Global Gauge Anomalies in Coset Models of Conformal Field Theory

Abstract

We study the occurrence of global gauge anomalies in the coset models of two-dimensional conformal field theory that are based on gauged WZW models. A complete classification of the non-anomalous theories for a wide family of gauged rigid adjoint or twisted-adjoint symmetries of WZW models is achieved with the help of Dynkin's classification of Lie subalgebras of simple Lie algebras.Considering each subgroup Z and the corresponding values of a 1 , a 2 ,ã 1 , andã 2 , and recalling the admissible values (3.33) of the level, we deduce Proposition 3.10 The twisted coset models corresponding to Lie algebra g = so(8), outer automorphism ω 4 and arbitrary subalgebra do not have anomalies for Z =Z (+ theory) and Z = Z 1 , Z 2 or Z diag . The models with h = g and Z =Z (-theory) is anomalous.The results for the twist ω −1 4 may be deduced from the above proposition if we observe that ω −1 4 may be obtained from ω 4 by the conjugation by any cyclic outer automorphism ω ′ of order 2. Hence the conditions for the absence or the presence of anomalies for the theory twisted by ω −1 4 are as for the ones for the twist ω 4 except for the exchange of the ± theories for Z =Z and k odd leading to Proposition 3.11 The twisted coset models corresponding to Lie algebra g = so(8), outer automorphism ω −1 4 and arbitrary subalgebra do not have anomalies for Z =Z ((−) k theory) and Z = Z 1 , Z 2 or Z diag . The models with h = g, and Z =Z ((−) k+1 theory) is anomalous.This may be confirmed by a direct calculation. 9

Bosonization with a background U(1) gauge field

“…Moreover, it is shown that the U (1) boson obtains mass term by gauging out background gauge field. This result itself is quite different from G/G coset WZW model description which results in topological field theory by gauging out the background gauge field 20,21 . Therefore, it is necessary to modify the bosonic gauged WZW models so that they can produce a consistent path integral with their fermionic counterparts.…”

Gauge-theoretic invariants for topological insulators: a bridge between Berry, Wess–Zumino, and Fu–Kane–Mele

“…Finally, the proof of Theorem 4.1 involves the adjoint Polyakov-Wiegmann formula from Theorem 3.5 concerning Wess-Zumino amplitudes for products of fields, which mostly relies on the homotopy classes of the considered maps, characterized by Lemmas A.1 and A.2. As it was pointed out in Remark 3.6, the Polyakov-Wiegmann formula and its adjoint version can be anomalous, so that that the Wess-Zumino amplitude of a product map gh is not easily related to the ones of g and h. This part of our work also constitutes a first step towards a classification of anomalies for U (N )-valued fields, that generalizes the one for simple Lie groups obtained using gerbe techniques [21,15], for what concerns the Polyakov-Wiegmann formula, its adjoint version, and beyond.…”

“…Such an obstruction, or anomaly, was already studied in detail for every closed compact Σ and every compact simple Lie group in [21], and Theorem 3.5 above states that the adjoint Polyakov-Wiegmann formula has no anomaly for Σ = T 2 and G = U (N ). More generally, a detailed classification for simple Lie groups in the context of gauged Wess-Zumino-Witten models shows that the corresponding adjoint version can also be anomalous in some cases [15]. ♦…”

Black hole microstates from the worldsheet

“…It is possible that the quantization conditions on k, m, n, p could alternatively be obtained by analyzing the global consistency of the gauging, see e.g. [61][62][63][64][65]. In the next section we will instead proceed to analyze the global geometry of the gauged target space.…”