  We characterize the global maximizers of a certain non-local functional
defined on the space of all positively curved metrics on an ample line bundle L
over a Kahler manifold X. This functional is an adjoint version, introduced by
Berndtsson, of Donaldson's L-functional and generalizes the Ding-Tian
functional whose critical points are Kahler-Einstein metrics of positive Ricci
curvature. Applications to (1) analytic torsions on Fano manifolds (2)
Chern-Simons-Higgs vortices on tori and (3) Kahler geometry are given. In
particular, proofs of conjectures of (1) Gillet-Soul\'e and Fang (concerning
the regularized determinant of Dolbeault Laplacians on the two-sphere) (2)
Tarantello and (3) Aubin (concerning Moser-Trudinger type inequalities) in
these three settings are obtained. New proofs of some results in Kahler
geometry are also obtained, including a lower bound on Mabuchi's K-energy and
the uniqueness result for Kahler-Einstein metrics on Fano manifolds of
Bando-Mabuchi. This paper is a substantially extended version of the preprint
arXiv:0905.4263 which it supersedes.
