Singular connection and Riemann theta function

TL;DR

This paper establishes a Chern-Weil formula for singular connections on SU(n+1) over four-manifolds with embedded surfaces, linking number theory and representation theory to analyze fundamental group representations.

Contribution

It introduces a new Chern-Weil formula for singular SU(n+1) connections and connects number theory with the study of fundamental group representations in four-manifolds.

Findings

Derived a Chern-Weil formula for singular connections

Connected number theory with representation theory of fundamental groups

Analyzed irreducible SU(n+1)-representations in four-manifolds

Abstract

We prove the Chern-Weil formula for SU(n+1)-singular connections over the complement of an embedded oriented surface in smooth four manifolds. The expression of the representation of a number as a sum of nonvanishing squares is given in terms of the representations of a number as a sum of squares. Using the number theory result, we study the irreducible SU(n+1)-representations of the fundamental group of the complement of an embedded oriented surface in smooth four manifolds.