We propose generalised $\mathcal{N}=1$ superconformal higher-spin (SCHS) gauge multiplets of depth $t$, $Υ_{α(n)\dotα(m)}^{(t)}$, with $n\geq m \geq 1$. At the component level, for $t>2$ they contain generalised conformal higher-spin (CHS) gauge fields with depths $t-1$, $t$ and $t+1$. The supermultiplets with $t=1$ and $t=2$ include both ordinary and generalised CHS gauge fields. Super-Weyl and gauge invariant actions describing the dynamics of $Υ_{α(n)\dotα(m)}^{(t)}$ on conformally-flat superspace backgrounds are then derived. For the case $n=m=t=1$, corresponding to the maximal-depth conformal graviton supermultiplet, we extend this action to Bach-flat backgrounds. Models for superconformal non-gauge multiplets, which are expected to play an important role in the Bach-flat completions of the models for $Υ^{(t)}_{α(n)\dotα(m)}$, are also provided. Finally we show that, on Bach-flat backgrounds, requiring gauge and Weyl invariance does not always determine a model for a CHS field uniquely.
We systematically analyze the operator content of unitary superconformal multiplets in $d > 3$ spacetime dimensions. We present a simple, general, and efficient algorithm that generates all of these multiplets by correctly eliminating possible null states. The algorithm is conjectural, but passes a vast web of consistency checks. We apply it to tabulate a large variety of superconformal multiplets. In particular, we classify and construct all multiplets that contain conserved currents or free fields, which play an important role in superconformal field theories (SCFTs). Some currents that are …
Within the framework of six-dimensional ${\cal N}=(1,0)$ conformal supergravity, we introduce new off-shell multiplets ${\cal O}{}^{*}(n)$, where $n=3,4,\dots,$ and use them to construct higher-rank extensions of the linear multiplet action. The ${\cal O}{}^{*}(n)$ multiplets may be viewed as being dual to well-known superconformal ${\cal O}(n)$ multiplets. We provide prepotential formulations for the ${\cal O}(n)$ and ${\cal O}{}^{*}(n)$ multiplets coupled to conformal supergravity. For every ${\cal O}{}^{*}(n)$ multiplet, we construct a higher derivative invariant which is superconformal on …
We study 2d N=4 superconformal field theories, focusing on its application on numerical bootstrap study. We derive the superconformal block by utilizing the global part of the super Virasoro algebra and set up the crossing equations for the non-BPS long-multiplet 4-point function. Along the way, we build global N=4 superconformal short and long multiplets, and compute all possible 2,3-point functions of long-multiplets that are needed to construct the superconformal blocks and the crossing equations. Since we consider a long-multiplet 4-point function, the number of crossing equations is huge …
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…
As an extension of the recent construction of generalised Fayet-Iliopoulos terms in ${\cal N}=1$ supergravity given in [1], we present self-interactions for a vector multiplet coupled to conformal supergravity. They are used to construct new models for spontaneously broken local supersymmetry.
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 explicitly construct and list all unitary superconformal multiplets, along with their index contributions, in five and six dimensions. From this data, we uncover various unifying themes in the representation theory of five- and six-dimensional superconformal field theories. At the same time, we provide a detailed argument for the complete classification of unitary irreducible representations in five dimensions using a combination of physical and mathematical techniques.
We study the short multiplet spectrum in 4d $\mathcal{N}=2$ superconformal theories of low rank using the full superconformal indices and the selection rules from the superconformal representation theory. We find a universal expression for the leading terms for the superconformal index of rank one $H_0, H_1, H_2, D_4, E_6, E_7$ theories. From this result, we argue that certain short multiplets appear in the operator product expansions involving stress-tensor, conserved current, and Coulomb branch operator vanish. We also apply the same procedure to 5d superconformal theories and find that $E_1…
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 …
On compactified harmonic/projective superspace, 5D superconformal theories, and all that
在超扭量方法中，我们分析了4D N = 2紧致调和/射影超空间的超共形结构。在5D超共形对称的情况下，我们推导出超共形Killing矢量和相关的积木出现在初级超场的变换律。各种离壳超共形多重态在5D调和超空间和射影超空间中都有出现，包括所谓的热带（向量）多重态和极（超）多重态。超共形作用的家庭被描述在5D调和和射影超空间设置。我们还提出了规范中心荷的5D超共形理论的例子。
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 $\mathcal{N}=1$ 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…
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…
We carry out a detailed superspace analysis of the OPE of two N=2 stress-tensor multiplets. Knowledge of the multiplets appearing in the expansion, together with the two-dimensional chiral algebra description of N=2 SCFTs, imply an analytic bound on the central charge c. This bound is valid for any N=2 SCFT regardless of its matter content and flavor symmetries, and is saturated by the simplest Argyres-Douglas fixed point. We also present a partial conformal block analysis for the scalar superconformal primary of the multiplet.
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 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 …
We compute the superconformal characters of various short multiplets in 4d N=2 superconformal algebra, from which selection rules for operator products are obtained. Combining with the superconformal index, we show that a particular short multiplet appearing in the n-fold product of stress-tensor multiplet is absent in the $(A_1, A_{2n})$ Argyres-Douglas (AD) theory. This implies that the operator product expansion (OPE) coefficients involving this multiplet vanish whenever the central charge $c$ is identical to that of the AD theory. Similarly, by considering the n-th power of the current mul…
We propose general relations between the conformal anomaly and the chiral (R-symmetry and gravitational) anomaly coefficients in 6d (1,0) superconformal theories. The suggested expressions for the three type B conformal anomaly c-coefficients complement the expression for the type A anomaly a-coefficient found in arXiv:1506.03807. We check them on several examples -- the standard (1,0) hyper and tensor multiplets as well as some higher derivative short multiplets containing vector fields that generalize the superconformal 6d vector multiplet discussed in arXiv:1506.08727. We also consider a fa…
We find and classify the simplest ${\cal N}=2$ SUSY multiplets on AdS$_4$ which contain partially massless fields. We do this by studying representations of the ${\cal N}=2$, $d=3$ superconformal algebra of the boundary, including new shortening conditions that arise in the non-unitary regime. Unlike the ${\cal N}=1$ case, the simplest ${\cal N}=2$ multiplet containing a partially massless spin-2 is short, containing several exotic fields. More generally, we argue that ${\cal N}=2$ supersymmetry allows for short multiplets that contain partially massless spin-$s$ particles of depth $t=s-2$.
Applications of the bootstrap program to superconformal field theories promise unique new insights into their landscape and could even lead to the discovery of new models. Most existing results of the superconformal bootstrap were obtained form correlation functions of very special fields in short (BPS) representations of the superconformal algebra. Our main goal is to initiate a superconformal bootstrap for long multiplets, one that exploits all constraints from superprimaries and their descendants. To this end, we work out the Casimir equations for four-point correlators of long multiplets o…
We assume the existence of a background vector field that enables us to make an ansatz for the superconformal transformations for the non-Abelian 6d $(1,0)$ tensor multiplet. Closure of supersymmetry on generators of the conformal algebra, requires that the vector field is Abelian, has scaling dimension minus one and that the supersymmetry parameter as well as all the fields in the tensor multiplet have vanishing Lie derivatives along this vector field. We couple the tensor multiplet to a hypermultiplet and obtain superconformal transformations that we close off-shell.
We study global subalgebras of superconformal algebras in two dimensions and their unitary representations. Global superconformal multiplets are decomposed into conformal multiplets using Racah-Speiser algorithm, revealing many essential aspects of superconformal theories such as stress-energy tensor, conserved current, supersymmetric deformation and supersymmetry enhancement. Character formulae for the representations are presented. We further find a collection of conserved charges that are $k$-forms under the R-symmetry, which must be part of the super Virasoro algebra with $\mathcal{N} \geq…
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 formulate off-shell N=1 superconformal higher spin multiplets in four spacetime dimensions and briefly discuss their coupling to conformal supergravity. As an example, we explicitly work out the coupling of the superconformal gravitino multiplet to conformal supergravity. The corresponding action is super-Weyl invariant for arbitrary supergravity backgrounds. However, it is gauge invariant only if the supersymmetric Bach tensor vanishes. This is similar to linearised conformal supergravity in curved background.
We extend the Mellin space techniques of [1] for computing holographic four-point correlation functions in maximally superconformal theories to theories with only eight Poincaré supercharges. The one-half BPS operators in these correlators are taken to be the superconformal primary in the $\mathcal{D}[k]$ multiplet (with $k=2$ corresponding to the flavor current multiplet), and transform in the adjoint representation of a flavor group $G$. Because of the smaller R-symmetry group $SU(2)$, each individual superconformal Ward identity is less powerful. On the other hand, the constraining power is…
We present a comprehensive analysis of supersymmetry anomalies in the free and massless Wess-Zumino (WZ) model in perturbation theory. At the classical level the model possesses ${\mathcal N}=1$ superconformal symmetry, which is partially broken by quantum anomalies. The form of the anomalies and the part of the symmetry they break depend on the multiplet of conserved currents used. It was previously shown that the R-symmetry anomaly of the conformal current multiplet induces an anomaly in Q-supersymmetry, which appears first in 4-point functions. Here we confirm this result by an explicit 1-l…
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 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…
We classify possible supersymmetry-preserving relevant, marginal, and irrelevant deformations of unitary superconformal theories in $d \geq 3$ dimensions. Our method only relies on symmetries and unitarity. Hence, the results are model independent and do not require a Lagrangian description. Two unifying themes emerge: first, many theories admit deformations that reside in multiplets together with conserved currents. Such deformations can lead to modifications of the supersymmetry algebra by central and non-central charges. Second, many theories with a sufficient amount of supersymmetry do not…
The problem of classifying all short multiplets of superconformal algebras still seems to be an open question. A generic short multiplet is non-unitary, which nevertheless is of interest in various contexts. Even if one is interested in unitarity theories only, non-unitary short multiplets are of use in the analysis of (super)conformal blocks. The classification problem is mathematically formulated in terms of the representation theory of parabolic Verma modules, whose theory is known to be more challenging than that of more standard Verma modules associated with the Borel subalgebra. We comme…
