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Unoriented WZW models and holonomy of bundle gerbes. (English) Zbl 1148.53057
To become a conformally invariant theory, the Langrangian description of a Wess-Zumino-Witten (WZW) model needs an additional term, known as the Wess-Zumino term. Its nature as a surface holonomy has been identified in [O. Alvarez, Commun. Math. Phys. 100, 279–309 (1985; Zbl 0612.55009) and K. Gawȩdzki, Topological actions in two-dimensional quantum field theories, in: Non-perturbative quantum field theory, London, Plenum Press (1988)], while the most natural differential-geometric object for the latter turned out to be a Hermitian \(U(1)\) bundle gerbe with connection and curving [A. L. Carey, S. Johnson and M. K. Murray, J. Geom. Phys. 52, No. 2, 186–216 (2004; Zbl 1092.81055)].
To extend the notion of surface holonomy to unoriented surfaces, the authors introduce an additional structure, called bundle gerbes with Jandl structure. A Jandl structure relates, roughly speaking, the pull back of the gerbe data under an involution of the base manifold to the local data of the dual gerbe. It is shown that this structure allows indeed an extension of the definition of the usual gerbe holonomy from oriented to unoriented surfaces. Furthermore, formulas for these holonomies are derived in local data which generalize known expressions for oriented surfaces. These results are finally applied to define WZW models on unoriented surfaces.

53C80 Applications of global differential geometry to the sciences
53C29 Issues of holonomy in differential geometry
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T45 Topological field theories in quantum mechanics
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[1] Alvarez O. (1985). Topological Quantization and Cohomology. Commun. Math. Phys. 100: 279–309 · Zbl 0612.55009 · doi:10.1007/BF01212452
[2] Bachas C., Couchoud N. and Windey P. (2001). Orientifolds of the 3-Sphere. JHEP 12: 003 · doi:10.1088/1126-6708/2001/12/003
[3] Berger, M., Gostiaux, B.: Differential Geometry: Manifolds, Curves, and Surfaces. Volume 115 of Graduate Texts in Mathematics, Berlin-Heidelberg-New York: Springer, 1988 · Zbl 0629.53001
[4] Bianchi M., Pradisi G. and Sagnotti A. (1992). Toroidal Compactification and Symmetry Breaking in Open-String Theories. Nucl. Phys. B 376: 365–386 · doi:10.1016/0550-3213(92)90129-Y
[5] Brunner I. (2002). On Orientifolds of wzw Models and their Relation to Geometry. JHEP 01: 007 · doi:10.1088/1126-6708/2002/01/007
[6] Brylinski, J.-L.: Loop spaces, Characteristic Classes and Geometric Quantization. Volume 107 of Progress in Mathematics, Basel: Birkhäuser, 1993 · Zbl 0823.55002
[7] Brylinski, J.-L.: Gerbes on complex reductive Lie Groups. http://arxiv.org/list/math/0002158, 2000
[8] Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology, Volume 82 of Graduate Texts in Mathematics, Berlin-Heidelberg-New York: Springer, 1982 · Zbl 0496.55001
[9] Carey A.L., Johnson S. and Murray M.K. (2002). Holonomy on D-Branes. J. Geom. Phys. 52(2): 186–216 · Zbl 1092.81055
[10] Fuchs J., Huiszoon L.R., Schellekens A.N., Schweigert C. and Walcher J. (2000). Boundaries, Crosscaps and simple Currents. Phys. Lett. B 495(3–4): 427–434 · Zbl 0976.81090 · doi:10.1016/S0370-2693(00)01271-5
[11] Fioravanti D., Pradisi G. and Sagnotti A. (1994). Sewing Constraints and non-orientable Open Strings. Phys. Lett. B 321: 349–354 · doi:10.1016/0370-2693(94)90255-0
[12] Fuchs J., Runkel I. and Schweigert C. (2004). TFT Construction of RCFT Correlators II: unoriented World Sheets. Nucl. Phys. B 678(3): 511–637 · Zbl 1097.81736 · doi:10.1016/j.nuclphysb.2003.11.026
[13] Gawedzki, K.: Topological Actions in two-dimensional Quantum Field Theories. In: Non- perturbative Quantum Field Theory, London: Plenum Press, 1988
[14] Gomi K. (2003). Equivariant smooth Deligne Cohomology. Osaka J. Math. 42(2): 309–337 · Zbl 1081.14030
[15] Gawedzki K. and Reis N. (2002). WZW Branes and Gerbes. Rev. Math. Phys. 14(12): 1281–1334 · Zbl 1033.81067 · doi:10.1142/S0129055X02001557
[16] Gawedzki K. and Reis N. (2003). Basic Gerbe over non-simply connected compact Groups. J. Geom. Phys. 50(1–4): 28–55 · Zbl 1067.22009 · doi:10.1016/j.geomphys.2003.11.004
[17] Huiszoon L.R. and Schellekens A.N. (2000). Crosscaps, Boundaries and T-Duality. Nucl. Phys. B 584(3): 705–718 · Zbl 0984.81119 · doi:10.1016/S0550-3213(00)00320-5
[18] Huiszoon L.R., Schellekens A.N. and Sousa N. (1999). Klein bottles and simple Currents. Phys. Lett. B 470(1): 95–102 · Zbl 0987.81096 · doi:10.1016/S0370-2693(99)01241-1
[19] Huiszoon L.R., Schalm K. and Schellekens A.N. (2002). Geometry of WZW orientifolds. Nucl. Phys. B 624(1–2): 219–252 · Zbl 0985.81057 · doi:10.1016/S0550-3213(02)00005-6
[20] Jandl E. (1995) Lechts und rinks. Munich, Luchterhand Literaturverlag
[21] Meinrenken E. (2002). The Basic Gerbe over a compact simple Lie Group. Enseign. Math., II. Sér. 49(3–4): 307–333 · Zbl 1061.53034
[22] Pradisi G., Sagnotti A. and Stanev Y.S. (1995). The Open descendants of nondiagonal SU(2) WZW models. Phys. Lett. B 356: 230–238 · doi:10.1016/0370-2693(95)00840-H
[23] Pradisi G., Sagnotti A. and Stanev Y.S. (1995). Planar duality in SU(2) WZW models. Phys. Lett. B 354: 279–286 · doi:10.1016/0370-2693(95)00532-P
[24] Sousa N. and Schellekens A.N. (2003). Orientation matters for NIMreps. Nucl. Phys. B 653(3): 339–368 · Zbl 1010.81073 · doi:10.1016/S0550-3213(02)01124-0
[25] Stevenson, D.: The Geometry of Bundle Gerbes, PhD thesis, University of Adelaide, http://arxiv.org/list/math.DG/0004117, 2000 · Zbl 1019.55009
[26] Witten E. (1984). Nonabelian Bosonization in two Dimensions. Commun. Math. Phys. 92: 455–472 · Zbl 0536.58012 · doi:10.1007/BF01215276
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