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Abelian and Non-Abelian Branes in WZW Models and Gerbes

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Abstract

We discuss how gerbes may be used to set up a consistent Lagrangian approach to the WZW models with boundary. The approach permits to study in detail possible boundary conditions that restrict the values of the fields on the worldsheet boundary to brane submanifolds in the target group. Such submanifolds are equipped with an additional geometric structure that is summarized in the notion of a gerbe module and includes a twisted Chan-Paton gauge field. Using the geometric approach, we present a complete classification of the branes that conserve the diagonal current-algebra symmetry in the WZW models with simple, compact but not necessarily simply connected target groups. Such symmetric branes are supported by a discrete series of conjugacy classes in the target group and may carry Abelian or non-Abelian twisted gauge fields. The latter situation occurs for the conjugacy classes with fundamental group ℤ2×ℤ2 in SO(4n)/ℤ2. The branes supported by such conjugacy classes have to be equipped with a projectively flat twisted U(2) gauge field in one of the two possible WZW models differing by discrete torsion. We show how the geometric description of branes leads to explicit formulae for the boundary partition functions and boundary operator product coefficients in the WZW models with non-simply connected target groups.

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References

  1. Alekseev, A., Meinrenken, E., Woodward, C.: The Verlinde formulas as fixed point formulas. J. Symplectic Geom. 1, 1–46 (2001) and 1, 427–434 (2002)

    Google Scholar 

  2. Alekseev, A. Yu., Schomerus, V.: D-branes in the WZW model. Phys. Rev. D 60, R061901-R061902 (1999)

    Google Scholar 

  3. Alvarez, O.: Topological quantization and cohomology. Commun. Math. Phys. 100, 279–309 (1985)

    Google Scholar 

  4. Alvarez-Gaumé, L., Gomez, C., Sierra, G.: Quantum group interpretation of some conformal field theories. Phys. Lett. B 220, 142–152 (1989)

    Google Scholar 

  5. Aspinwall, P. S.: D-branes on Calabi-Yau manifolds. http://arxiv.org/abs/list/hep-th/0403166, 2004

  6. Bachas, C., Douglas, M. Schweigert, C.: Flux stabilization of D-branes. JHEP 05, 048 (2000)

    Google Scholar 

  7. Behrend, R. E., Pearce, P. A., Petkova, V. B., Zuber, J.-B.: Boundary conditions in rational conformal field theories. Nucl. Phys. B 579, 707–773 (2000)

    Google Scholar 

  8. Behrend, K., Xu, P. Zhang, B.: Equivariant gerbes over compact simple Lie groups. C. R. Acad. Sci. Paris 336 Sér. I , 251–256 (2003)

  9. Birke, L., Fuchs, J., Schweigert, C.: Symmetry breaking boundary conditions and WZW orbifolds. Adv. Theor. Math. Phys. 3, 671–726 (1999)

    Google Scholar 

  10. Bouwknegt, P., Carey, A. L., Mathai, V., Murray, M. K., Stevenson, D.: Twisted K-theory and K-theory of bundle gerbes. Commun. Math. Phys. 228, 17–45 (2002)

    Google Scholar 

  11. Bouwknegt, P., Dawson, P., Ridout, D.: D-branes on group manifolds and fusion rings. JHEP 12, 065 (2002)

    Google Scholar 

  12. Bouwknegt, P., Ridout, D.: A note on the equality of algebraic and geometric D-brane charges in WZW models. JHEP 05, 029 (2004)

    Google Scholar 

  13. Braun, V.: Twisted K-theory of Lie groups. JHEP 03, 029 (2004)

    Google Scholar 

  14. Braun, V., Schafer-Nameki, S.: Supersymmetric WZW models and twisted K-theory of SO(3). http://arxiv.org/abs/hep-th/0403287, 2004

  15. Bruner, I., Schomerus, V.: On Superpotentials for D-branes in Gepner models. JHEP 10, 016 (2000)

    Google Scholar 

  16. Brylinski, J.-L.: Loop Spaces, Characteristic Classes and Geometric Quantization. Prog. Math. 107, Boston: Birkhäuser, 1993

  17. Brylinski, J.-L.: Gerbes on complex reductive Lie groups. http://arxiv/org/list/math.DG/0002158, 2000

  18. Cardy, J. L.: Boundary conditions, fusion rules and the Verlinde formula. Nucl. Phys. B 324, 581–598 (1989)

    Google Scholar 

  19. Carey, A. L., Johnson, S., Murray, M. K.: Holonomy on D-Branes. http://arxiv.org/list/hep-th/0204199, 2002

  20. Carter, J. S., Flath, D. E., Saito, M.: The Classical and Quantum 6j-Symbols. Princeton, NJ: Princeton U. Press, 1995

  21. Chatterjee, D. S.: On gerbs. Ph.D. thesis, Trinity College, Cambridge, 1998

  22. Di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal Field Theory. New York: Springer-Verlag, 1997

  23. Douglas, M.: D-branes and discrete torsion. http://arxiv.org/list/hep-th/9807235, 1998

  24. Douglas, M.: Topics in D-geometry. Class. Quant. Grav. 17, 1057–1070 (2000)

    Article  CAS  Google Scholar 

  25. Douglas, M.: Lectures on D-branes on Calabi-Yau manifolds. ICTP Lect. Notes, VII, Trieste 2002, http://www.ictp.trieste.it/~pub_off/lectures/vol7.html

  26. Douglas, M., Fiol, B.: D-branes and discrete torsion II. http://arxiv.org/list/hep-th/9903031, 1999

  27. Felder, G., Fröhlich, J., Fuchs, J., Schweigert, C.: Conformal boundary conditions and three-dimensional topological field theory. Phys. Rev. Lett. 84, 1659–1662 (2000)

    CAS  PubMed  Google Scholar 

  28. Elitzur, S., Sarkissian, G.: D-Branes on a gauged WZW model. Nucl.Phys. B 625, 166–178 (2002)

    Google Scholar 

  29. Felder, G., Gawedzki, K., Kupiainen, A.: Spectra of Wess-Zumino-Witten models with arbitrary simple groups. Commun. Math. Phys. 117, 127–158 (1988)

    Google Scholar 

  30. Fredenhagen, S., Schomerus, V.: Branes on group manifolds, gluon condensates, and twisted K-theory. JHEP 04, 007 (2001)

    Google Scholar 

  31. Freed, D. S., Witten, E.: Anomalies in string theory with D-branes. Asian J. Math. 3, 819–851 (1999)

    Google Scholar 

  32. Fuchs, J., Huiszoon, L. R., Schellekens, A. N., Schweigert, C., Walcher, J.: Boundaries, crosscaps and simple currents. Phys. Lett. B 495, 427–434 (2000)

    CAS  Google Scholar 

  33. Fuchs, J., Kaste, P., Lerche, W., Lutken, C., Schweigert, C., Walcher, J: Boundary fixed points, enhanced Gauge symmetry and singular bundles on K3. Nucl. Phys. B598, 57–72 (2001)

    Google Scholar 

  34. Fuchs, J., Runkel, I., Schweigert, C.: TFT construction of RCFT correlators III: Simple currents. Nucl. Phys. B694, 277–353 (2004)

    Google Scholar 

  35. Fuchs, J., Schweigert, C.: The action of outer automorphisms on bundles of chiral blocks. Commun. Math. Phys. 206, 691–736 (1999)

    Google Scholar 

  36. Gaberdiel, M. R., Gannon, T.: Boundary states for WZW models. Nucl. Phys. B 639, 471–501 (2002)

    Google Scholar 

  37. Gaberdiel, M. R., Gannon, T.: D-brane charges on non-simply connected groups. JHEP 04, 030 (2004)

    Google Scholar 

  38. K. Gawedzki, Topological actions in two-dimensional quantum field theories. In: Non-perturbative Quantum Field Theory, eds. ‘t Hooft, G., Jaffe, A., Mack, G., Mitter, P. K., Stora, R., New York: Plenum Press, 1988, pp. 101–142

  39. Gawedzki, K.: Conformal field theory: a case study. In: Conformal Field Theory: New Non-Perturbative Methods in String and Field Theory, eds. Nutku, Y., Saclioglu, C., Turgut, T., London: Perseus, 2000, pp. 1–55

  40. Gawedzki, K.: Boundary WZW, G/H, G/G and CS theories. Ann. Henri Poincaré 3, 847–881 (2002)

    Google Scholar 

  41. Gawedzki, K., Reis, N.: WZW branes and gerbes. Rev. Math. Phys. 14, 1281–1334 (2002)

    Google Scholar 

  42. Gawedzki, K., Reis, N.: Basic gerbe over non-simply connected compact groups. J. Geom. Phys. 50, 28–55 (2004)

    MathSciNet  Google Scholar 

  43. Gawedzki, K., Todorov, I., Tran-Ngoc-Bich, P.: Canonical quantization of the boundary Wess-Zumino-Witten model. Commun. Math. Phys: 248, 217–254 (2004)

    MathSciNet  Google Scholar 

  44. Gepner, D., E. Witten, E.: String Theory on Group Manifolds. Nucl. Phys. B 278, 493–549 (1986)

    Google Scholar 

  45. Giraud, J.: Cohomologie non-abélienne. Grundl. 179, Berlin-Heidelberg-New York: Springer, 1971

  46. Halpern, M. B., Helfgott, C.: The general twisted open WZW string. http://arxiv.org/list/hep-th/0406003, 2004 and references therein

  47. Hitchin, N. J.: Lectures on special Lagrangian submanifolds. In: Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifold, eds. Vafa, C., Yau, S.-T., AMS/IP Stud. Adv. Math. Vol. 23, Providence, RI: Amer. Math. Soc., 2001, pp. 151–182

  48. Kac, V. G.: Infinite Dimensional Lie Algebras. Cambridge: Cambridge University Press, 1985

  49. Kapustin, A.: D-branes in a topologically nontrivial B-field. Adv. Theor. Math. Phys. 4, 127–154 (2000)

    Google Scholar 

  50. Klimčik, C., Ševera, P.: Open Strings and D-branes in WZNW model. Nucl. Phys. B488, 653–676 (1997)

    Google Scholar 

  51. Kontsevich, M.: Mirror symmetry in dimension 3. Séminaire Bourbaki, 801, Astérisque 237, 275–293 (1996)

    Google Scholar 

  52. B. Kostant: Quantization and unitary representations. Lecture Notes in Math., Vol. 170, Berlin: Springer, 1970, pp. 87–207

  53. Kreuzer, M, Schellekens, A., N.: Simple currents versus orbifolds with discrete torsion - a complete classification. Nucl. Phys. B 411, 97–121 (1994)

    Google Scholar 

  54. Mackaay, M.: A note on the holonomy of connections in twisted bundles. http://arxiv.org/list/math.DG/0106019, 2001

  55. Matsubara, K., Schomerus, V., Smedback, M.: Open strings in simple current orbifolds. Nucl. Phys. B O626, 53–72 (2002)

    Google Scholar 

  56. Meinrenken, E.: The basic gerbe over a compact simple Lie group. L’Enseignement Mathematique 49, 307–333 (2003)

    Google Scholar 

  57. Moore, G.: K-Theory from a physical perspective. http://arxiv.org/list/hep-th/0304018, 2003

  58. Moore, G., Seiberg, N.: Lectures on RCFT. Physics, Geometry, and Topology, New York: Plenum Press, 1990

  59. Murray, M. K.: Bundle gerbes. J. London Math. Soc. (2) 54, 403–416 (1996)

  60. Murray, M. K., Stevenson, D.: Bundle gerbes: stable isomorphisms and local theory. J. London Math. Soc. (2) 62, 925–937 (2000)

    Google Scholar 

  61. Pawe lczyk, J.: SU(2) WZW D-branes and their non-commutative geometry from DBI action. JHEP 08, 006 (2000)

    Google Scholar 

  62. Petkova, V. B., Zuber, J.-B.: Conformal boundary conditions and what they teach us. In: Proceedings of Nonperturbative Quantum Field Theoretic Methods and their Applications, Horvath, Z., Palla, L. eds., Singapore: World Scientific, 2001, pp. 1–35

  63. Picken, R.: TQFT’s and gerbes. In: Algebraic and Geometric Topology 4, 243–272 (2004)

    Google Scholar 

  64. Polchinski, J.: TASI lectures on D-branes. http://arxiv.org/list/hep-th/9611050, 1996

  65. Reis, N.: Geometric interpretation of boundary conformal field theories. Ph.D. thesis. ENS-Lyon 2003

  66. Runkel, I.: Boundary structure constants for the A-series Virasoro minimal models. Nucl. Phys. B 549, 563–578 (1999)

    Google Scholar 

  67. Runkel, I.: Structure constants for the D-series Virasoro minimal models. Nucl. Phys. B 579, 561–589 (2000)

    Google Scholar 

  68. Schellekens, A. N.: The program Kac. http://www.nikhef.nl/~t58/kac, 1996

  69. Schweigert, C., Fuchs, J., Walcher, J.: Conformal field theory, boundary conditions and applications to string theory. In: Non-Perturbative QFT Methods and Their Applications, Horvath, Z., Palla, L. eds., Singapore: World Scientific, 2001, pp. 37–93

  70. Schomerus, V.: Lectures on branes in curved backgrounds. Class. Quant. Grav. 19, 5781–5847 (2002)

    Google Scholar 

  71. Sharpe, E. R.: Discrete torsion and gerbes I, II. http://arxiv.org/list/hep-th/9909108, and http://arxiv.org/list/hep-th/9909120, 1999

  72. Vafa, C.: Modular invariance and discrete torsion on orbifolds. Nucl. Phys. B 273, 592–606 (1986)

    Google Scholar 

  73. Walcher, J.: Worldsheet boundaries, supersymmetry, and quantum geometry. ETH dissertation No. 14225, 2001

  74. Witten, E.: Non-abelian bosonization in two dimensions. Commun. Math. Phys. 92, 455–472 (1984)

    Google Scholar 

  75. Witten, E.: Overview of K-theory applied to strings. J. Mod. Phys. A16, 693–706 (2001)

    Google Scholar 

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  1. Laboratoire de Physique, ENS-Lyon, 46, Allée d’Italie, 69364, Lyon, France

    Krzysztof Gawedzki

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Communicated by M.R. Douglas

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Gawedzki, K. Abelian and Non-Abelian Branes in WZW Models and Gerbes. Commun. Math. Phys. 258, 23–73 (2005). https://doi.org/10.1007/s00220-005-1301-1

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