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

A bstract Using the $$ \textrm{SO}\left(\mathcal{N}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>SO</mml:mi> <mml:mfenced> <mml:mi>N</mml:mi> </mml:mfenced> </mml:math> superspace formulation for $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> -extended conformal supergravity in three dimensions, we derive all maximally supersymmetric backgrounds in the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 case. The specific feature of this choice is that the so-called super Cotton tensor X IJKL = X [ IJKL ] , which exists for $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> ≥ 4, is equivalent to the scalar X defined by X IJKL = ε 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, p ≥ q , it is known that X ≠ 0 exists only if p = 4 and q = 0. In general, the (4, 0) AdS superspaces are characterised by the structure group SL(2, ℝ ) × 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 + 2 S ) and ( X – 2 S ) in the left and right sectors of SU(2) L × SU(2) R , respectively. Setting S = 0 leads to the so-called deformed $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 Minkowski superspace $$ {\mathbbm{M}}_X^{\left.3\right|8} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>X</mml:mi> <mml:mrow> <mml:mfenced> <mml:mn>3</mml:mn> </mml:mfenced> <mml:mn>8</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> introduced thirteen years ago. We use projective-superspace techniques to construct general interacting supersymmetric field theories in $$ {\mathbbm{M}}_X^{\left.3\right|8} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>X</mml:mi> <mml:mrow> <mml:mfenced> <mml:mn>3</mml:mn> </mml:mfenced> <mml:mn>8</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> and demonstrate that they originate as massive deformations of the following two families of $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 theories in standard Minkowski superspace $$ {\mathbbm{M}}^{\left.3\right|8} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>M</mml:mi> <mml:mrow> <mml:mfenced> <mml:mn>3</mml:mn> </mml:mfenced> <mml:mn>8</mml:mn> </mml:mrow> </mml:msup> </mml:math> : (i) $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 superconformal field theories; and (ii) $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 supersymmetric gauge theories in $$ {\mathbbm{M}}^{\left.3\right|8} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Mat