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

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.