Abstract:We revisit the gauging of rigid symmetries in two-dimensional bosonic sigma models with a Wess-Zumino term in the action. Such a term is related to a background closed 3-form H on the target space. More exactly, the sigma-model Feynman amplitudes of classical fields are associated to a bundle gerbe with connection of curvature H over the target space. Under conditions that were unraveled more than twenty years ago, the classical amplitudes may be coupled to the topologically trivial gauge fields of the symmetry group in a way which assures infinitesimal gauge invariance. We show that the resulting gauged Wess-Zumino amplitudes may, nevertheless, exhibit global gauge anomalies that we fully classify. The general results are illustrated on the example of the WZW and the coset models of conformal field theory. The latter are shown to be inconsistent in the presence of global anomalies. We introduce a notion of equivariant gerbes that allow an anomaly-free coupling of the Wess-Zumino amplitudes to all gauge fields, including the ones in non-trivial principal bundles. Obstructions to the existence of equivariant gerbes and their classification are discussed. The choice of different equivariant structures on the same bundle gerbe gives rise to a new type of discrete-torsion ambiguities in the gauged amplitudes. An explicit construction of gerbes equivariant with respect to the adjoint symmetries over compact simply connected simple Lie groups is given.
We define the sigma-model action for world-sheets with embedded defect networks in the presence of a three-form field strength. We derive the defect gluing condition for the sigma-model fields and their derivatives, and use it to distinguish between conformal and topological defects. As an example, we treat the WZW model with defects labelled by elements of the centre Z(G) of the target Lie group G; comparing the holonomy for different defect networks gives rise to a 3-cocycle on Z(G). Next, we describe the factorisation properties of two-dimensional quantum field theories in the presence of defects and compare the correlators for different defect networks in the quantum WZW model. This, again, results in a 3-cocycle on Z(G). We observe that the cocycles obtained in the classical and in the quantum computation are cohomologous. *
Bundle gerbes with connection and their modules play an important role in the theory of two-dimensional sigma models with a background Wess-Zumino flux: their holonomy determines the contribution of the flux to the Feynman amplitudes of classical fields. We discuss additional structures on bundle gerbes and gerbe modules needed in similar constructions for orientifold sigma models describing closed and open strings. BUNDLE GERBES FOR ORIENTIFOLD SIGMA MODELS 625the k-fold fibre product of Y with itself (composed of the elements in the Cartesian product whose components have the same projection to M ). The fibre products are, again, smooth manifolds in such a way that the canonical projections π i 1 ...ir : Y [k] G G Y [r] are smooth maps.In the following, we collect the basic definitions.Definition 2.1 ([18]). A bundle gerbe G over a smooth manifold M is a surjective submersion π : Y G G M , a line bundle L over Y [2] , a 2-form C ∈ Ω 2 (Y ), and an isomorphismof line bundles over Y [3] , such that two axioms are satisfied:(G1) The curvature of L is fixed by
The simplest orientifolds of the WZW models are obtained by gauging a Z 2 symmetry group generated by a combined involution of the target Lie group G and of the worldsheet. The action of the involution on the target is by a twisted inversion g → (ζg) −1 , where ζ is an element of the center of G. It reverses the sign of the KalbRamond torsion field H given by a bi-invariant closed 3-form on G. The action on the worldsheet reverses its orientation. An unambiguous definition of Feynman amplitudes of the orientifold theory requires a choice of a gerbe with curvature H on the target group G, together with a so-called Jandl structure introduced in [31]. More generally, one may gauge orientifold symmetry groups Γ = Z 2 ⋉Z that combine the Z 2 -action described above with the target symmetry induced by a subgroup Z of the center of G. To define the orientifold theory in such a situation, one needs a gerbe on G with a Z-equivariant Jandl structure. We reduce the study of the existence of such structures and of their inequivalent choices to a problem in group-Γ cohomology that we solve for all simple simply-connected compact Lie groups G and all orientifold groups Γ = Z 2 ⋉Z.
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