Singular Monopoles and Supersymmetric Gauge Theories in Three Dimensions

TL;DR

The paper demonstrates that the Coulomb branches of $D=3,N=4$ $SU(n)$ gauge theories with $k$ fundamental hypermultiplets are, precisely, the centered moduli spaces of $n$ copies of $U(2)$ monopoles with $k$ singularities on the $A_{k-1}$ ALF space. It constructs these spaces as infinite-dimensional hyperkähler quotients of Nahm data, providing an implicit but rigorous metric description, and verifies complex-structure data against Seiberg–Witten predictions for $SU(2)$ with up to four hypermultiplets. The results yield explicit ALF gravitational instantons of type $D_k$ and illuminate brane realizations, including absence of phase transitions, with a path toward potential $E_k$ extensions. Overall, the work ties three-dimensional Coulomb branches to monopole moduli spaces, offering a geometric framework aligned with field theory and string-theoretic dualities.

Abstract

According to the proposal of Hanany and Witten, Coulomb branches of N=4 SU(n) gauge theories in three dimensions are isometric to moduli spaces of BPS monopoles. We generalize this proposal to gauge theories with matter, which allows us to describe the metrics on their spaces of vacua by means of the hyperkähler quotient construction. To check the identification of moduli spaces a comparison is made with field theory predictions. For SU(2) theory with k fundamental hypermultiplets the Coulomb branch is expected to be the D_k ALF gravitational instanton, so our results lead to a construction of such spaces. In the special case of SU(2) theory with four or fewer fundamental hypermultiplets we calculate the complex structures on the moduli spaces and compare them with field-theoretical results. We also discuss some puzzles with brane realizations of three-dimensional N=4 theories.

Figures (6)

• Figure 1: (a) Parallel NS5-branes (vertical lines) with D3-branes (horizontal lines) suspended between them. The horizontal direction corresponds to $x^6$, while the vertical direction corresponds to $x^3,\,x^4,$ and $x^5$ collectively. (b) The same, with additional D5-branes (crosses) connected by D3-branes to NS5-branes.
• Figure 2: Starting with configuration (a) and moving the D5-brane to the right one gets configuration (b). In both configurations the low-energy gauge theory on the D3-brane has one hypermultiplet.
• Figure 3: Starting with configuration (a) and moving the D5-brane to the left one gets configurations (b) and (c).
• Figure 4: The brane configuration corresponding to singular $U(2)$ monopoles can be obtained as a limit of that corresponding to regular $SU(3)$ monopoles.
• Figure 5: A $U(2)$ 2-monopole with three singularities is a limit of a regular $(1,2,2)$$SU(4)$ monopole.