POLYAKOV SPIN FACTORS AND LAPLACIANS ON HOMOGENEOUS SPACES

We derive the Laplacian Δ(A) associated with parallel transport of spin on the bundle SN ← E+. This Laplacian, previously used to construct the generalization of the Polyakov spin factor to an arbitrary number of dimensions N + 1, involves a nonabelian SO(N) monopole connection A. Using methods of differential geometry and harmonic analysis on homogeneous spaces, we prove that Δ(A) is given by the difference of two quadratic Casimir operators associated with Spin(N + 1) and Spin(N), respectively. We then employ the Gel'fand-Zetlin method to characterize the spectrum and eigenfunctions of -Δ(A). In particular, we prove the previously conjectured degeneracy of the ground state. © 1992.