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Applied Mathematicsematics

Communications in Mathematical Physics - Volume 258 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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One would like, however, to assure that the boundary version of the theory still possesses rich symmetry. The fundamental symmetry of the bulk WZW model is provided by two chiral copies of the current algebra g associated to the Lie algebra g of G. The simplest family of branes (called symmetric) assures that the diagonal current algebra is not broken in the presence of boundaries. Such requirement restricts the brane supports D ⊂ G to coincide with a conjugacy class in G [2] and imposes further conditions on the G-branes supported by D that we shall describe now.

3) Abelian and Non-Abelian Branes in WZW Models and Gerbes 33 for a sufficiently fine triangulation (split) of S 1 by intervals b with vertices v such that ϕ(b) ⊂ Oib and ϕ(v) ∈ Oiv . e. in agreement with the standard orientation of S 1 ) from right to left, starting from the vertex 1. Again, the matrix Giv ib should be inverted if v has negative orientation. Below, we shall use the same expression to define the parallel transport along open lines. For loops, let W(φ) = tr H(φ) be the corresponding Wilson loop “observable”.

The action of Z on the vector bundles D1 satisfies zz = zz and induces the action on the sections: ED 0 (U(z) )(g) = z for (z−1 g) D1 ∈ (ED0z ). One obtains in this way a representation U of Z in the space z HDD0 = 1 ⊕ (D 0 ,D 1 ) D HD 0 1 1 containing all the G-theory states compatible with the ones in H D0 . Operators U(z) D1 D D . 2) they induce the maps 1 λ1 UDD0 λ (z) : CN0 ⊗ CN1 ⊗ Mλ0zλ −→ CN0 ⊗ CN1 ⊗ Mλλ0 λ 1 1 z on the multiplicity spaces where zλ = k zτ for λ = kτ and λz = kτz = k z−1 τ .

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