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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.

MSC:
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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