Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 012, 51 pages      arXiv:2006.10048
Contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi

Topological T-Duality for Twisted Tori

Paolo Aschieri abc and Richard J. Szabo abdef
a) Dipartimento di Scienze e Innovazione Tecnologica, Università del Piemonte Orientale, Viale T. Michel 11, 15121 Alessandria, Italy
b) Arnold-Regge Centre, Via P. Giuria 1, 10125 Torino, Italy
c) Istituto Nazionale di Fisica Nucleare, Torino, Via P. Giuria 1, 10125 Torino, Italy
d) Department of Mathematics, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, UK
e) Maxwell Institute for Mathematical Sciences, Edinburgh, UK
f) Higgs Centre for Theoretical Physics, Edinburgh, UK

Received June 30, 2020, in final form January 22, 2021; Published online February 05, 2021

We apply the $C^*$-algebraic formalism of topological T-duality due to Mathai and Rosenberg to a broad class of topological spaces that include the torus bundles appearing in string theory compactifications with duality twists, such as nilmanifolds, as well as many other examples. We develop a simple procedure in this setting for constructing the T-duals starting from a commutative $C^*$-algebra with an action of ${\mathbb R}^n$. We treat the general class of almost abelian solvmanifolds in arbitrary dimension in detail, where we provide necessary and sufficient criteria for the existence of classical T-duals in terms of purely group theoretic data, and compute them explicitly as continuous-trace algebras with non-trivial Dixmier-Douady classes. We prove that any such solvmanifold has a topological T-dual given by a $C^*$-algebra bundle of noncommutative tori, which we also compute explicitly. The monodromy of the original torus bundle becomes a Morita equivalence among the fiber algebras, so that these $C^*$-algebras rigorously describe the T-folds from non-geometric string theory.

Key words: noncommutative $C^*$-algebraic T-duality; nongeometric backgrounds; Mostow fibration of almost abelian solvmanifolds; $C^*$-algebra bundles of noncommutative tori.

pdf (809 kb)   tex (66 kb)  


  1. Andriot D., New supersymmetric vacua on solvmanifolds, J. High Energy Phys. 2016 (2016), no. 2, 112, 43 pages, arXiv:1507.00014.
  2. Bock C., On low-dimensional solvmanifolds, Asian J. Math. 20 (2016), 199-262, arXiv:0903.2926.
  3. Bouwknegt P., Evslin J., Mathai V., $T$-duality: topology change from $H$-flux, Comm. Math. Phys. 249 (2004), 383-415, arXiv:hep-th/0306062.
  4. Bouwknegt P., Pande A.S., Topological $T$-duality and $T$-folds, Adv. Theor. Math. Phys. 13 (2009), 1519-1539, arXiv:0810.4374.
  5. Brace D., Morariu B., Zumino B., Dualities of the Matrix model from $T$-duality of the type II string, Nuclear Phys. B 545 (1999), 192-216, arXiv:hep-th/9810099.
  6. Brodzki J., Mathai V., Rosenberg J., Szabo R.J., D-branes, RR-fields and duality on noncommutative manifolds, Comm. Math. Phys. 277 (2008), 643-706, arXiv:hep-th/0607020.
  7. Brodzki J., Mathai V., Rosenberg J., Szabo R.J., Non-commutative correspondences, duality and D-branes in bivariant $K$-theory, Adv. Theor. Math. Phys. 13 (2009), 497-552, arXiv:0708.2648.
  8. Bunke U., Rumpf P., Schick T., The topology of $T$-duality for $T^n$-bundles, Rev. Math. Phys. 18 (2006), 1103-1154, arXiv:math.GT/0501487.
  9. Bunke U., Schick T., On the topology of $T$-duality, Rev. Math. Phys. 17 (2005), 77-112, arXiv:math.GT/0405132.
  10. Console S., Macrì M., Lattices, cohomology and models of 6-dimensional almost abelian solvmanifolds, Rend. Semin. Mat. Univ. Politec. Torino 74 (2016), 95-119, arXiv:1206.5977.
  11. Corwin L.J., Greenleaf F.P., Representations of nilpotent Lie groups and their applications. Part I. Basic theory and examples, Cambridge Studies in Advanced Mathematics, Vol. 18, Cambridge University Press, Cambridge, 1990.
  12. Cowen C.C., MacCluer B.D., Linear fractional maps of the ball and their composition operators, Acta Sci. Math. (Szeged) 66 (2000), 351-376.
  13. Dabholkar A., Hull C., Duality twists, orbifolds, and fluxes, J. High Energy Phys. 2003 (2003), no. 9, 054, 25 pages, arXiv:hep-th/0210209.
  14. Echterhoff S., Crossed products and the Mackey-Rieffel-Green machine, Oberwolfach Sem. 47 (2017), 5-79, arXiv:1006.4975.
  15. Echterhoff S., Nest R., Oyono-Oyono H., Principal non-commutative torus bundles, Proc. Lond. Math. Soc. 99 (2009), 1-31, arXiv:0810.0811.
  16. Ellwood I., Hashimoto A., Effective descriptions of branes on non-geometric tori, J. High Energy Phys. 2006 (2006), no. 12, 025, 22 pages, arXiv:hep-th/0607135.
  17. Fack T., Skandalis G., Connes' analogue of the Thom isomorphism for the Kasparov groups, Invent. Math. 64 (1981), 7-14.
  18. Fröhlich J., Gawędzki K., Conformal field theory and geometry of strings, in Mathematical Quantum Theory. I. Field Theory and Many-Body Theory (Vancouver, BC, 1993), CRM Proc. Lecture Notes, Vol. 7, Amer. Math. Soc., Providence, RI, 1994, 57-97, arXiv:hep-th/9310187.
  19. Grange P., Schäfer-Nameki S., $T$-duality with $H$-flux: non-commutativity, $T$-folds and $G\times G$ structure, Nuclear Phys. B 770 (2007), 123-144, arXiv:hep-th/0609084.
  20. Green P., The structure of imprimitivity algebras, J. Funct. Anal. 36 (1980), 88-104.
  21. Hannabuss K.C., Mathai V., Noncommutative principal torus bundles via parametrised strict deformation quantization, in Superstrings, Geometry, Topology, and $C^\ast$-Algebras, Proc. Sympos. Pure Math., Vol. 81, Amer. Math. Soc., Providence, RI, 2010, 133-147, arXiv:0911.1886.
  22. Hannabuss K.C., Mathai V., Parametrized strict deformation quantization of $C^\ast$-bundles and Hilbert $C^\ast$-modules, J. Aust. Math. Soc. 90 (2011), 25-38, arXiv:1007.4696.
  23. Havas G., Majewski B.S., Integer matrix diagonalization, J. Symbolic Comput. 24 (1997), 399-408.
  24. Hori K., D-branes, $T$-duality, and index theory, Adv. Theor. Math. Phys. 3 (1999), 281-342, arXiv:hep-th/9902102.
  25. Hull C.M., A geometry for non-geometric string backgrounds, J. High Energy Phys. 2005 (2005), no. 10, 065, 30 pages, arXiv:hep-th/0406102.
  26. Hull C.M., Reid-Edwards R.A., Flux compactifications of string theory on twisted tori, Fortschr. Phys. 57 (2009), 862-894, arXiv:hep-th/0503114.
  27. Hull C.M., Reid-Edwards R.A., Non-geometric backgrounds, doubled geometry and generalised $T$-duality, J. High Energy Phys. 2009 (2009), no. 9, 014, 79 pages, arXiv:0902.4032.
  28. Hull C.M., Szabo R.J., Noncommutative gauge theories on D-branes in non-geometric backgrounds, J. High Energy Phys. 2019 (2019), no. 9, 051, 40 pages, arXiv:1903.04947.
  29. Landi G., Lizzi F., Szabo R.J., String geometry and the noncommutative torus, Comm. Math. Phys. 206 (1999), 603-637, arXiv:hep-th/9806099.
  30. Lizzi F., Szabo R.J., Duality symmetries and noncommutative geometry of string spacetimes, Comm. Math. Phys. 197 (1998), 667-712, arXiv:hep-th/9707202.
  31. Lowe D.A., Nastase H., Ramgoolam S., Massive IIA string theory and matrix theory compactification, Nuclear Phys. B 667 (2003), 55-89, arXiv:hep-th/0303173.
  32. Mathai V., Rosenberg J., $T$-duality for torus bundles with $H$-fluxes via noncommutative topology, Comm. Math. Phys. 253 (2005), 705-721, arXiv:hep-th/0401168.
  33. Mathai V., Rosenberg J., $T$-duality for torus bundles with $H$-fluxes via noncommutative topology. II. The high-dimensional case and the $T$-duality group, Adv. Theor. Math. Phys. 10 (2006), 123-158, arXiv:hep-th/0508084.
  34. Milnor J., Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976), 293-329.
  35. Mostow G.D., Factor spaces of solvable groups, Ann. of Math. 60 (1954), 1-27.
  36. Nikolaus T., Waldorf K., Higher geometry for non-geometric T-duals, Comm. Math. Phys. 374 (2020), 317-366, arXiv:1804.00677.
  37. Phillips N.C., Every simple higher-dimensional noncommutative torus is an AT algebra, arXiv:math.OA/0609783.
  38. Pioline B., Schwarz A., Morita equivalence and $T$-duality (or $B$ versus $\Theta$), J. High Energy Phys. 1999 (1999), no. 8, 021, 16 pages, arXiv:hep-th/9908019.
  39. Raeburn I., Rosenberg J., Crossed products of continuous-trace $C^\ast$-algebras by smooth actions, Trans. Amer. Math. Soc. 305 (1988), 1-45.
  40. Raghunathan M.S., Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 68, Springer-Verlag, New York - Heidelberg, 1972.
  41. Rieffel M.A., Induced representations of $C^{\ast}$-algebras, Adv. Math. 13 (1974), 176-257.
  42. Rieffel M.A., Strong Morita equivalence of certain transformation group $C^*$-algebras, Math. Ann. 222 (1976), 7-22.
  43. Rieffel M.A., $C^{\ast} $-algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415-429.
  44. Rieffel M.A., The cancellation theorem for projective modules over irrational rotation $C^{\ast} $-algebras, Proc. London Math. Soc. 47 (1983), 285-302.
  45. Rieffel M.A., Noncommutative tori - a case study of noncommutative differentiable manifolds, in Geometric and Topological Invariants of Elliptic Operators (Brunswick, ME, 1988), Contemp. Math., Vol. 105, Amer. Math. Soc., Providence, RI, 1990, 191-211.
  46. Rieffel M.A., Schwarz A., Morita equivalence of multidimensional noncommutative tori, Internat. J. Math. 10 (1999), 289-299, arXiv:math.QA/9803057.
  47. Schwarz A., Morita equivalence and duality, Nuclear Phys. B 534 (1998), 720-738, arXiv:hep-th/9805034.
  48. Seiberg N., Witten E., String theory and noncommutative geometry, J. High Energy Phys. 1999 (1999), no. 9, 032, 93 pages, arXiv:hep-th/9908142.
  49. Steenrod N., The Topology of Fibre Bundles, Princeton Mathematical Series, Vol. 14, Princeton University Press, Princeton, N.J., 1951.
  50. Tralle A., Oprea J., Symplectic manifolds with no Kähler structure, Lecture Notes in Mathematics, Vol. 1661, Springer-Verlag, Berlin, 1997.
  51. Williams D.P., The structure of crossed products by smooth actions, J. Austral. Math. Soc. Ser. A 47 (1989), 226-235.
  52. Williams D.P., Crossed products of $C{^\ast}$-algebras, Mathematical Surveys and Monographs, Vol. 134, Amer. Math. Soc., Providence, RI, 2007.

Previous article  Next article  Contents of Volume 17 (2021)