Singular connections and Riemann theta functions

We prove the Chern-Weil formula forSU(n + l)-singular connections over the complement of an embedded oriented surface in a smooth four-manifold. The number of representations of a positive integer n as a sum of nonvanishing squares is given in terms of the number of its representations as a sum of squares. Using this number-theoretic result, we study the irreducible SU(n +1)-representations of the fundamental group of the complement of an embedded oriented surface in smooth four-manifold.