Localization and Abelianization of Strings on Group Manifolds: The Non-simply Connected Case

Abstract

We study the partition functions of Wess--Zumino--Witten (WZW) models with compact connected simple Lie group manifolds that are not simply connected. Starting from the modular-invariant partition function of Felder--Gaw\k{e}dzki--Kupiainen (FGK), we derive a localization formula, which can also be obtained directly by supersymmetric localization of the corresponding supersymmetric WZW model. Our results reveal a variety of topological effects associated with the nontrivial topology of the target group. In particular, the Wess--Zumino amplitude is governed by FGK cocycles, while the fermion Pfaffians exhibit global anomalies associated with the holonomies of Pfaffian line bundles, captured by relative Rochlin invariants. Together, these results provide a unified description of the topological contributions to the semiclassical localization of WZW models with non-simply connected target groups.