We present a new model of $\mathcal{N}{=}\,8$ mechanics with semi-dynamic supermultiplets. The model is constructed as an interaction of $\mathcal{N}{=}\,4$ supermultiplets which carry an implicit $\mathcal{N}{=}\,4$ supersymmetry. The initial field content consists of three dynamical $({\bf 1, 4, 3})$ multiplets: one bosonic and two fermionic. To ensure implicit $\mathcal{N}{=}\,4$ supersymmetry, we introduce the superfields describing three semi-dynamical $({\bf 4, 4, 0})$ multiplets: one fermionic and two bosonic. To avoid the second-order Lagrangian for fermions from the fermionic $({\bf 1, 4, 3})$ multiplets, the conversion of their velocities into new auxiliary fields is carried out. After conversion, these multiplets turn into semi-dynamical mirror $({\bf 4, 4, 0})$ multiplets without non-canonical terms in the $\mathcal{N}{=}\,8$ Lagrangian at the component level. The final $\mathcal{N}{=}\,8$ multiplet content is $({\bf 1, 8, 7}) \oplus ({\bf 8, 8, 0})$. As a first step to the ultimate $\mathcal{N}{=}\,4$ superfield formulation of the model, we remind a natural description of the standard and mirror $({\bf 4, 4, 0})$ multiplets in the framework of $\mathcal{N}{=}\,4, d{=}\,1$ biharmonic superspace.
The main ingredient for local superconformal methods is the multiplet of gauge fields: the Weyl multiplet. We construct the transformations of this multiplet for $\mathcal{N}=3$, $D = 4$. The construction is based on a supersymmetry truncation from the $\mathcal{N}=4$ Weyl multiplet, on coupling with a current multiplet, and on the implementation of a soft algebra at the nonlinear level, extending su$(2, 2|3)$. This is the first step towards a superconformal calculus for $\mathcal{N}=3$, $D = 4$.
We compute general expressions for two types of three-point functions of (semi-)short multiplets in four-dimensional $\mathcal{N}=2$ superconformal field theories. These (semi-)short multiplets are called "Schur multiplets" and play an important role in the study of associated chiral algebras. The first type of the three-point functions we compute involves two half-BPS Schur multiplets and an arbitrary Schur multiplet, while the second type involves one stress tensor multiplet and two arbitrary Schur multiplets. From these three-point functions, we read off the corresponding OPE selection rule…
We provide a tensor calculus for $n$-number of ${\mathcal N}=(1,0)$ linear multiplets in six dimensions. The coupling of linear multiplets is encoded in a function ${\mathcal F}_{IJ}$ that is subject to certain constraints. We provide various rigid and local supersymmetric models depending on the choice of the function ${\mathcal F}_{IJ}$ and provide an interesting off-diagonal superinvariant, which leads to an $R^2$ supergravity upon elimination of auxiliary fields.
We compute the superconformal partial waves of the four-point correlator $\langle JJJJ\rangle$, in which the external operator $J$ is the superconformal primary of the $4D$ $\mathcal{N}=2$ stress-tensor multiplet $\mathcal{J}$. We develop the superembedding formalism for the superconformal field theories (SCFTs) with extended supersymmetry. In $\mathcal{N}=2$ SCFTs, the three-point functions $\langle \mathcal{J}\mathcal{J}\mathcal{O}\rangle$ with general multiplet $\mathcal{O}$ contain two independent nilpotent superconformal invariants and new superconformal tensor structures, which can be ni…
Any 4d theory possessing $\mathcal{N}=1$ supersymmetry admits a so called $\mathcal{S}$-multiplet, containing the conserved energy-momentum tensor and supercurrent. When a defect is introduced into such a theory, the $\mathcal{S}$-multiplet receives contributions localised on the defect, which indicate the breaking of some translation symmetry and consequently also some supersymmetries. We call this the defect multiplet. We classify such terms corresponding to half-BPS defects which can be either three-dimensional, preserving 3d $\mathcal{N}=1$, or two-dimensional, preserving $\mathcal{N}=(0,2…
We describe in detail the derivation of a superconformal off-shell formulation of the alternative $\mathcal{N}=2$, $d=5$ ungauged supergravity of Nishino and Rajpoot, coupled to $n$ Abelian vector multiplets, using a general dilaton-Weyl multiplet. We generalize the vector multiplet coupling available in the literature and show under which assumptions that the scalar manifold reduces to the known case of SO(1,1) $\times$ SO(1,n)/SO(n). As an application of the formalism we propose generalized vector multiplet coupled higher curvature terms, whose construction we sketch briefly.
We study the vacuum structures of four-dimensional $\mathcal{N} = 1$ old minimal supergravity with higher derivative corrections. We find that $\mathcal{N} = 1$ supergravity with Riemann curvature square corrections and higher derivative matter chiral multiplets induces a non-trivial de Sitter vacuum, even in the absence of superpotentials. This vacuum generically breaks supersymmetry. We show that the auxiliary fields in the gravity and the chiral multiples play important roles to generate a potential in supersymmetric higher derivative theories.
Supersymmetric partition function of $\mathcal N=1$ superconformal theories on $S^1_β \times S^3$ is related to the superconformal index receiving contributions from short representations. The leading coefficients in the small $β$ (high "temperature") expansion of the index were previously related to the conformal anomaly coefficients of the theory. Assumptions underlying universality of these relations were tested only for simplest low-spin unitary multiplets. Here we consider examples of higher derivative non-unitary $\mathcal N=1$ multiplets that naturally appear in the context of extended …
We compute the supersymmetric partition function of an $\mathcal{N}=1$ chiral multiplet coupled to an external Abelian gauge field on complex manifolds with $T^2 \times S^2$ topology. The result is locally holomorphic in the complex structure moduli of $T^2\times S^2$. This computation illustrates in a simple example some recently obtained constraints on the parameter dependence of supersymmetric partition functions. We also devise a simple method to compute the chiral multiplet partition function on any four-manifold $\mathcal{M}_4$ preserving two supercharges of opposite chiralities, via sup…
In this short note we use the flat space limit and the relation between the 4-pt correlation function of the bottom and top components of the stress tensor multiplet to constraint its stringy corrections at strong coupling in the planar limit. Then we use this four point function to compute corrections to the anomalous dimension of double trace operators of the Lagrangian density and to compute energy-energy correlators at strong coupling.
Using superspace techniques, we compute the mixed OPE between an ${\mathcal N}=2$ stress-tensor multiplet, a chiral multiplet and a flavor current multiplet. We perform a detailed analysis of the three-point function between two of the mentioned multiplets and a third arbitrary operator. We then solve all the constraints coming from the ${\mathcal N}=2$ superconformal symmetry and from the equations of motion and/or conservation equations, and obtain all the possible operators that can appear in the expansion. This calculation is the first step towards a more general superconformal block analy…
In this work we construct the crossing symmetry equations for mixed correlators of two long and two BPS operators in 4D $\mathcal{N}=1$ SCFTs. The analysis presented here illustrates how our general group theoretic approach to long superblocks and tensor structures of superconformal algebras can be applied to give explicit ready-to-use expressions. In the case at hand, we obtain a system of four crossing symmetry equations for the relevant OPE coefficients. One of these four equations coincides with the equation found and analysed by Li, Meltzer and Stergiou by restricting to the superprimary …
We will use the covariant superform approach to develop a new density formula for $\mathcal{N}=2$ conformal supergravity which is based on a fermionic multiplet whose lowest component is a dimension-5/2 spinor. We will show that this density formula admits an embedding of the real scalar multiplet of [arXiv:1712.02309]. Upon using the embedding of the tensor multiplet into the real scalar multiplet, we will construct a new higher derivative action of the tensor multiplet in $\mathcal{N}=2$ conformal supergravity.
We consider $\mathcal{N=1}$ superconformal field theories in three-dimensions possessing a conserved current multiplet $\mathcal{F}_{ (α_{1} α_{2} α_{3} α_{4}) }$ which we refer to as the superspin-2 current multiplet. At the component level it contains a conserved spin-2 current different from the energy-momentum tensor and a conserved fermionic higher spin current of spin 5/2. Using a superspace formulation, we calculate correlation functions involving $\mathcal{F}$, focusing particularly on the three-point function $ \langle \mathcal{F} \mathcal{F} \mathcal{F} \rangle $. After imposing the …
In this paper, we study the general form of three-point functions of conserved current multiplets $S_{α(k)}= S_{(α_1 \dots α_k)}$ of arbitrary rank in four-dimensional ${\mathcal N}=1$ superconformal theory. We find that the correlation function of three such operators $\langle \bar{S}_{\dotα(k)} (z_1) S_{β(k+l)} (z_2) \bar{S}_{\dotγ(l)} (z_3) \rangle$ is fixed by the superconformal symmetry up to a single complex coefficient though the precise form of the correlator depends on the values of $k$ and $l$. In addition, we present the general structure of mixed correlators of the form $\langle \b…
In 4d $\mathcal{N}=1$ superconformal field theories (SCFTs) the R-symmetry current, the stress-energy tensor, and the supersymmetry currents are grouped into a single object, the Ferrara-Zumino multiplet. In this work we study the most general form of three-point functions involving two Ferrara-Zumino multiplets and a third generic multiplet. We solve the constraints imposed by conservation in superspace and show that non-trivial solutions can only be found if the third multiplet is R-neutral and transforms in suitable Lorentz representations. In the process we give a prescription for counting…
We calculate the first three Seeley-DeWitt coefficients for fluctuation of the massless fields of an $\mathcal{N}=2$ Einstein-Maxwell supergravity theory (EMSGT) distributed into different multiplets in $d=4$ space-time dimensions. By utilizing the Seeley-DeWitt data in the quantum entropy function formalism, we then obtain the logarithmic correction contribution of individual multiplets to the entropy of extremal Kerr-Newman family of black holes. Our results allow us to find the logarithmic entropy corrections for the extremal black holes in a fully matter coupled $\mathcal{N}=2,d=4$ EMSGT, …
We describe a new 1+1 dimensional N=(2,2) vector multiplet that naturally couples to semi chiral superfields in the sense that the gauged supercovariant derivative algebra is only consistent with imposing covariantly semi chiral superfield constraints. It has the advantages that its prepotentials shift by semi chiral superfields under gauge transformations. We also see that the multiplet relates the chiral vector multiplet with the twisted chiral vector multiplet by reducing to either multiplet under appropriate limits without being reducible in terms of the chiral and twisted chiral vector mu…
In $\mathcal{N}=1$ superconformal theories in four dimensions the two-point function of superconformal multiplets is known up to an overall constant. A superconformal multiplet contains several conformal primary operators, whose two-point function coefficients can be determined in terms of the multiplet's quantum numbers. In this paper we work out these coefficients in full generality, i.e. for superconformal multiplets that belong to any irreducible representation of the Lorentz group with arbitrary scaling dimension and R-charge. From our results we recover the known unitarity bounds, and al…
Theories with 3D $\mathcal{N}=2$ bulk supersymmetry may preserve a 2D $\mathcal{N}=(0,2)$ subalgebra when a boundary is introduced, possibly with localized degrees of freedom. We propose generalized supercurrent multiplets with bulk and boundary parts adapted to such setups. Using their structure, we comment on implications for the $\overline{Q}_+$-cohomology. As an example, we apply the developed framework to Landau-Ginzburg models. In these models, we study the role of boundary degrees of freedom and matrix factorizations. We verify our results using quantization.
We consider D3 branes in presence of an S-fold plane. The latter is a non perturbative object, arising from the combined projection of an S-duality twist and a discrete orbifold of the R-symmetry group. This construction naively gives rise to 4d $\mathcal{N}=3$ SCFTs. Nevertheless it has been observed that in some cases supersymmetry is enhanced to $\mathcal{N}=4$. In this paper we study the explicit counting of degrees of freedom arising from vector multiplets associated to strings suspended between the D3 branes probing the S-fold. We propose that, for trivial discrete torsion, there is no v…
We present a new multiparticle model of $\mathcal{N}{=}\,8$ mechanics with superconformal $F(4)$ symmetry. The system is constructed in terms of two matrix $\mathcal{N}{=}\,4$ multiplets. One of them is a bosonic matrix $({\bf 1, 4, 3})$ multiplet and another is a fermionic $({\bf 0, 4, 4})$ one. Off-diagonal bosonic components of the $({\bf 1, 4, 3})$ multiplet are chosen to take values in the flag manifold $\mathrm{U}(n)/[\mathrm{U}(1)]^n$ and they carry additional gauge symmetries. The explicit form of the $F(4)$ supersymmetry generators is found. We demonstrate that the $F(4)$ superalgebra…
We define a new dilaton Weyl multiplet of ${\mathcal{N}}=2$ conformal supergravity in four dimensions. This is constructed by reinterpreting the equations of motion of an on-shell hypermultiplet as constraints that render some of the fields of the standard Weyl multiplet composite. The independent bosonic components include four scalar fields and a triplet of gauge two-forms. The resulting, so-called, hyper-dilaton Weyl multiplet defines a $24+24$ off-shell representation of the local ${\mathcal{N}}=2$ superconformal algebra. By coupling the hyper-dilaton Weyl multiplet to an off-shell vector …
We formulate $\mathcal{N}=2$ global supersymmetric Lagrangians of self-interacting vector multiplets in terms of variant multiplets, whose non-propagating fields are replaced with gauge three-forms. Setting the three-forms on-shell results in a dynamical generation of the parameters entering the scalar potential. As an application, we study how gauge three-forms may determine the partial breaking of $\mathcal{N}=2$ supersymmetry and how they affect the low energy effective description.
We study the supersymmetry breaking patterns in four-dimensional $\mathcal{N}=2$ gauged supergravity. The model contains multiple (Abelian) vector multiplets and a single hypermultiplet which parametrizes SO$(4,1)/{\rm{SO}}(4)$ coset. We derive the expressions of two gravitino masses under {\it{general}} gaugings and prepotential based on the embedding tensor formalism, and discuss their behaviors in some concrete models. Then we confirm that in a single vector multiplet case, the partial breaking always occurs when the third derivative of the prepotential exists at the vacuum, which is consis…
We present $\mathcal{N}=2$ superconformal $\mathsf{U}(1)$ duality-invariant models for an Abelian vector multiplet coupled to conformal supergravity. In a Minkowski background, such a nonlinear theory is expected to describe (the planar part of) the low-energy effective action for the $\mathcal{N}=4$ $\mathsf{SU}(N)$ super-Yang-Mills (SYM) theory on its Coulomb branch where (i) the gauge group $\mathsf{SU}(N)$ is spontaneously broken to $\mathsf{SU}(N-1) \times \mathsf{U}(1)$; and (ii) the dynamics is captured by a single $\mathcal{N}=2$ vector multiplet associated with the $\mathsf{U}(1)$ fac…
We study the $T^2$ compactification of a class of 6d $\mathcal{N}{=}(1,0)$ theories that is Higgsable to $\mathcal{N}{=}(2,0)$ theories. We show that the resulting 4d $\mathcal{N}{=}2$ theory at the origin of the Coulomb branch and the parameter space is generically given by two superconformal matter sectors coupled by an infrared-free gauge multiplet and another conformal gauge multiplet. Our analysis utilizes the 5d theories obtained by putting the same class of 6d theories on $S^1$. Our class includes, among others, the 6d theories describing multiple M5 branes on an ALE singularity, and we…
Possible short and semi-short representations for $\N=2$ and $\N=4$ superconformal symmetry in four dimensions are discussed. For $\N=4$ the well known short supermultiplets whose lowest dimension conformal primary operators correspond to $\half$-BPS or ${1\over 4}$-BPS states and are scalar fields belonging to the $SU(4)_r$ symmetry representations $[0,p,0]$ and $[q,p,q]$ and having scale dimension $Δ=p$ and $Δ= 2q+p$ respectively are recovered. The representation content of semi-short multiplets, which arise at the unitarity threshold for long multiplets, is discussed. It is shown how, at th…
We construct SU$(2|1)$, $d=1$ supersymmetric models based on the coupling of dynamical and semi-dynamical (spin) multiplets, where the interaction term of both multiplets is defined on the generalized chiral superspace. The dynamical multiplet is defined as a chiral multiplet ${\bf (2,4,2)}$, while the semi-dynamical multiplet is associated with a multiplet ${\bf (4,4,0)}$ of the mirror type.
We revisit the study of singular points in the Coulomb branch of $\mathcal{N}=2$ SQCD in four dimensions with gauge group $SU(N)$ and odd number of flavors. For certain choices of the mass parameters these vacua are not lifted by a mass term for the chiral multiplet in the adjoint representation. By using recent results about the M5 brane description of $\mathcal{N}=1$ theories we study the resulting vacua and argue that the low-energy effective theory has a simple Lagrangian description involving a free chiral multiplet in the adjoint representation of the flavor symmetry group, a system some…
The half-BPS boundary conditions preserving $\mathcal{N}=(2,2)$ and $\mathcal{N}=(0,4)$ supersymmetry in 3d $\mathcal{N}=4$ supersymmetric gauge theories are examined. The BPS equations admit decomposition of the bulk supermultiplets into specific boundary supermultiplets of preserved supersymmetry. Nahm-like equations arise in the vector multiplet BPS boundary condition preserving $\mathcal{N}=(0,4)$ supersymmetry and Robin-type boundary conditions appear for the hypermultiplet coupled to vector multiplet when $\mathcal{N}=(2,2)$ supersymmetry is preserved. The half-BPS boundary conditions ar…
By extending the recent analysis of arXiv:2203.12203 for ${\mathcal{N}}=2$ conformal supergravity in four dimensions, we define new hyper-dilaton Weyl multiplets for five-dimensional ${\mathcal{N}}=1$, and six-dimensional ${\mathcal{N}}=(1,0)$ conformal supergravities. These are constructed by coupling the five- and six-dimensional standard Weyl multiplets to on-shell hypermultiplets and reinterpreting the systems as new multiplets of conformal supergravity. In the five-dimensional case, we also construct a new hyper-dilaton Poincaré supergravity by coupling to an off-shell vector multiplet co…
We derive the partition function of an $\mathcal N=2$ chiral multiplet on topologically twisted $H^2\times S^1$. The chiral multiplet is coupled to a background vector multiplet encoding a real mass deformation. We consider an $ H^2\times S^1$ metric containing two parameters: one is the $S^1$ radius, while the other gives a fugacity $q$ for the angular momentum on $H^2$. The computation is carried out by means of supersymmetric localization, which provides a finite answer written in terms of $q$-Pochammer symbols and multiple Zeta functions. Especially, the partition function of normalizable …
We compute $\mathcal O(1)$ corrections to the holographic Weyl anomaly for six-dimensional $\mathcal N=(1,0)$ and $(2,0)$ theories using the functional Schrödinger method that is conjectured to work for supersymmetric theories on Ricci-flat backgrounds. We show that these corrections vanish for long representations of the $\mathcal N=(1,0)$ theory, and we obtain an expression for $δ(c-a)$ for short representations with maximum spin two. We also confirm that the one-loop corrections to the $\mathcal N=(2,0)$ M5-brane theory are equal and opposite to the anomaly for the free tensor multiplet. Fi…
We study the general deformation of $\mathcal N=2$ supersymmetry transformations of a vector multiplet that forms a (constant) triplet under the $SU(2)$ R-symmetry corresponding to the magnetic dual of the triplet of the Fayet-Iliopoulos (FI) parameters. We show that in the presence of both triplets, the induced scalar potential of a vector multiplet with generic prepotential has always a minimum that realises partial breaking of $\mathcal N=2\to \mathcal N=1$ supersymmetry. We then consider the impact of the deformation in the Dirac-Born-Infeld (DBI) action where one supersymmetry is non-line…
Adinkras are graphs that encode a supersymmetric representation's transformation laws that have been reduced to one dimension, that of time. A goal of the supersymmetry ``genomics'' project is to classify all 4D, $\mathcal{N}=1$ off-shell supermultiplets in terms of their adinkras. In~previous works, the genomics project uncovered two fundamental isomer adinkras, the cis- and trans-adinkras, into which all multiplets investigated to date can be decomposed. The number of cis- and trans-adinkras describing a given multiplet define the isomer-equivalence class to which the multiplet belongs. A fu…
Surface operators are among the most important observables of the 6d $\mathcal{N} = (2,0)$ theory. Here we apply the tools of defect CFT to study local operator insertions into the 1/2-BPS plane. We first relate the 2-point function of the displacement operator to the expectation value of the bulk stress tensor and translate this relation into a constraint on the anomaly coefficients associated with the defect. Secondly, we study the defect operator expansion of the stress tensor multiplet and identify several new operators of the defect CFT. Technical results derived along the way include the…
On anomaly free 4d $\mathcal{N}$=4 and 6d (2,0) conformal supergravities and UV finiteness of Poincaré supergravities
Leinweber Institute for Theoretical Physics at Stanford（斯坦福大学莱因韦伯理论物理研究所）；Imperial College London（帝国理工学院）；ITMP（伊萨克·牛顿数学物理研究所）；Lebedev Institute（列别捷夫物理研究所）；Abdus Salam Centre for Theoretical Physics（阿卜杜斯·萨拉姆理论物理中心）
We discuss the conditions for extra supersymmetry of the N=(2,2) supersymmetric vector multiplets described in arXiv:0705.3201 [hep-th] and in arXiv:0808.1535 [hep-th]. We find (4,4) supersymmetry for the semichiral vector multiplet but not for the Large Vector Multiplet.
We discuss the signature of the anomalous breaking of the superconformal symmetry in $\mathcal{N}=1$ super Yang Mills theory, mediated by the Ferrara-Zumino hypercurrent ($\mathcal{J}$) with two vector ($\mathcal V$) supercurrents $(\mathcal{JVV})$ and its manifestation in the anomaly action, in the form of anomaly poles. This allows to investigate in a unified way both conformal and chiral anomalies. The analysis is performed in parallel to the Standard Model, for comparison. We investigate, in particular, massive deformations of the $\mathcal{N}=1$ theory and the spectral densities of the an…
We use the numerical conformal bootstrap to study six-dimensional $\mathcal{N}=(1,0)$ superconformal field theories with flavor symmetry $\mathfrak{so}_{4k}$. We present evidence that minimal $(D_k, D_k)$ conformal matter saturates the unitarity bounds for arbitrary $k$. Furthermore, using the extremal-functional method, we check that the chiral-ring relations are correctly reproduced, extract the anomalous dimensions of low-lying long superconformal multiplets, and find hints for novel OPE selection rules involving type-$\mathcal{B}$ multiplets.
We study the two-point function of the stress-tensor multiplet of $\mathcal{N}=4$ SYM in the presence of a line defect. To be more precise, we focus on the single-trace operator of conformal dimension two that sits in the $20'$ irrep of the $\mathfrak{so}(6)_\text{R}$ R-symmetry, and add a Maldacena-Wilson line to the configuration which makes the two-point function non-trivial. We use a combination of perturbation theory and defect CFT techniques to obtain results up to next-to-leading order in the coupling constant. Being a defect CFT correlator, there exist two (super)conformal block expans…
We calculate the $\mathcal{S}$-multiplets for two-dimensional Euclidean $\mathcal{N}=(0,2)$ and $\mathcal{N} = (2,2)$ superconformal field theories under the $T\overline{T}$ deformation at leading order of perturbation theory in the deformation coupling. Then, from these $\mathcal{N} = (0, 2)$ deformed multiplets, we calculate two- and three-point correlators. We show the $\mathcal{N} = (0,2)$ chiral ring's elements do not flow under the $T\overline{T}$ deformation. Specializing to integrable supersymmetric seed theories, such as $\mathcal{N} = (2,2)$ Landau-Ginzburg models, we use the thermod…
Within the framework of ${\mathcal N}=1$ anti-de Sitter (AdS) supersymmetry in four dimensions, we derive superspin projection operators (or superprojectors). For a tensor superfield $\mathfrak{V}_{α(m)\dotα (n)} := \mathfrak{V}_{(α_1...α_m) (\dot α_1...\dot α_n)}$ on AdS superspace, with $m$ and $n$ non-negative integers, the corresponding superprojector turns $\mathfrak{V}_{α(m)\dot α(n)} $ into a multiplet with the properties of a conserved conformal supercurrent. It is demonstrated that the poles of such superprojectors correspond to (partially) massless multiplets, and the associated gaug…
We present a systematic method to expand in components four dimensional superconformal multiplets. The results cover all possible $\mathcal{N} = 1$ multiplets and some cases of interest for $\mathcal{N} = 2$. As an application of the formalism we prove that certain $\mathcal{N} = 2$ spinning chiral operators (also known as "exotic" chiral primaries) do not admit a consistent three-point function with the stress tensor and therefore cannot be present in any local SCFT. This extends a previous proof in the literature which only applies to certain classes of theories. To each superdescendant we a…
There exists two distinct off-shell ${\mathcal{N}}=2$ supergravities in three dimensions. They are also referred to as ${\mathcal{N}}=(1,1)$ and ${\mathcal{N}}=(2,0)$ supergravities, and they arise from the coupling of the Weyl multiplet to a compensating scalar or vector multiplet, respectively, followed by fixing of conformal symmetries. The ${\mathcal{N}} =(p,q)$ terminology refers to the underlying anti-de Sitter superalgebras $OSp(2,p) \oplus OSp(2,q)$ with $R$-symmetry group $SO(p) \times SO(q)$. We construct off-shell invariants of these theories up to fourth order in derivatives. As an…
