Twisted circle compactification of $\mathcal{N}=4$ SYM and its Holographic Dual

TL;DR

This paper analyzes a twisted circle compactification of $SU(N)$\ ${\cal N}=4$ SYM on $\mathbb{R}^{2,1}\times S^1$ with anti-periodic fermions and a background $U(1)$ gauge field in the $R$-symmetry, preserving four supercharges. The low-energy 3D theory features both gapped and ungapped phases: the gapped phase corresponds to an IR $3$D ${\cal N}=2$ YM-Chern-Simons theory at level $N$, while the ungapped phase sits at the root of a Higgs branch and is dual to a quotiented AdS$_5\times S^5$ geometry. The holographic description involves two backgrounds, the SUSY AdS$_5$ soliton with a shrinking circle and the smooth AdS quotient with a non-shrinking circle, with their SUSY and Polyakov-loop properties matching the field-theoretic expectations. The authors further extend the twisting procedure to maximally SUSY YM theories in other even dimensions (2D and 6D) and discuss extensions and limitations to odd dimensions, as well as probe brane dynamics that illuminate Higgs-branch physics and domain-wall structure.

Abstract

We consider a compactification of 4D $\mathcal{N}=4$ SYM, with $SU(N)$ gauge group, on a circle with anti-periodic boundary conditions for the fermions. We couple the theory to a constant background gauge field along the circle for an abelian subgroup of the $R$-symmetry which allows to preserve four supersymmetries. The 3D effective theory exhibits gapped and ungapped phases, which we argue are holographically dual, respectively, to a supersymmetric soliton in AdS$_{5}\times S^{5}$, and a particular quotient of AdS$_5\times S^5$. The gapped phase corresponds to an IR 3D $\mathcal{N}=2$ supersymmetric Yang-Mills-Chern-Simons theory at level $N$, while the ungapped phase is naturally identified with the root of a Higgs branch in the 3D theory. We discuss the extension of the twisting procedure to maximally SUSY Yang-Mills theories in different dimensions, obtaining the relevant duals for 2D and 6D, and comment on the odd dimensional cases.