# Index-Like Theorems from Line Defect Vevs

## TL;DR

This paper establishes a new index-like theorem linking Dirac operator indices on singular monopole moduli spaces to characteristic numbers of vector bundles over Kronheimer-Nakajima quiver varieties, using spectral network coordinates and expectation values of 't Hooft defects.

## Contribution

It introduces a novel index theorem connecting Dirac operator indices on singular monopole moduli spaces with characteristic numbers of vector bundles over quiver varieties, via spectral network techniques.

## Key findings

- Derived an index-like theorem for Dirac operators on singular monopole moduli spaces.
- Expressed Dirac operator indices in terms of characteristic numbers of vector bundles.
- Connected complexified Fenchel-Nielsen coordinates with spectral network coordinates.

## Abstract

In this paper we investigate the relation between complexified Fenchel-Nielsen coordinates and spectral network coordinates on Seiberg-Witten moduli space. The main technique is the comparison of exact expressions for the expectation value of 't Hooft defects in certain 4D $SU(2)$ $\mathcal{N}=2$ gauge theories. We derive an index-like theorem for a class of Dirac operators on singular monopole moduli spaces. Our expression determines the indices of Dirac operators on singular monopole moduli spaces in terms of characteristic numbers for vector bundles over certain Kronheimer-Nakajima quiver varieties.