Anomalies for compact simple Lie groups via G-kernels for loop group nets
Abstract
Let G be a compact simply connected simple Lie group and k ≥ 1. Jump loops give an outer homomorphism θk from G to the outer automorphisms of the local factor of the level-k loop group net, which is the hyperfinite type III1 factor. We show that its Sutherland obstruction is k times the positive basic class in H3Borel(G, 𝕋) ≅ H4(BG, ℤ) and that θk is faithful, so it is a G-kernel. Central elements act by the outer automorphisms of the simple currents. In types A, D, E the level-one class is computed by comparing maximal-torus jump loops with the lattice α-induction construction of the companion paper on both current and charged generators, followed by integral degree-four Cartan restriction. Higher levels follow by tensor additivity, and the remaining types by restriction along index-one embeddings into simply laced groups. As a consequence, for every compact simply connected semisimple G, every class in H4(BG, ℤ) is the obstruction of a G-kernel on the hyperfinite type III1 factor.
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BibTeX
@misc{bischoff2026anomalies,
title = {{Anomalies for compact simple Lie groups via $G$-kernels for loop group nets}},
author = {Marcel Bischoff and Pradyut Karmakar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23321},
eprint = {2610.23321},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23321},
}
RIS
TY - UNPB
TI - Anomalies for compact simple Lie groups via G-kernels for loop group nets
AU - Marcel Bischoff
AU - Pradyut Karmakar
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23321
UR - https://arxiv.org/abs/2610.23321
AB - Let G be a compact simply connected simple Lie group and k ≥ 1. Jump loops give an outer homomorphism θ_k from G to the outer automorphisms of the local factor of the level-k loop group net, which is the hyperfinite type III_1 factor. We show that its Sutherland obstruction is k times the positive basic class in H^3_Borel(G, 𝕋) ≅ H^4(BG, ℤ) and that θ_k is faithful, so it is a G-kernel. Central elements act by the outer automorphisms of the simple currents. In types A, D, E the level-one class is computed by comparing maximal-torus jump loops with the lattice α-induction construction of the companion paper on both current and charged generators, followed by integral degree-four Cartan restriction. Higher levels follow by tensor additivity, and the remaining types by restriction along index-one embeddings into simply laced groups. As a consequence, for every compact simply connected semisimple G, every class in H^4(BG, ℤ) is the obstruction of a G-kernel on the hyperfinite type III_1 factor.
ER -