Localization and Abelianization of Strings on Group Manifolds: The Simply Connected Case

Abstract: We study the localization formula for the partition function of the Wess--Zumino--Witten (WZW) model on the torus for compact, connected, and simply connected simple Lie groups. We identify a missing factor in the original localization treatment of Murthy and Witten and show that it agrees with the result obtained from the Hamiltonian formulation. We trace its origin to the abelianization of the Wess--Zumino amplitude, which reduces to a flat Kalb--Ramond B--field holonomy in a Narain lattice CFT associated with the maximal torus. This establishes a direct relation between the localization result and the lattice description of the corresponding abelian theory. We further analyze the point--particle limit of the torus partition function, in which the WZW model reduces to quantum mechanics on the group manifold, and reproduce Frenkel's heat--kernel trace formula.