Off-shell cubic hypermultiplet couplings to $$ \mathcal{N} $$ = 2 higher spin gauge superfields
Ioseph Buchbinder, Evgeny Ivanov, Nikita Zaigraev
Abstract
A bstract We construct manifestly 4 D , $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 supersymmetric and gauge invariant off-shell cubic couplings of matter hypermultiplets to the higher integer spin gauge $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 multiplets introduced in arXiv:2109.07639. The hypermultiplet is described by an analytic harmonic 4 D , $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 superfield q + with the physical component spins $$ \mathbf{s}=\left(\frac{1}{2},0\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mfenced> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mn>0</mml:mn> </mml:mfenced> </mml:math> and an infinite number of auxiliary fields. The cubic coupling constructed has the schematic structure $$ {q}^{+}{\hat{\mathrm{\mathscr{H}}}}_{(s)}^{++}{q}^{+} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msubsup> <mml:mover> <mml:mi>ℋ</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mfenced> <mml:mi>s</mml:mi> </mml:mfenced> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msubsup> <mml:msup> <mml:mi>q</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:math> , where $$ {\hat{\mathrm{\mathscr{H}}}}_{(s)}^{++} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mover> <mml:mi>ℋ</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mfenced> <mml:mi>s</mml:mi> </mml:mfenced> <mml:mrow> <mml:mo>+</mml:mo> <mml:mo>+</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> is a differential analytic operator of the highest degree ( s − 1) accommodating the massless gauge $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 multiplet with the highest spin s . For odd s the gauge group generators and couplings are proportional to U(1) PG generator of the internal SU(2) PG symmetry of the hypermultiplet and so do not exist if SU(2) PG is unbroken. If this U(1) PG is identified with the central charge of 4 D , $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 supersymmetry, a mass for the hypermultiplet is generated and the odd s couplings vanish in the proper massless limit. For even s the higher-spin gauge transformations and cubic superfield couplings can be defined for both massive and massless (central-charge neutral) hypermultiplets without including U(1) PG generator. All these features directly extend to the case of n hypermultiplets with the maximal internal symmetry USp(2 n ) × SU(2).