Revisiting the Multi-Monopole Point of $SU(N)$ $\mathcal{N} = 2$ Gauge Theory in Four Dimensions

TL;DR

This paper revisits the multi-monopole point in pure SU(N) N=2 gauge theory, calculating threshold corrections and clarifying previous inconsistencies, with results applicable to various N and connections to integrable systems.

Contribution

It provides an explicit calculation of threshold corrections at the multi-monopole point, extending previous results to finite N and comparing with multiple approaches.

Findings

Broad agreement with existing literature

Explicit extension of results to finite N

Clarification of previous inconsistencies

Abstract

Motivated by applications to soft supersymmetry breaking, we revisit the expansion of the Seiberg-Witten solution around the multi-monopole point on the Coulomb branch of pure $SU(N)$ $\mathcal{N}=2$ gauge theory in four dimensions. At this point $N-1$ mutually local magnetic monopoles become massless simultaneously, and in a suitable duality frame the gauge couplings logarithmically run to zero. We explicitly calculate the leading threshold corrections to this logarithmic running from the Seiberg-Witten solution by adapting a method previously introduced by D'Hoker and Phong. We compare our computation to existing results in the literature; this includes results specific to $SU(2)$ and $SU(3)$ gauge theories, the large-$N$ results of Douglas and Shenker, as well as results obtained by appealing to integrable systems or topological strings. We find broad agreement, while also clarifying some lingering inconsistencies. Finally, we explicitly extend the results of Douglas and Shenker to finite $N$, finding exact agreement with our first calculation.