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Topology and Flux of T-Dual Manifolds with Circle Actions

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Abstract

We present an explicit formula for the topology and H-flux of the T-dual of a general type II, compactification, significantly generalizing earlier results. Our results apply to T-dualities with respect to any circle action on spacetime X. As before, T-duality exchanges type IIA and type IIB string theories. A new consequence is that the T-dual spacetime is a singular space when the fixed point set \({X^\mathbb{T}}\) is non-empty; the singularities correspond to Kaluza-Klein monopoles. We propose that the Ramond-Ramond charges of type II string theories on the singular dual are classified by twisted equivariant cohomology groups. We also discuss the K-theory approach.

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References

  1. Buscher, T.: A symmetry of the string background field equations. Phys. Lett. B194, 59–62 (1987); Path integral derivation of quantum duality in nonlinear sigma models, ibid. B201, 466–472 (1988)

    Google Scholar 

  2. Alvarez E., Alvarez-Gaumé L., Barbón J.L.F., Lozano Y.: Some global aspects of duality in string theory. Nucl. Phys. B 415, 71–100 (1994)

    Article  ADS  MATH  Google Scholar 

  3. Duff M.J., Lü H., Pope C.N.: AdS 5 × S 5 untwisted. Nucl. Phys. B 532, 181–209 (1998)

    Article  ADS  MATH  Google Scholar 

  4. Dasgupta K., Rajesh G., Sethi S.: M theory, orientifolds and G-flux. J. High Energy Phys. 08, 023 (1999)

    Article  MathSciNet  ADS  Google Scholar 

  5. Gurrieri S., Louis J., Micu A., Waldram D.: Mirror symmetry in generalized Calabi-Yau compactifications. Nucl. Phys. B 654, 61–113 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  7. Bouwknegt, P., Evslin, J., Mathai, V.: On the Topology and Flux of T-Dual Manifolds. Phys. Rev. Lett. 92, 181601 (2004); T-duality: Topology Change from H-flux. Commun. Math. Phys. 249, 383-415 (2004)

    Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  11. Bunke, U., Schick, T.: T-duality for non-free circle actions. In: B. Booß-Bavnbek et al. (eds.), Analysis, geometry and topology of elliptic operators, Hackensack, NJ: World Sci. Publ., 2006, pp. 429–466

  12. Pande A.S.: Topological T-duality and Kaluza-Klein monopoles. Adv. Theor. Math. Phys. 12, 185–215 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Goresky M., MacPherson R.: Intersection homology theory. Topology 19, 135–162 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sorkin R.D.: Kaluza-Klein Monopole. Phys. Rev. Lett. 51, 87–90 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  15. Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., Zaslow, E.: Mirror symmetry, Clay Mathematics Monographs, 1. Providence, RI: Amer. Math. Soc. and Cambridge, MA: Clay Mathematics Institute, 2003

  16. Bouwknegt P., Hannabuss K., Mathai V.: T-duality for principal torus bundles. J. High Energy Phys. 03, 018 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  17. Atiyah M.F., Bott R.: The moment map and equivariant cohomology. Topology 23, 1–28 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Mundeti Riera I.: Lifts of smooth group actions to line bundles. Bull. London Math. Soc. 33, 351–361 (2001)

    Article  MathSciNet  Google Scholar 

  20. Guillemin V., Ginzburg V., Karshon Y.: Moment maps, cobordisms, and Hamiltonian group actions, Math. Surveys and Monographs, 98, Providence, RI: Amer. Math. Soc., 2002

  21. Harada M., Landweber G.: The K-theory of abelian symplectic quotients. Math. Res. Lett. 15, 57–72 (2008)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  23. Phillips N.C.: Representable K-theory for σ-C *-algebras. K-Theory 3, 441–478 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  25. Phillips N.C., Schweitzer L.: Representable K-theory of smooth crossed products by \({\mathbb{R}}\) and \({\mathbb{Z}}\) . Trans. Amer. Math. Soc. 344, 173–201 (1994)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  27. Phillips N.C.: The Atiyah-Segal completion theorem for C *-algebras. K-Theory 3, 479–504 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Adelaide, Adelaide, 5005, Australia

    Varghese Mathai

  2. Department of Mathematics, University of Hong Kong, Pokfulam Road, Hong Kong, China.

    Siye Wu

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  1. Varghese Mathai
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  2. Siye Wu
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Correspondence to Varghese Mathai.

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Communicated by N. A. Nekrasov

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Mathai, V., Wu, S. Topology and Flux of T-Dual Manifolds with Circle Actions. Commun. Math. Phys. 316, 279–286 (2012). https://doi.org/10.1007/s00220-012-1542-8

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