D-branes, Monopoles and Nahm Equations

TL;DR

The paper addresses how to realize the moduli space of BPS monopoles within the D-brane realization of $D=4$ $N=4$ SYM and connect it to the Nahm/ADHMN construction. It analyzes the $SU(2)$ case in depth, deriving Nahm data on $s\in(-1,1)$ with pole boundary conditions and showing a precise one-to-one correspondence with monopole moduli, aided by a probe analysis that reconstructs the monopole gauge field as a world-sheet coupling. It then generalizes the framework to $SU(n)$, formulating interval-based Nahm data, boundary/matching conditions, and D-brane configurations that realize the corresponding monopole moduli spaces, including embeddings along simple roots. The results reveal that Nahm boundary conditions are the physically appropriate constraint in the brane setup, expose intriguing pole behavior and noncommutative transverse coordinates at the boundary, and suggest deep dualities (e.g., T-duality and instanton reciprocity) linking monopoles, branes, and broader soliton moduli spaces.

Abstract

We study the correspondence between IIb solitonic 1-branes and monopoles in the context of the 3-brane realization of $D=4$ $N=4$ super Yang-Mills theory. We show that a bound state of 1-branes stretching between two separated 3-branes exhibits a family of super-symmetric ground states that can be identified with the ADHMN construction of the moduli space of $SU(2)$ monopoles.. This identification is supported by the construction of the monopole gauge field as a space-time coupling in the quantum mechanical effective action of a 1-brane used as a probe. The analysis also reveals an intriguing aspect of the 1-brane theory:the transverse oscillations of the 1-branes in the ground states are described by non-commuting matrix valued fields which develop poles at the boundary. Finally, the construction is generalized to $SU(n)$ monopoles with arbitrary $n>2$.