Complete Intersection Moduli Spaces in N=4 Gauge Theories in Three Dimensions

TL;DR

The work classifies and analyzes a broad family of three-dimensional ${ m N}=4$ gauge theories labeled by ordered partitions $(oldsymbol{\sigma},oldsymbol{ ho})$, realized via brane intervals and linear quivers. By leveraging mirror symmetry, the authors compute the Coulomb branches as Higgs branches of the mirrors and derive explicit Hilbert series, establishing three infinite families whose Coulomb branches are complete intersections. They provide concrete generators and defining relations at various orders in the Hilbert-series expansion, and demonstrate consistency with monopole moduli space descriptions and hyperkähler quotient constructions. The results give exact algebraic presentations of Coulomb branches, enabling direct checks of mirror symmetry and offering detailed insights into the geometry of 3D ${ m N}=4$ moduli spaces with potential implications for brane dynamics and string-theory realizations.

Abstract

We study moduli spaces of a class of three dimensional N=4 gauge theories which are in one-to-one correspondence with a certain set of ordered pairs of integer partitions. It was found that these theories can be realised on brane intervals in Type IIB string theory and can therefore be described using linear quiver diagrams. Mirror symmetry was known to act on such a theory by exchanging the partitions in the corresponding ordered pair, and hence the quiver diagram of the mirror theory can be written down in a straightforward way. The infrared Coulomb branch of each theory can be studied using moment map equations for a hyperKahler quotient of the Higgs branch of the mirror theory. We focus on three infinite subclasses of these singular hyperKahler spaces which are complete intersections. The Hilbert series of these spaces are computed in order to count generators and relations, and they turn out to be related to the corresponding partitions of the theories. For each theory, we explicitly discuss the generators of such a space and relations they satisfy in detail. These relations are precisely the defining equations of the corresponding complete intersection space.

Figures (33)

• Figure 1: (a) The quiver diagram of the $(1)-(2)- \cdots -(n-1)-[n]$ theory. (b) The corresponding brane configuration. (c) The D5-branes are moved to the right of all NS5-branes. The D3-branes are created according to Hanany:1996ie. The partitions $\sigma = (1, \ldots,1)$ and $\rho = (1, \ldots,1)$ (with $n$ one's) are in one-to-one correspondence with this diagram.
• Figure 2: The mirror of the $(1)-(2)- \cdots -(n-1)-[n]$ theory. (a) From diagram (c) in Figure \ref{['fig:123nquiv']}, the NS5-branes and the D5-branes are exchanged and the directions $x^4, x^5, x^6$ are rotated into $x^7, x^8, x^9$ and vice-versa. (b) The D5-branes are moved across the NS5-branes. The D3-brane creation and annihilation are according to Hanany:1996ie. The corresponding quiver diagram is also given next to the brane configuration. Observe that this is actually the $(1)-(2)- \cdots -(n-1)-[n]$ theory. Thus, the theory is self-mirror.
• Figure 3: The quiver diagram of the mirror of the $(1)-(2)- \cdots -(n-1)-[n]$ theory, with the labels of bi-fundamental chiral multiplets.
• Figure 4: (a) The quiver diagram of the $(1)-(2)-[4]$ theory. (b) The corresponding brane configuration. (c) The D5-branes are moved to the right of all NS5-branes. The D3-branes are created according to Hanany:1996ie. The partitions $\sigma = (1,1,1,1)$ and $\rho = (2,1,1)$ are in one-to-one correspondence with this diagram.
• Figure 5: The mirror of the $(1)-(2)-[4]$ theory. (a) From diagram (c) in Figure \ref{['fig:124quiv']}, the NS5-branes and the D5-branes are exchanged and the directions $x^4, x^5, x^6$ are rotated into $x^7, x^8, x^9$ and vice-versa. The partitions $\sigma = (2,1,1)$ and $\rho = (1,1,1,1)$ are in one-to-one correspondence with this diagram. (b) The D5-branes are moved across the NS5-branes. The D3-brane creation and annihilation are according to Hanany:1996ie. The corresponding quiver diagram is also given next to the brane configuration.