The demand to know the structure of functionally independent invariants of tensor fields arises in many problems of theoretical and mathematical physics, for instance for the construction of interacting higher-order tensor field actions. In mathematical terms the problem can be formulated as follows. Given a semi-simple finite-dimensional Lie algebra g and a g-module V, one may ask about the structure of the sub-ring of g-invariants inside the ring freely generated by the module. We point out how some information about the ring of invariants may be obtained by studying an extended Lie algebra. Numerous examples are given, with particular focus on the difficult problem of classifying invariants of a self-dual 5-form in 10 dimensions.

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

Using the SO(𝒩) superspace formulation for 𝒩 -extended conformal supergravity in three dimensions, we derive all maximally supersymmetric backgrounds in the 𝒩 = 4 case. The specific feature of this choice is that the so-called super Cotton tensor XIJKL = X[IJKL], which exists for 𝒩 ≥ 4, is equivalent to the scalar X defined by XIJKL = εIJKLX. 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 + 2S) and (X – 2S) in the left and right sectors of SU(2)L × SU(2)R, respectively. Setting S = 0 leads to the so-called deformed 𝒩 = 4 Minkowski superspace M_X^.3|8 introduced thirteen years ago. We use projective-superspace techniques to construct general interacting supersymmetric field theories in M_X^.3|8 and demonstrate that they originate as massive deformations of the following two families of 𝒩 = 4 theories in standard Minkowski superspace M^.3|8 : (i) 𝒩 = 4 superconformal field theories; and (ii) 𝒩 = 4 supersymmetric gauge theories in M^.3|8 which are not superconformal but possess the R-symmetry group SU(2)L × SU(2)R. Extensions of the theories in (ii) to M_X^.3|8 necessarily contain Chern-Simons terms at the component level. We also demonstrate the generation of topologically massive 𝒩 = 4 supersymmetric gauge theories from radiative corrections in the hypermultiplet sector.

We propose an on-shell ${\mathcal N}=3$ 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 $\mathsf{SL}(2,\mathbb{C})$, 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-$1/2$ superfields: (i) the super-Weyl spinor $W_\alpha$; and (ii) a spinor isospinor $\chi_\alpha^i$.

The standard geometric description of d-dimensional anti-de Sitter (AdS) space is a quadric in ℝd−1,2 defined by (X0)2 − (X1)2 − ⋯ − (Xd−1)2 + (Xd)2 = ℓ2 = const. In this paper we provide a supersymmetric generalisation of this embedding construction in the d = 5 case. Specifically, a bi-supertwistor realisation is given for the 𝒩 -extended AdS superspace AdS^5| 8𝒩 , with 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 AdS^5| 8𝒩 which is manifestly invariant under the AdS isometry supergroup SU(2, 2| 𝒩 ) and involves two independent two-derivative terms.

We propose an 𝒩 = 3 nonlinear multiplet coupled to conformal supergravity and use it to formulate the equations of motion for 𝒩 = 3 Poincaré supergravity. These equations, which are naturally described in a new curved supergeometry with structure group SL(2, ℂ), imply that the 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 𝒩 = 3 Einstein superspace, is described in terms of two dimension-1/2 superfields: (i) the super-Weyl spinor Wα; and (ii) a spinor isospinor χ_α^i .

Gauge p-forms in diverse dimensions are ubiquitous in supergravity and string theory. This work reviews novel covariant formulations designed to generate arbitrary interacting duality-invariant or chiral (self-dual) p-form theories in d = 2p + 2 space-time dimensions. For odd p, such theories possess 𝖴(1) duality invariance and include the Born-Infeld and ModMax theories. For even p, they describe a self-interacting chiral p-form with its gauge-invariant field strength obeying a nonlinear self-duality condition. We provide a complete description of TT̅-like deformations of 𝖴(1) duality-invariant models for nonlinear electrodynamics in four dimensions and their six-dimensional counterparts – interacting chiral two-form field theories. We also elaborate on consistent flows in the spaces of duality-invariant or chiral (self-dual) p-form theories beyond six dimensions.

We provide a unified description of the three covariant superspace approaches to $$\mathscr {N}=2$$ conformal supergravity in three dimensions: (i) conformal superspace, (ii) U(2) superspace, and (iii) SU(2) superspace. Each of them can be used to formulate general supergravity-matter systems, although conformal superspace has the largest structure group and is intimately related to the superconformal tensor calculus. We review the structure of covariant projective multiplets and demonstrate how they are used to describe pure and matter-coupled supergravity, including locally superconformal off-shell sigma models. Higher-derivative invariants, topological invariants, and super-Weyl anomalies are also briefly discussed.

The Freedman-Townsend model is quantized using the Batalin-Vilkovisky approach to Lagrangian quantization of gauge theories with linearly dependent generators. Path integral arguments are then applied to demonstrate the quantum equivalence of the Freedman-Townsend model to the principal chiral σ-model.

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 TT -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 TT -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.

The superspace formalism for $$\mathscr {N} = 1$$ supergravity in four dimensions is a powerful geometric setting to engineer off-shell supergravity-matter theories, including higher-derivative couplings. This review provides a unified description of the three superspace approaches to $$\mathscr {N}=1$$ conformal supergravity: (i) conformal superspace, (ii) U(1) superspace, and (iii) the Grimm-Wess-Zumino formalism. The prepotential formulation for the latter is discussed. We briefly describe the known off-shell formulations for Poincaré and anti-de Sitter supergravity theories as conformal supergravity coupled to certain compensators. As simple applications of the formalism, we present the superfield equations of motion for various off-shell formulations for pure Poincaré and anti-de Sitter supergravity and show that every solution of these equations is also a solution of the equations of motion for conformal supergravity.

Abstract We develop a superspace formulation for $$ \mathcal{N} $$ 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 $$ {\nabla}_A=\left({\nabla}_a,{\nabla}_{\alpha}^i,{\nabla}_i^{\overset{\cdot }{\alpha }}\right) $$ ∇ A = ∇ a ∇ α i ∇ i α ⋅ is shown to be determined in terms of a single primary chiral spinor superfield, the super-Weyl spinor Wα of dimension +1/2 and its conjugate. Associated with Wα is its primary descendant Bij 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 $$ \mathcal{N} $$ N = 3 conformal supergravity. We describe the model for linearised $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ N = 3 conformal supergravity. Our conformal superspace setting opens up the possibility to formulate the dynamics of the off-shell $$ \mathcal{N} $$ N = 3 super Yang-Mills theory coupled to conformal supergravity.

There are several approaches to formulate gauge-invariant models for massive integer-spin fields in dimensions including the following: (i) in terms of symmetric tensor fields 1..., with =, -1,...,0, restricted to be double traceless for >= 4; and (ii) in terms of a quartet of traceful symmetric tensor fields 1..., of rank =, -1, -2, -3. We demonstrate that these formulations in Minkowski space are equivalent to the gauge-invariant theory for a massive integer-spin field proposed in 1989 by Pashnev. We also make use of the Klishevich-Zinoviev theory in to derive a unique generalization of the Singh-Hagen model for a massive integer-spin field in >4 dimensions.

We present higher-derivative deformations of the ModMax theory which preserve both $\mathsf{U}(1)$ 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 $\mathcal{N}=1$ super ModMax theory.

We develop a superspace formulation for 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 ∇_A=(∇_a,∇_α^i,∇_i^·α) 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 𝒩 = 3 conformal supergravity. We describe the model for linearised 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 𝒩 = 3 conformal supergravity. Our conformal superspace setting opens up the possibility to formulate the dynamics of the off-shell 𝒩 = 3 super Yang-Mills theory coupled to conformal supergravity.

The Batalin-Vilkovisky formalism is applied to quantise the 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 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.

Three years ago, we proposed free off-shell models for 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 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.

We develop a general formalism of duality rotations for N-extended superconformal gauge multiplets in conformally flat backgrounds as an extension of the approach given in arXiv:2107.02001. Additionally, we construct U(1) duality-invariant models for the N=2 superconformal gravitino multiplet recently described in arXiv:2305.16029. Each of them is automatically self-dual with respect to a superfield Legendre transformation. A method is proposed to generate such self-dual models, including a family of superconformal theories.

We elaborate on the structure of higher-spin $\mathcal{N}=2$ supercurrent multiplets in four dimensions. It is shown that associated with every conformal supercurrent $J_{\alpha(m) \dot{\alpha}(n)}$ (with $m,n$ non-negative integers) is a descendant $J^{ij}_{\alpha(m+1) \dot{\alpha}(n+1)}$ with the following properties: (a) it is a linear multiplet with respect to its $\mathsf{SU}(2)$ indices, that is $ D_\beta^{(i} J^{ jk)}_{\alpha(m+1) \dot{\alpha}(n+1) }=0$ and $ \bar D_{\dot \beta}^{(i} J^{jk)}_{ \alpha(m+1) \dot{\alpha}(n+1) }=0$; and (b) it is conserved, $\partial^{\beta \dot{\beta}} J^{ij}_{\beta \alpha(m) \dot{\beta} \dot{\alpha}(n)}=0$. Realisations of the conformal supercurrents $J_{\alpha(s) \dot{\alpha}(s)}$, with $s=0,1, \dots$, are naturally provided by a massless hypermultiplet and a vector multiplet. It turns out that such supercurrents and their linear descendants $J^{ij}_{\alpha(s+1) \dot{\alpha}(s+1)}$ do not occur in the harmonic-superspace framework recently described in arXiv:2212.14114. Making use of a massive hypermultiplet, we derive non-conformal higher-spin $\mathcal{N}=2$ supercurrent multiplets. Additionally, we derive the higher symmetries of the kinetic operators for both a massive and massless hypermultiplet. Building on this analysis, we sketch the construction of higher-derivative gauge transformations for the off-shell arctic multiplet $\Upsilon^{(1)}$, which are expected to be vital in the framework of consistent interactions between $\Upsilon^{(1)}$ and superconformal higher-spin gauge multiplets.

We study the quantum dynamics of a system of n Abelian 𝒩 = 1 vector multiplets coupled to 1/2n(n+1) chiral multiplets which parametrise the Hermitian symmetric space Sp(2n, ℝ)/U(n). In the presence of supergravity, this model is super-Weyl invariant and possesses the maximal non-compact duality group Sp(2n, ℝ) at the classical level. These symmetries should be respected by the logarithmically divergent term (the “induced action”) of the effective action obtained by integrating out the vector multiplets. In computing the effective action, one has to deal with non-minimal operators for which the known heat kernel techniques are not directly applicable, even in flat (super)space. In this paper we develop a method to compute the induced action in Minkowski superspace. The induced action is derived in closed form and has a simple structure. It is a higher-derivative superconformal sigma model on Sp(2n, ℝ)/U(n). The obtained 𝒩 = 1 results are generalised to the case of 𝒩 = 2 local supersymmetry: a system of n Abelian 𝒩 = 2 vector multiplets coupled to 𝒩 = 2 chiral multiplets XI parametrising Sp(2n, ℝ)/U(n). The induced action is shown to be proportional to ∫d^4xd^4θd^4θE𝔎(X,X) , where 𝔎(X,X) is the Kähler potential for Sp(2n, ℝ)/U(n). We also apply our method to compute DeWitt’s a2 coefficients in some non-supersymmetric theories with non-minimal operators.