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On three-dimensional ${\cal N}=4$ supersymmetry: maximally supersymmetric backgrounds and massive deformations (2504.20712v1)

Published 29 Apr 2025 in hep-th, math-ph, and math.MP

Abstract: Using the $SO ({\cal N})$ superspace formulation for $\cal N$-extended conformal supergravity in three dimensions, we derive all maximally supersymmetric backgrounds in the ${\cal N} =4$ case. The specific feature of this choice is that the so-called super Cotton tensor $X{IJKL} = X{[IJKL]}$, which exists for ${\cal N} \geq 4$, is equivalent to the scalar $X$ defined by $X{IJKL} = \varepsilon{IJKL} X$. This scalar may be used as a deformation parameter. In the family of $(p,q)$ anti-de Sitter (AdS) superspaces with $p+q=4$, it is known that $X\neq 0$ exists only if $p=4$ and $q=0$. In general, the $(4,0)$ AdS superspaces are characterised by the structure group $SL(2,{\mathbb R}) \times SO (4)$ and their geometry is determined by two constant parameters, $S$ and $X$, of which the former determines the AdS curvature, while the $R$-symmetry curvature is determined by the parameters $(X+2S)$ and $(X-2S)$ in the left and right sectors of $SU(2){\rm L} \times SU(2){\rm R}$, respectively. Setting $S=0$ leads to the so-called deformed ${\cal N}=4$ Minkowski superspace ${\mathbb M}{3|8}_X$ introduced thirteen years ago. We construct general interacting supersymmetric field theories in ${\mathbb M}{3|8}_X$ and demonstrate that they originate as massive deformations of the following two families of ${\cal N} =4$ theories in standard Minkowski superspace ${\mathbb M}{3|8}$: (i) ${\cal N}=4$ superconformal field theories; and (ii) ${\cal N}=4$ supersymmetric gauge theories in ${\mathbb M}{3|8}$ which are not superconformal but possess the $R$-symmetry group $SU(2){\rm L} \times SU(2){\rm R}$. Extensions of the theories in (ii) to ${\mathbb M}{3|8}_X$ necessarily contain Chern-Simons terms at the component level. We also demonstrate the generation of topologically massive ${\cal N}=4$ supersymmetric gauge theories from radiative corrections in the hypermultiplet sector.

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