.
  33. 33.
    Hořava, P.: UnpublishedGoogle Scholar
  34. 34.
    Vafa, C.: Evidence for F-Theory. Nucl. Phys. B469, 403–418 (1996)Google Scholar
  35. 35.
    Adams, A., Evslin, J., Varadarajan, U.: To appearGoogle Scholar
  36. 36.
    Rosenberg, J.: Continuous trace C*-algebras from the bundle theoretic point of view. J. Aust. Math. Soc. A47, 368 (1989)Google Scholar
  37. 37.
    Bouwknegt, P., Mathai, V.: D-branes, B-fields and twisted K-theory. J. High Energy Phys. 03, 007 (2000)zbMATHGoogle Scholar
  38. 38.
    Freed, D., Hopkins, M., Telemann, C.: Unpublished; Freed, D.S.: The Verlinde algebra is twisted equivariant K-theory. Turkish J. Math. 25, 159–167 (2001)zbMATHGoogle Scholar
  39. 39.
    Atiyah, M. F., Singer, I. M.: The index of elliptic operators, IV. Ann. of Math. (2) 93, 119–138 (1971)Google Scholar
  40. 40.
    Mathai, V., Quillen, D. G.: Superconnections, Thom classes and equivariant differential forms. Topology 25(1), 85–110 (1986)CrossRefzbMATHGoogle Scholar
  41. 41.
    Maldacena, J., Moore, G., Seiberg, N.: D-Brane Instantons and K-Theory Charges. J. High Energy Phys. 11, 062 (2001)Google Scholar
  42. 42.
    Tong, D.: NS5-branes, T-duality and worldsheet fermions. J. High Energy Phys. 07, 013 (2002)CrossRefGoogle Scholar
  43. 43.
    David, J., Gutperle, M., Headrick, M., Minwalla, S.: Closed String Tachyon Condensation on Twisted Circles. J. High Energy Phys. 04, 041 (2002)CrossRefGoogle Scholar
  44. 44.
    Evslin, J., Varadarajan, U.: K-Theory and S-Duality: Starting over from Square 3. J. High Energy Phys. 03, 026 (2003)CrossRefGoogle Scholar
  45. 45.
    Witten, E.: Phases of N=2 Theories in 2 Dimensions. Nucl. Phys. B403, 159–222 (1993)Google Scholar
  46. 46.
    Hori, K., Vafa, C.: Mirror Symmetry., 2000

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Peter Bouwknegt
    • 1
    • 2
  • Jarah Evslin
    • 3
  • Varghese Mathai
    • 2
    Email author
  1. 1.Department of PhysicsSchool of Chemistry and PhysicsAustralia
  2. 2.Department of Pure MathematicsUniversity of AdelaideAdelaideAustralia
  3. 3.INFN Sezione di PisaPisaItaly

Personalised recommendations