We present the harmonic superspace formulation of <math altimg='si1.svg'><mi mathvariant='script'>N</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>2</mn></math> gravitino multiplet, the simplest <math altimg='si1.svg'><mi mathvariant='script'>N</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>2</mn></math> half-integer spin gauge supermultiplet. It is shown that, quite similar to other <math altimg='si1.svg'><mi mathvariant='script'>N</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>2</mn></math> gauge multiplets, the gravitino supermultiplet is described by unconstrained analytic prepotentials <math altimg='si134.svg'><msup><mrow><mi>h</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mi>&#945;</mi></mrow></msup></math> and <math altimg='si3.svg'><msup><mrow><mi>h</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup></math> which contain a conformal gravitino in the Wess-Zumino gauge. The analytic prepotentials naturally come out from the study of <math altimg='si1.svg'><mi mathvariant='script'>N</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>2</mn></math> supercurrents associated with hidden symmetries of <math altimg='si1.svg'><mi mathvariant='script'>N</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>2</mn></math> vector-hypermultiplet system. We construct the covariant <math altimg='si1.svg'><mi mathvariant='script'>N</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>2</mn></math> superfield strengths and the invariant <math altimg='si1.svg'><mi mathvariant='script'>N</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>2</mn></math> superfield actions and sketch their component contents. We observe that, at cost of introducing new auxiliary coordinates <math altimg='si199.svg'><msup><mrow><mi mathvariant='normal'>&#936;</mi></mrow><mrow><mi>&#945;</mi></mrow></msup></math> and <math altimg='si5.svg'><msup><mrow><mi>&#969;</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup></math>, the gravitino analytic prepotentials acquire a nice geometric interpretation as extra vielbeins of the covariant harmonic derivative <math altimg='si6.svg'><msup><mrow><mi mathvariant='fraktur'>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup></math>. We speculate on a possible origin of the additional coordinates, including their relationship with <math altimg='si1.svg'><mi mathvariant='script'>N</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>2</mn></math> supertwistors.

Russia

Evgeny Ivanov

Nikita Zaigraev

<math altimg='si1.svg'><mi mathvariant='script'>N</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>2</mn></math> superconformal gravitino in harmonic superspace

We have constructed conserved <math><mi>N</mi><mo>=</mo><mn>2</mn></math> higher-spin supercurrents for an arbitrary integer spin through the unconstrained superfield formulation of <math><mn>4</mn><mi>D</mi><mo>,</mo><mspace width="0.2em"></mspace><mi>N</mi><mo>=</mo><mn>2</mn></math> massless higher-spin theories in the harmonic superspace. The obtained supercurrents are gauge-invariant and determine consistent cubic interactions in the gauge superfield × supercurrent form for massless spin <math><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and two massless spin <math><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></math> <math><mi>N</mi><mo>=</mo><mn>2</mn></math> supermultiplets. Such interactions and <math><mi>N</mi><mo>=</mo><mn>2</mn></math> supercurrents exist only for <math><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mn>2</mn><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></math>. These supercurrents are the <math><mi>N</mi><mo>=</mo><mn>2</mn></math> supersymmetric extension of Berends-Burgers-van Dam higher-spin currents and generalize the linearized Bel-Robinson tensor. The relevant <math><mi>N</mi><mo>=</mo><mn>2</mn></math> supercurrents can be considered as the descendants of the <math><mi>N</mi><mo>=</mo><mn>2</mn></math> principle supercurrent, which has a simple universal structure. An important feature of our outcomes is that, ultimately, all <math><mi>N</mi><mo>=</mo><mn>2</mn></math> supercurrents are built from the <math><mi>N</mi><mo>=</mo><mn>2</mn></math> superfield strengths, which are constructed from the unconstrained analytical higher-spin prepotentials.

JINR

Higher-spin Supercurrents

Using the harmonic superspace approach, we construct, at the linearized level, <math><mi>N</mi><mo>=</mo><mn>2</mn></math> supersymmetric curvatures generalizing scalar curvature, Ricci curvature and Weyl tensor. These supercurvatures are the building blocks of various linearized <math><mn>4</mn><mi>D</mi><mo>,</mo><mi>N</mi><mo>=</mo><mn>2</mn></math> Einstein supergravity invariants. The supercurvatures involving the scalar and Ricci curvatures are analytic harmonic <math><mi>N</mi><mo>=</mo><mn>2</mn></math> superfields, while the Weyl supertensor is a chiral <math><mi>N</mi><mo>=</mo><mn>2</mn></math> superfield. As the basic distinguished feature of our construction, all these objects are expressed through the fundamental analytic gauge prepotentials <math><msup><mi>h</mi><mrow><mo>+</mo><mo>+</mo><mi>M</mi></mrow></msup><mo>,</mo><mi>M</mi><mo>=</mo><mo>(</mo><mi>α</mi><mover><mi>α</mi><mo>˙</mo></mover><mo>,</mo><mi>α</mi><mo>+</mo><mo>,</mo><mover><mi>α</mi><mo>˙</mo></mover><mo>+</mo><mo>,</mo><mn>5</mn><mo>)</mo></math>. The related characteristic features are the heavy use of harmonic derivatives and harmonic zero-curvature equations. On a number of instructive examples, we describe the component reduction of the superfield invariants constructed.

Off-shell invariants of linearized <math><mn>4</mn><mi>D</mi><mo>,</mo><mi>N</mi><mo>=</mo><mn>2</mn></math> supergravity in the harmonic approach

We construct an off-shell   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal cubic vertex for the hypermultiplet coupled to an arbitrary integer higher spin s gauge   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 supermultiplet in a general   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 conformal supergravity background. We heavily use   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2, 4D harmonic superspace that provides an unconstrained superfield Lagrangian description. We start with   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 global superconformal symmetry transformations of the free hypermultiplet model and require invariance of the cubic vertices of general form under these transformations and their gauged version. As a result, we deduce   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2, 4D unconstrained analytic superconformal gauge potentials for an arbitrary integer s. These are the basic ingredients of the approach under consideration. We describe the properties of the gauge potentials, derive the corresponding superconformal and gauge transformation laws, and inspect the off-shell contents of the thus obtained   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal higher-spin s multiplets in the Wess-Zumino gauges. The spin s multiplet involves 8(2s − 1) B  + 8(2s − 1) F  essential off-shell degrees of freedom. The cubic vertex has the generic structure higher spin gauge superfields × hypermultiplet supercurrents. We present the explicit form of the relevant supercurrents.

Ioseph Buchbinder

<math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superconformal higher-spin multiplets and their hypermultiplet couplings

As a continuation of our previous papers  arXiv:2109.07639   and  arXiv:2202.08196  , we study the linearized structure of the manifestly 4D,   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 supersymmetric theory of the cubic couplings of the higher spin gauge superfields to the matter hypermultiplets. We consider in detail the superfield equations of motion, construct the conserved hypermultiplet superfield currents, explore their component structure (basically in the bosonic sector) and compare it with the corresponding currents in the conventional higher-spin bosonic theory. We thoroughly study the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 spin 2 and 3 models as instructive examples.

<math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 higher spins: superfield equations of motion, the hypermultiplet supercurrents, and the component structure

We construct manifestly 4D,   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 supersymmetric and gauge invariant off-shell cubic couplings of matter hypermultiplets to the higher integer spin gauge   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 multiplets introduced in arXiv:2109.07639. The hypermultiplet is described by an analytic harmonic 4D,   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superfield q + with the physical component spins   <math> <mi>s</mi> <mo>=</mo> <mfenced> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mn>0</mn> </mfenced> </math>  $$ \mathbf{s}=\left(\frac{1}{2},0\right) $$  and an infinite number of auxiliary fields. The cubic coupling constructed has the schematic structure   <math> <msup> <mi>q</mi> <mo>+</mo> </msup> <msubsup> <mover> <mi>ℋ</mi> <mo>̂</mo> </mover> <mfenced> <mi>s</mi> </mfenced> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msubsup> <msup> <mi>q</mi> <mo>+</mo> </msup> </math>  $$ {q}^{+}{\hat{\mathrm{\mathscr{H}}}}_{(s)}^{++}{q}^{+} $$ , where   <math> <msubsup> <mover> <mi>ℋ</mi> <mo>̂</mo> </mover> <mfenced> <mi>s</mi> </mfenced> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msubsup> </math>  $$ {\hat{\mathrm{\mathscr{H}}}}_{(s)}^{++} $$  is a differential analytic operator of the highest degree (s − 1) accommodating the massless gauge   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 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 4D,   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 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(2n) × SU(2).

Off-shell cubic hypermultiplet couplings to   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 higher spin gauge superfields

We present, for the first time, the complete off-shell 4D,   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 superfield actions for any free massless integer spin s ≥ 2 fields, using the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 harmonic super-space approach. The relevant gauge supermultiplet is accommodated by two real analytic bosonic superfields   <math> <msubsup> <mi>h</mi> <mrow> <mi>α</mi> <mfenced> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> <mover> <mi>α</mi> <mo>̇</mo> </mover> <mfenced> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> </mrow> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msubsup> </math>  $$ {h}_{\alpha \left(s-1\right)\dot{\alpha}\left(s-1\right)}^{++} $$ ,   <math> <msubsup> <mi>h</mi> <mrow> <mi>α</mi> <mfenced> <mrow> <mi>s</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfenced> <mover> <mi>α</mi> <mo>̇</mo> </mover> <mfenced> <mrow> <mi>s</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfenced> </mrow> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msubsup> </math>  $$ {h}_{\alpha \left(s-2\right)\dot{\alpha}\left(s-2\right)}^{++} $$  and two conjugated complex analytic spinor superfields   <math> <msubsup> <mi>h</mi> <mrow> <mi>α</mi> <mfenced> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> <mover> <mi>α</mi> <mo>̇</mo> </mover> <mfenced> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> </mrow> <mrow> <mo>+</mo> <mn>3</mn> </mrow> </msubsup> </math>  $$ {h}_{\alpha \left(s-1\right)\dot{\alpha}\left(s-1\right)}^{+3} $$ ,   <math> <msubsup> <mi>h</mi> <mrow> <mi>α</mi> <mfenced> <mrow> <mi>s</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfenced> <mover> <mi>α</mi> <mo>̇</mo> </mover> <mfenced> <mrow> <mi>s</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> </mrow> <mrow> <mo>+</mo> <mn>3</mn> </mrow> </msubsup> </math>  $$ {h}_{\alpha \left(s-2\right)\dot{\alpha}\left(s-1\right)}^{+3} $$ , where α(s) := (α 1 . . . α  s ),   <math> <mover> <mi>α</mi> <mo>̇</mo> </mover> </math>  $$ \dot{\alpha} $$ (s) := (  <math> <mover> <mi>α</mi> <mo>̇</mo> </mover> </math>  $$ \dot{\alpha} $$  1 . . .   <math> <mover> <mi>α</mi> <mo>̇</mo> </mover> </math>  $$ \dot{\alpha} $$   s ). Like in the harmonic superspace formulations of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 Maxwell and supergravity theories, an infinite number of original off-shell degrees of freedom is reduced to the finite set (in WZ-type gauge) due to an infinite number of the component gauge parameters in the analytic superfield parameters. On shell, the standard spin content (s , s − 1 / 2 , s − 1 / 2 , s − 1) is restored. For s = 2 the action describes the linearized version of “minimal”   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 Einstein supergravity.