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Abelian and non-abelian branes in WZW models and gerbes. (English) Zbl 1094.81047
The paper extends the geometric classification of WZW models with boundary which was initiated by K. Gawedzki and N. Reis [Rev. Math. Phys. 14, 1281–1334 (2002; Zbl 1033.81067); J. Geom. Phys. 50, 28–55 (2004; Zbl 1067.22009)]. These models are a special class of boundary conformal field theories, viz. conformally invariant sigma models with a group manifold as target. The presence of boundaries is of particular interest in string theory where it manifests itself in the existence of Dirichlet branes (D-branes). An understanding of the “stringy” or D-geometry that arises in this context may be furthered by a Lagrangian description of these models. This is achieved by using an additional geometric structure on the brane that is summarized in the notion of a gerbe module and includes a twisted Chan-Paton gauge field. The gerbe construction is motivated by a characteristic feature of the WZW model in the Lagrangian formulation: the presence of the coupling to a topologically non-trivial Kalb-Ramond background 2-form field \(B\) on the target group, which is defined only locally, whereas \(H=dB\) is defined globally (and equals the standard invariant 3-form \(\text{tr}(g^{-1}dg)^3\) on the group), but not exact.
Using this geometric approach the paper presents a complete classification of the branes that conserve the diagonal current-algebra symmetry of the WZW models with simple compact, but not necessarily 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 geometric description of branes leads to explicit formulae for the boundary partition functions and boundary operator coefficients in the WZW models with non-simply connected target groups.

MSC:
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
53C80 Applications of global differential geometry to the sciences
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