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

Australia

Sergei Kuzenko

Emmanouil Raptakis

Igor Samsonov

Gabriele Tartaglino-Mazzucchelli

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

We propose an $$ \mathcal{N} $$ = 3 nonlinear multiplet coupled to conformal supergravity and use it to formulate the equations of motion for $$ \mathcal{N} $$ = 3 Poincaré supergravity. These equations, which are naturally described in a new curved supergeometry with structure group SL(2$\textit{,}$ ℂ), imply that the $$ \mathcal{N} $$ = 3 super-Bach tensor vanishes, and thus every solution of Poincaré supergravity is a solution of conformal supergravity. The aforementioned superspace formulation, which we refer to as $$ \mathcal{N} $$ = 3 Einstein superspace, is described in terms of two dimension-1$\textit{/}$2 superfields: (i) the super-Weyl spinor $\textit{W}$ ; and (ii) a spinor isospinor $$ {\chi}_{\alpha}^i $$ .

$$ \mathcal{N} $$ = 3 nonlinear multiplet and supergravity

The standard geometric description of $\textit{d}$-dimensional anti-de Sitter (AdS) space is a quadric in ℝ defined by ($\textit{X}$ ) − ($\textit{X}$ ) − ⋯ − ($\textit{X}$ ) + ($\textit{X}$ ) = $\textit{ℓ}$ = const. In this paper we provide a supersymmetric generalisation of this embedding construction in the $\textit{d}$ = 5 case. Specifically, a bi-supertwistor realisation is given for the $$ \mathcal{N} $$ -extended AdS superspace $$ {\textrm{AdS}}^{5\mid 8\mathcal{N}} $$ , with $$ \mathcal{N} $$ ≥ 1. The proposed formalism offers a simple construction of AdS super-invariants. As an example, we present a new model for a massive superparticle in $$ {\textrm{AdS}}^{5\mid 8\mathcal{N}} $$ which is manifestly invariant under the AdS isometry supergroup SU(2, 2| $$ \mathcal{N} $$ ) and involves two independent two-derivative terms.

Nowar Koning

Embedding formalism for AdS superspaces in five dimensions

In a supergravity framework, the $$ \mathcal{N} $$ -extended anti-de Sitter (AdS) superspace in four spacetime dimensions, $$ {\textrm{AdS}}^{4\left|4\mathcal{N}\right.} $$ , is a maximally symmetric background that is described by a curved superspace geometry with structure group SL(2, $\textit{ℂ}$) × $$ \textrm{U}\left(\mathcal{N}\right) $$ . On the other hand, within the group-theoretic setting, $$ {\textrm{AdS}}^{4\left|4\mathcal{N}\right.} $$ is realised as the coset superspace $$ \textrm{O}\textrm{Sp}\left(\left.\mathcal{N}\right|4;\mathbb{R}\right)/\left[\textrm{SL}\left(2,\mathbb{C}\right)\times \textrm{O}\left(\mathcal{N}\right)\right] $$ , with its structure group being SL(2, $\textit{ℂ}$) × $$ \textrm{O}\left(\mathcal{N}\right) $$ . Here we explain how the two frameworks are related. We give two explicit realisations of $$ {\textrm{AdS}}^{4\left|4\mathcal{N}\right.} $$ as a conformally flat superspace, thus extending the $$ \mathcal{N} $$ = 1 and $$ \mathcal{N} $$ = 2 results available in the literature. As applications, we describe: (i) a two-parameter deformation of the $$ {\textrm{AdS}}^{4\left|4\mathcal{N}\right.} $$ interval and the corresponding superparticle model; (ii) some implications of conformal flatness for superconformal higher-spin multiplets and an effective action generating the $$ \mathcal{N} $$ = 2 super-Weyl anomaly; and (iii) $\textit{κ}$-symmetry of the massless AdS superparticle.

The anti-de Sitter supergeometry revisited

Three years ago, we proposed free off-shell models for $$ \mathcal{N} $$ = 2 superconformal higher-spin multiplets in arbitrary conformally flat backgrounds, constructed conserved conformal higher-spin supercurrents for a massless hypermultiplet, and sketched the Noether procedure to generate its cubic couplings to the superconformal higher-spin multiplets. This paper is devoted to completing the Noether procedure. Specifically, we: (i) describe the unique off-shell primary extensions of the conformal higher-spin supercurrents; (ii) embed the off-shell superconformal prepotentials into primary unconstrained isotwistor multiplets; and (iii) present the unique gauge transformations of the hypermultiplet and the isotwistor prepotentials. An extension of the Noether procedure beyond the cubic level is also sketched, following the earlier $$ \mathcal{N} $$ = 1 superconformal approach developed by the authors and Ponds in 2022. Our construction is based on making use of the polar hypermultiplet within the projective-superspace setting.

Towards $$ \mathcal{N} $$ = 2 superconformal higher-spin theory

The Batalin-Vilkovisky formalism is applied to quantise the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supersymmetric generalisation of the Freedman-Townsend (FT) model, which was proposed by Lindström and Roček in 1983 in Minkowski superspace and is lifted to a supergravity background in this paper. This super FT theory describes a non-Abelian tensor multiplet and is known to be classically equivalent to a supersymmetric nonlinear sigma model. Using path integral considerations, we demonstrate that this equivalence holds at the quantum level in the sense that the quantum supercurrents in the two theories coincide. A modified Faddeev-Popov procedure is employed to quantise models for the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 tensor multiplet in harmonic superspace. The obtained results agree with those derived by applying the Batalin-Vilkovisky scheme within the harmonic superspace setting.

Covariant quantisation of tensor multiplet models

We present higher-derivative deformations of the ModMax theory which preserve both U(1) duality symmetry and Weyl invariance. In particular, we single out a class of deformations expected to describe a low-energy effective action for the ModMax theory. We also elaborate on (higher-derivative) deformations of the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 super ModMax theory.

Higher-derivative deformations of the ModMax theory

In this paper we initiate the study of six-dimensional non-linear chiral two-form gauge theories as deformations of free chiral two-form gauge theories driven by stress-tensor   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ T\overline{T} $$ -like flows. To lay the background for this study, we elaborate on the relationship between different Lagrangian formulations of duality-invariant p-form theories and corresponding   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ T\overline{T} $$ -like flows in various dimensions. To this end we propose a new formulation which (i) is a generalization of the four-dimensional construction by Ivanov, Nurmagambetov and Zupnik (INZ) and (ii) turns into the PST formulation upon integrating out an auxiliary self-dual field. We elucidate space-time covariant properties of the PST formulation by clarifying and making use of its relation to the INZ-type formulation and to a so-called “clone” construction.

United States

Christian Ferko

Italy

Kurt Lechner

Dmitri Sorokin

Interacting chiral form field theories and   <math> <mi>T</mi> <mover> <mi>T</mi> <mo>¯</mo> </mover> </math>  $$ T\overline{T} $$ -like flows in six and higher dimensions

We develop a superspace formulation for   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 3 conformal supergravity in four spacetime dimensions as a gauge theory of the superconformal group SU(2, 2|3). Upon imposing certain covariant constraints, the algebra of conformally covariant derivatives   <math> <msub> <mo>∇</mo> <mi>A</mi> </msub> <mo>=</mo> <mfenced> <msub> <mo>∇</mo> <mi>a</mi> </msub> <msubsup> <mo>∇</mo> <mi>α</mi> <mi>i</mi> </msubsup> <msubsup> <mo>∇</mo> <mi>i</mi> <mover> <mi>α</mi> <mo>⋅</mo> </mover> </msubsup> </mfenced> </math>  $$ {\nabla}_A=\left({\nabla}_a,{\nabla}_{\alpha}^i,{\nabla}_i^{\overset{\cdot }{\alpha }}\right) $$  is shown to be determined in terms of a single primary chiral spinor superfield, the super-Weyl spinor W  α  of dimension +1/2 and its conjugate. Associated with W  α  is its primary descendant B  i   j  of dimension +2, the super-Bach tensor, which determines the equation of motion for conformal supergravity. As an application of this construction, we present two different but equivalent action principles for   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 3 conformal supergravity. We describe the model for linearised   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 3 conformal supergravity in an arbitrary conformally flat background and demonstrate that it possesses U(1) duality invariance. Additionally, upon degauging certain local symmetries, our superspace geometry is shown to reduce to the U(3) superspace constructed by Howe more than four decades ago. Further degauging proves to lead to a new superspace formalism, called SU(3) superspace, which can also be used to describe   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 3 conformal supergravity. Our conformal superspace setting opens up the possibility to formulate the dynamics of the off-shell   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 3 super Yang-Mills theory coupled to conformal supergravity.

<math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 3 conformal superspace in four dimensions

We propose a new family of U(1) duality-invariant models for nonlinear   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 supersymmetric electrodynamics coupled to supergravity. It includes the Cribiori-Farakos-Tournoy-van Proeyen supergravity-matter theory for spontaneously broken local supersymmetry with a novel Fayet-Iliopoulos term without gauged R-symmetry. We present superconformal duality-invariant models, as well as new U(1) duality-invariant models for spontaneously broken local supersymmetry.