Relate Duistermaat–Heckman and supersymmetric localization

Establish a precise relation between Duistermaat–Heckman equivariant localization and supersymmetric localization for the Wess–Zumino–Witten torus partition function, and determine whether the agreement between their localization formulas can be proved directly at the level of fixed-point contributions, including a geometric interpretation of the global Wess–Zumino phase in the underlying symplectic data.

Background

The paper derives the supersymmetric localization formula for the torus partition function of the Wess–Zumino–Witten model and identifies its localization configurations with Weyl-group and coroot-lattice data. It observes that a related Duistermaat–Heckman description arises from a Hamiltonian action on a coadjoint orbit of a centrally extended double loop group.

The unresolved issue is to construct a systematic geometric correspondence between the two localization frameworks. In particular, the authors ask how the global Wess–Zumino holonomy, which appears as the phase factor $(-1)^{\kappa(m,w)}$ in supersymmetric localization, is represented in the symplectic data of the Duistermaat–Heckman formulation, and whether the two formulas can be matched term by term at their fixed points.