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T-Duality: Topology Change from H-Flux

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Abstract

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 Pn, and the AdS5×S5 to AdS5×P2×S1 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.

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References

  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. Roček, M., Verlinde, E.: Duality, quotients, and currents. Nucl. Phys. 373, 630–646 (1992)

    Article  MathSciNet  Google Scholar 

  3. Álvarez, E., Álvarez-Gaumé, L., Lozano, Y.: An introduction to T-duality in string theory. Nucl. Phys. Proc. Suppl. 41, 1–20 (1995)

    Article  MathSciNet  Google Scholar 

  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. 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. Duff, M.J. , Lü, H., Pope, C.N.: AdS5× S5 untwisted. Nucl. Phys. B532, 181–209 (1998)

  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. Kachru, S., Schulz, M., Tripathy, P., Trivedi, S.: New supersymmetric string compactifications. J. High Energy Phys. 03, 061 (2003)

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  10. Gukov, S.: K-theory, reality, and orientifolds. Commun. Math. Phys. 210, 621–639 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Sharpe, E.R.: D-branes, derived categories, and Grothendieck groups. Nucl. Phys. B561, 433–450 (1999)

    Google Scholar 

  12. Olsen, K., Szabo, R.J.: Constructing D-Branes from K-Theory. Adv. Theor. Math. Phys. 3, 889–1025 (1999)

    MathSciNet  MATH  Google Scholar 

  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. Strominger, A., Yau, S.-T., Zaslow, E.: Mirror symmetry is T-duality. Nucl. Phys. B479, 243–259 (1996)

    Google Scholar 

  15. Bott, R., Tu, L.: Differential forms in algebraic topology. Graduate Texts in Mathematics 82, New York: Springer Verlag, 1982

  16. Brylinski, J.-L.: Loop spaces, characteristic classes and geometric quantization. Prog. Math. 107, Boston: Birkhäuser Boston, 1993

  17. Mathai, V., Melrose, R.B., Singer, I.M.: The index of projective families of elliptic operators. http://arXiv.org/abs/math.DG/0206002, 2002

  18. Mathai, V., Melrose, R.B., Singer, I.M.: Work in progress

  19. Minasian, R., Moore, G.: K-theory and Ramond-Ramond charge. J. High Energy Phys. 11, 002 (1997)

    Google Scholar 

  20. Witten, E.: D-Branes and K-Theory. J. High Energy Phys. 12, 019 (1998)

    MATH  Google Scholar 

  21. Moore, G., Witten, E.: Self duality, Ramond-Ramond fields, and K-theory. J. High Energy Phys. 05, 032 (2000)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Mathai, V., Stevenson, D.: Chern Character in Twisted K-Theory: Equivariant and Holomorphic Cases. Commun. Math. Phys. 236, 161–186 (2003)

    MathSciNet  MATH  Google Scholar 

  24. 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 

  25. Connes, A.: An analogue of the Thom isomorphism for crossed products of a C* algebra by an action of . Adv. Math. 39, 31–55 (1981)

    Google Scholar 

  26. Álvarez-Gaumé, L., Ginsparg, P.: The Structure of Gauge and Gravitational Anomalies. Ann. Phys. 161, 423 (1985) Erratum-ibid. 171, 233 (1986)

    MathSciNet  Google Scholar 

  27. Witten, E.: On Flux Quantization in M-Theory and the Effective Action. J. Geom. Phys. 22, 1–13 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  28. Evslin, J.: Twisted K-Theory from Monodromies. J. High Energy Phys. 05, 030 (2003)

    Article  Google Scholar 

  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. Gomez, C., Manjarin, J. J.: Dyons, K-theory and M-theory.http://arXiv.org/abs/hep-th/0111169, 2001

  31. Adams, A., Evslin, J.: The Loop Group of E8 and K-Theory from 11d. J. High Energy Phys. 02, 029 (2003)

    Article  Google Scholar 

  32. Morrison, D. R.: Half K3 surfaces. Talk at Strings 2002, Cambridge. http://www.damtp.cam.ac.uk/strings02/avt/morrison/

  33. Hořava, P.: Unpublished

  34. Vafa, C.: Evidence for F-Theory. Nucl. Phys. B469, 403–418 (1996)

    Google Scholar 

  35. Adams, A., Evslin, J., Varadarajan, U.: To appear

  36. Rosenberg, J.: Continuous trace C*-algebras from the bundle theoretic point of view. J. Aust. Math. Soc. A47, 368 (1989)

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

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  39. Atiyah, M. F., Singer, I. M.: The index of elliptic operators, IV. Ann. of Math. (2) 93, 119–138 (1971)

    Google Scholar 

  40. Mathai, V., Quillen, D. G.: Superconnections, Thom classes and equivariant differential forms. Topology 25(1), 85–110 (1986)

    Article  MATH  Google Scholar 

  41. Maldacena, J., Moore, G., Seiberg, N.: D-Brane Instantons and K-Theory Charges. J. High Energy Phys. 11, 062 (2001)

    Google Scholar 

  42. Tong, D.: NS5-branes, T-duality and worldsheet fermions. J. High Energy Phys. 07, 013 (2002)

    Article  Google Scholar 

  43. David, J., Gutperle, M., Headrick, M., Minwalla, S.: Closed String Tachyon Condensation on Twisted Circles. J. High Energy Phys. 04, 041 (2002)

    Article  Google Scholar 

  44. Evslin, J., Varadarajan, U.: K-Theory and S-Duality: Starting over from Square 3. J. High Energy Phys. 03, 026 (2003)

    Article  Google Scholar 

  45. Witten, E.: Phases of N=2 Theories in 2 Dimensions. Nucl. Phys. B403, 159–222 (1993)

    Google Scholar 

  46. Hori, K., Vafa, C.: Mirror Symmetry. http://arXiv.org/abs/hep-th/0002222, 2000

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Authors and Affiliations

  1. Department of Physics, School of Chemistry and Physics, University of Adelaide, Adelaide, SA 5005, Australia

    Peter Bouwknegt

  2. Department of Pure Mathematics, University of Adelaide, Adelaide, SA 5005, Australia

    Peter Bouwknegt & Varghese Mathai

  3. INFN Sezione di Pisa, Via Buonarroti, 2, Ed. C, 56127, Pisa, Italy

    Jarah Evslin

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  1. Peter Bouwknegt
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  2. Jarah Evslin
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  3. Varghese Mathai
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Correspondence to Varghese Mathai.

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Communicated by M.R. Douglas

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Bouwknegt, P., Evslin, J. & Mathai, V. T-Duality: Topology Change from H-Flux. Commun. Math. Phys. 249, 383–415 (2004). https://doi.org/10.1007/s00220-004-1115-6

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  • DOI: https://doi.org/10.1007/s00220-004-1115-6

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