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Superembedding methods for 4d N=1 SCFTs

We extend SO(4,2) covariant lightcone embedding methods of four-dimensional CFTs to N=1 superconformal field theory (SCFT). Manifest superconformal SU(2,2|1) invariance is achieved by realizing 4D superconformal space as a surface embedded in the projective superspace spanned by certain complex chiral supermatrices. Because SU(2,2|1) acts linearly on the ambient space, the constraints on correlators implied by superconformal Ward identities are automatically solved in this formalism. Applications include new, compact expressions for correlation functions containing one anti-chiral superfield and arbitrary chiral superfield insertions, and manifestly invariant expressions for the superconformal cross-ratios that parametrize the four-point function of two chiral and two anti-chiral fields. Superconformal expressions for the leading singularities in the OPE of chiral and anti-chiral operators are also given. Because of covariance, our expressions are valid in any superconformally flat background, e.g., AdS_4 or R times S^3.


I. INTRODUCTION
The dynamics of four-dimensional conformal field theory (CFT) is tightly constrained by its fifteen parameter group SO(4, 2) of spacetime symmetries. These symmetries completely fix the form of the two-point and three-point correlators of the theory, and reduce the four-point correlators to a function of two conformally invariant cross ratios. Additional constraints on correlators follow from crossing symmetry, the operator product expansion (OPE), and unitarity, which sets lower bounds on scaling dimensions of fields.
In light of the possible applications of conformal symmetry to strongly coupled models of TeV scale physics and to condensed matter systems, it becomes relevant to fully understand its implications. A central question [1] is to what extent does the symmetry algebra, together with general properties such as crossing and unitarity, determine the space of allowed theories, parametrized by the spectrum of primary operators and their OPE coefficients.
While it is not yet known if such general principles are sufficient to fix the dynamics of CFTs in spacetime dimensions greater than two, there has been recent work that employs four-dimensional conformal invariance to place non-trivial constraints on the physics. One such example was put forward in [2], which uses the conformal block structure of four-point functions to derive a sum rule that can be used to place upper bounds on the scaling dimensions of certain operators in the theory (see [3][4][5] for extensions of this approach). These bounds have consequences for model building, as they constrain the types of CFTs that are consistent with the stability of the electroweak scale against large ultraviolet radiative corrections.
Ref. [6] provides another recent example of this type of analysis. This work considers large N four-dimensional CFTs whose spectrum of primary operators contains a relatively small number of fields with scaling dimensions close to the unitarity bounds (e.g. conserved currents). The remaining primaries have dimensions that are parametrically large, so that there is a gap in the spectrum of scaling dimensions (these types of CFTs have been further analyzed by [7], which dubs them "effective CFTs"). As shown in ref. [6], it follows from conformal symmetry that the space of CFTs of this type is in one-to-one correspondence with weakly coupled theories propagating in a background anti-deSitter (AdS) space of one higher dimension. This result therefore provides a sort of converse to AdS/CFT, and begins to address the question of which type of CFTs can have (even without supersymmetry) weakly coupled gravity duals.
The results uncovered in [2,6] make heavy use of the technology of four-dimensional conformal symmetry, in particular the conformal block decomposition of four-point correlators.
If it is indeed possible to sharpen the constraints imposed by SO(4, 2) invariance on quantum field theories, it is necessary to develop more efficient methods for representing the observables of a CFT. A useful language for CFT correlation functions is provided by embedding four-dimensional Minkowski spacetime into the projective lightcone of a six-dimensional flat spacetime with signature (4, 2) metric. The group SO(4, 2) acts linearly on the ambient space coordinates, and thus CFT correlators expressed in this language automatically exhibit manifest conformal symmetry. In addition, results written in terms of the embedding space coordinates are valid not just in Minkowski spacetime, but in any other curved but conformally flat space, as different slicings through the projective lightcone correspond to conformal transformations of flat space. This approach goes as far back as ref. [8] for free fields, with applications to more general CFTs given in [9,10]. In the mathematical literature, there exsits a very similar approach to conformal transformations, known as tractor calculus, see e.g [11].
The embedding methods have more recently been used to obtain novel results for CFTs.
These include a simple closed-form expression for the conformal block expansion of fourpoint functions in spacetimes of dimension d > 2 [12] (streamlining an earlier approach by the same authors [13]), the formulation of correlators and conformal blocks [14] for symmetric traceless primary operators, applications to graviton correlators in conformally flat backgrounds [15], and to the Mellin representation of AdS/CFT amplitudes [16].
In this paper we extend the six-dimensional embedding space methods to describe N = 1 superconformal field theory (SCFT). Apart from possible applications of SCFTs to TeV supersymmetry model building, our primary motivation for focusing on such theories is simply that most of the known four-dimensional interacting exactly conformal theories are also supersymmetric. Even if there are four-dimensional non-supersymmetric exactly conformal theories, the program of [1] seems more likely to succeed in the supersymmetric case, where symmetry constraints are stronger.
After reviewing the embedding formalism for SO(4, 2) invariant field theories in Sec. II A, setting up notation for the N = 1 superconformal group SU(2, 2|1) in Sec. II B and some of its linear representations in Sec. II C, we turn to the construction of an embedding super-space in Sec. III. Our superspace is constructed in such a way that SU(2, 2|1) transformations act linearly on the coordinates. This necessitates the introduction of complex superspace coordinates (X AB ,X AB ) constructed in terms of tensor products containing the SU(2, 2|1) fundamental V A and anti-fundamentalV A . In Sec. III A we introduce the SU(2, 2|1) invariant surface that generalizes the analogous SO(4, 2) invariant structure, and show that points on this surface span the standard N = 1 superspace (x µ , θ,θ) of four-dimensional supersymmetric field theory. In Sec. IV we focus our attention on projectively defined holomorphic fields Φ(X AB ) and show that they correspond to four-dimensional N = 1 chiral superfields whose θ =θ = 0 component is an N = 1 SCFT chiral primary operator (an embedding formalism for N = 1 superconformal multiplets with manifest invariance under SO(4, 2) but not SU(2, 2|1) has been previously introduced in ref. [17]). A supersymmetric version of the tractor calculus can be found in [18], and a related superembedding formalism for free fields has appeared in ref. [19].
Some of the consequences for correlators of chiral and anti-chiral supermultiplets are given in Sec. V. There, we give manifestly SU(2, 2|1) covariant expressions for two-point, threepoint, and in some cases (with only one anti-chiral insertion) N-point correlators that are completely fixed by superconformal invariance. We also introduce an SU(2, 2|1) invariant formulation of the OPE of chirals and anti-chirals, and give explicit formulas for the leading OPE singularities (the N = 1 superconformal invariant OPE of currents has been developed, using four-dimensional language, in [20]). In particular, we recover the additive property of dimensions of operators in the chiral ring. Finally, we discuss the four-point function with two chiral and two anti-chiral insertions, including its general parameterization in terms of independent coordinate superconformal invariants, and asymptotic behavior in various OPE limits.
In Sec. VI we outline possible extensions of the framework presented here. In particular, many of the results in this paper should have natural generalizations to extended superconformal symmetry, to supersymmetry in AdS 5 , as well to the study of SCFTs in curved, conformally flat background spacetimes.

A. Lightcone methods for CFT
The conformal group SO(4, 2) realized on four-dimensional Minkowski space is generated by Poincare transformations (M µν , P µ ), special conformal transformations K µ and dilatations D. While Lorentz transformations and dilatations act linearly on the coordinates x µ , translations P µ and special conformal transformations do not. For instance, a finite special conformal transformation parametrized by a four-vector b µ sends x µ to the point Because conformal transformations act non-linearly on the Poincare coordinates x µ , the consequences of SO(4, 2) invariance for CFT correlation functions can sometimes be obscure in this representation. This motivates [8][9][10] the introduction of a set of auxiliary coordinates, which include the Minkowski coordinates x µ , on which SO(4, 2) acts linearly.
Specifically, one introduces a six-dimensional flat spacetime of signature (4, 2) with coordinates X m , m = +, µ, −. The conformal group SO(4, 2) acts linearly on these coordinates, with η mn Λ m p Λ n q = η pq , where the metric is given by For infinitesimal transformations Λ m n = δ m n + ω m n , we have δX m = ω mn X n ≡ i 2 ω pq L pq X m , where the differential operators generate the SO(4, 2) algebra. Four-dimensional conformally flat spacetime is then recovered as the set of points on the lightcone subject to identification of points under re-scalings, X m ∼ λX m for arbitrary real parameter λ.
To see how four-dimensional Minkowski space is embedded in the projective lightcone, consider a slice through the surface X 2 = 0 defined by constant X + . Introducing the coordinate x µ = X µ /X + , we have X − = −X + x 2 on the lightcone. The set of points spanned by all possible coordinate values of x µ is then Minkowski space, as can be verified by computing the SO(4, 2) transformations of x µ induced by those of X m . First, note that L µν generates a standard SO(3, 1) Lorentz transformation with parameter ω µν while generates the Poincare translation δx µ = 1 2 ω µ− . Similarly, L µ− generates special conformal translations, so that b µ = −ω µ+ /2. Finally, generates dilatations. Note that the slice with constant X + defines a good set of coordinates near the "origin" X m = (X + , X µ = 0, X − = 0). To describe points near conformal infinity x µ → ∞, it is more suitable to look at slices of constant X − , defining four-dimensional coordinates z µ = X µ /X − . These new coordinates are related to x µ by the inversion z µ = −x µ /x 2 . It is clear that, given two points on the projective lightcone X 2 = 0, there exists an SO(4, 2) transformation that sends one to the origin x µ = 0 and the other to infinity It is possible to uplift the action of SO(4, 2) not just on the coordinates, but also on the operator algebra that defines the CFT. For instance, consider an SO(4, 2) scalar operator Φ(X) defined on the projective lightcone. Under SO(4, 2) transformations, and in order for Φ(X) to be well defined projectively, it must be a homogeneous function of the coordinates for some real parameter ∆. Performing an SO(4, 2) rotation in the +− plane with ω +− = −2 ln λ, we have X m = (X + , X µ , X − ) → X ′m = (λ −1 X + , X µ , λX − ). By Eqs. (9) and (10), this implies that Φ ′ (X + , λX µ , λ 2 X − ) = λ −∆ Φ(X). Therefore, the four-dimensional operator transforms under such transformations as so that it corresponds to a CFT operator of definite scaling dimension ∆. It is also straightforward to check, by applying an SO(4, 2) transformation with only ω µ+ = 0, that L µ− · φ(0) = 0. It follows that φ(x) defined by Eq. (11) corresponds to a CFT primary field of scaling dimension ∆. This construction has been generalized [9,10] to fields in non-trivial Lorentz representations. One concludes that the spectrum of primary operators of the CFT is in one-to-one correspondence with fields defined on the projective lightcone.
One advantage of the lightcone formalism is that conformal invariants and covariants of several points have a simple expression in the six-dimensional embedding space. The invariants constructed from points X i=1...,N are generated by the scalar products X i · X j = η mn X m i X n j (we will sometimes use the notation 1·2 = X 1 ·X 2 ). Due to the scaling properties in Eq. (10), the N-point correlator can only depend on  independent invariant cross ratios, e.g.
etc. for N ≥ 4. (We have used 1 · 2 = − 1 2 X + 1 X + 2 (x 1 − x 2 ) 2 for points on the lightcone.) For N = 2, the two-point function Φ 1 (X 1 )Φ 2 (X 2 ) is a function of the invariant X 1 · X 2 . The functional form is fixed by applying a scale transformation X 1 → λX 1 which gives On the other hand, using Eq. (10) with respect to Φ 2 , implies that this equation is only consistent if c 12 = 0 whenever ∆ 1 = ∆ 2 (in unitary CFTs c 12 ≥ 0). For ∆ 1 = ∆ 2 , the coefficient depends on the normalization of the operators and can be set to unity by a suitable choice of operator basis. Finally for N = 3, similar arguments constrain the correlator to have the form It follows from this result together with the form of the two-point function (normalized to c 12 = 1) that the OPE contains a term Thus, we see how the basic properties of CFTs are recovered in the lightcone embedding approach.
In what follows we formulate the lightcone construction in SU(2, 2) rather than SO(4, 2) language because it will be useful later on. The vector X m of SO(4, 2) is equivalent to an antisymmetric tensor of SU(2, 2). If SU(2, 2) acts on a defining four-component spinor V α=1,...,4 (our spinor conventions are summarized in appendix A) by one obtains the vector of SO(4, 2) as the antisymmetric SU(2, 2) tensor X αβ = −X βα , with each index transforming as in the above equation. An explicit map between X m in SO (4,2) and X αβ transforming as an SU(2, 2) tensor can be constructed by introducing a set of where ǫ αβρσ is the invariant epsilon tensor of SU(2, 2). Then the correspondence is Useful properties of the matrices Γ mαβ are collected in appendix A. In SU(2, 2) language, the projective lightcone is the surface with X αβ = 1 2 ǫ αβρσ X ρσ . The SU(2, 2) generators acting on the coordinates are and can be related to the operator L mn defined above through The N = 1 superconformal group in four spacetime dimensions is the supergroup SU(2, 2|1) consisting of supermatrices with commuting 4 × 4 and 1 × 1 blocks U α β and z, respectively, and anticommuting entries φ α , ψ β . An object in the fundamental of SU(2, 2|1) is given by and transforms as V A → U A B V B under superconformal transformations. In order to construct a superspace we will assume later on that V α is fermionic (anticommuting) and ψ is bosonic. To formulate the condition that U A B is an element of SU(2, 2|1), note that the bosonic subgroup SU(2, 2) is defined to leave invariant the bilinear combination where Wα = (W α ) † and Aα α is the SU(2, 2) invariant metric (our conventions for Aα α are given in appendix A). Thus we introduce the invariant metric for SU(2, 2|1) and define SU(2, 2|1) by the condition where UḂȦ = (U B A ) † . This equation defines U(2, 2|1). Its subgroup SU(2, 2|1) consists of supermatrices with unit superdeterminant, To construct the superalgebra [21], write an infinitesimal transformation in the fundamental representation as with generator T A B given by In terms of T A B , Eq. (29) becomes For indices A, B = α this equation just says that T α β is a U(2, 2) generator, i.e.
For mixed indices, we obtain the reality condition Finally, the condition sdetU = 1 implies that T A B has vanishing supertrace: Redefining T α β to remove its trace, so that it becomes a generator of SU(2, 2), we end up with the expression for an SU(2, 2|1) generator acting on the fundamental representation. Defining we get component transformation rules In particular, one can define supersymmetry generators Q α andQ β such that Thus as supermatrices in the fundamental representation If in addition we define U(1) R acting on the fundamental representation by with or R(V α ) = 1/4, and R(ψ) = 1. The normalization chosen for U(1) R is an arbitrary convention. We will relate this to the standard normalization for R-charge in Sec. IV.
For SU(2, 2) a convenient basis of generators is Given the explicit matrix representation, the algebra follows. The commutator of two J's just gives the SU(2, 2) algebra, while the other relations are and We will relate this manifestly SU(2, 2) covariant form of the N = 1 superconformal generators to the more conventional (Poincare covariant) form in Sec. III A.

C. Representations and invariants
Starting from the fundamental representation V A , transforming as and its conjugateV A = A AḂ VḂ, with transformation law we can form several product representation that will be useful in our construction of the superspace below. A more complete discussion of linear SU(2, 2|1) representations can be found in refs. [22][23][24] (see [25] for a review).
From the definitions, the product Z A B = V AW B of a fundamental V A and an anti-fundamentalW B transforms as a tensor This is a reducible representation, containing the singlet where we have introduced In particular, the supertrace is given by str The product also contains the adjoint representation, the super-traceless part of Z A B , In addition, one can construct tensors X AB andX AB which transform in the same way as the products V A V B andV AV B respectively. Their transformation properties under SU(2, 2|1) are whereÛ A B is obtained from U A B by settingÛ ψ δ = −U ψ δ and keeping all other matrix It then follows that X AC λ D CX DB transforms as in Eq. (51), and due to the property strM 1 M 2 = strM 2 M 1 , the is superconformally invariant.

III. SUPERSPACE
Our goal in this section is to construct a superconformal extension of the six-dimensional projective lightcone. In order to do this, we use the equivalence between the SO(4, 2) vector X m and the SU(2, 2) anti-symmetric tensor X αβ . The latter transforms under SU(2, 2) as the product V α V β of an anticommuting SU(2, 2) spinor V α . This suggests that the supersymmetric extension of X αβ is a tensor X AB transforming as the product where V A is the object introduced in Eq. (26). The coordinate supermultiplet X AB contains the usual SU(2, 2) coordinate X αβ = −X βα , as well as its fermionic superpartner which transforms as an SU(2, 2) spinor. In addition, X AB contains the bosonic coordinate The objects V A , X AB introduced here are analogous to quantities appearing in the SU(2, 2|3) superconformal twistor construction of Ref. [26], see also Ref. [27]. It is not possible to impose an SU(2, 2|1) invariant reality condition on X AB and we are forced to introduce the conjugate, denoted byX AB , as well. It transforms as the productV AV B of two SU(2, 2|1) anti-fundamental representations, introduced in Eq. (50). The complete superspace is then the space spanned by the pair (X AB ,X AB ). We impose the identification (X AB ,X AB ) ∼ (λX AB ,λX AB ) for arbitrary λ ∈ C. The coordinates X AB andX AB are independent, although below we will identifyX AB with the conjugate of X AB . In components, this means where The objects X AB andX AB transform under SU(2, 2|1) according to Eqs. (55) and (56), respectively. It follows that there is an SU(2, 2|1) invariant inner product between points on our superspace, given by This scalar product will play an important role in constructing superconformally covariant correlation functions later on. In components, it is given by The existence of a scalar product also implies that our superspace has a natural invariant supermetric whose specific role we will not explore further in this paper.
From the transformation of V A in the previous section, or from Eqs. (55) and (56), it is possible to work out the superconformal transformation of the component coordinates Likewise, One can check that these transformations are consistent with the superalgebra in Eqs. (47) and (48). From these expressions we can also obtain expressions for the differential operators realizing the algebra on superspace. These results can be found in appendix B.

A. 4D superspace
Just as Minkowski spacetime arises as (a portion) of the six-dimensional projective lightcone X m X m = 0, four-dimensional superspace can be described as a superconformally invariant surface embedded in the superspace (X AB ,X AB ). The correct surface should be the supersymmetric extension of an SU(2, 2)×U(1) R invariant constraint such as X αβ X αβ = 0 or X αβX αβ = 0. In order to determine the correct constraints we will start with the definition of four-dimensional superspace (x µ , θ a ,θ˙a) as the coset SU(2, 2|1)/H [28], where H consists of the N = 1 super-Poincare group with generators J µν , K µ , S a ,S˙a, together with D and R. This coset consists of the set of points generated by applying all possible SU(2, 2|1) transformations to the "origin" and likewise for the complex coordinateX AB .
Explicitly, note that the isotropy group of the origin consists of SU(2, 2) transformations with ω µ− = 0 (i.e. Lorentz and special conformal transformations) and by supersymmetry transformations parametrized by spinors of the form Note that the origin is also projectively invariant under dilatations (generated by ω +− = 0) and by U(1) R transformations. Thus non-trivial solutions are parameterized by translations ω µ− = 2x µ and by supersymmetry transformations with parameter The explicit SU(2, 2|1) transformation that generates the point labeled by (x µ , θ a ,θ˙a) can be decomposed as with and It follows that where we have introduced y µ = x µ + iθσ µθ . We therefore find with so that X αβ obeys the ordinary lightcone constraint X αβ X αβ = 0. In addition, and Likewise, we obtain for the complex conjugates, withȲ m = (Y m ) † , The solutions X AB (y µ , θ),X AB (ȳ µ ,θ) written above can be obtained as points obeying a set of algebraic constraints in the embedding superspace. First of all, X AB (y µ , θ) satisfies the equations X αβ X αβ = 0 (86) and similarly for the conjugate coordinates. These constraints can be generated by applying the differential operatorsQ α to the lightcone constraint Z = X αβ X αβ = 0. In fact the constraints fit into a sixteen dimensional irreducible representation of SU(2, 2|1) transforming as the product V A V B V C V D . Thus the constraints satisfied by X AB can be expressed covariantly as [X AB X CD ] 16 = 0 and similarly for the conjugates.
It is also possible to start with the constraints in Eqs. (86)-(90) and show that they their most general solution is exactly of the form given in Eqs. (80)-(82), and likewise for the conjugates. However, this by itself is not sufficient to prove that the coset space SU(2, 2|1)/H is given by the equations [X AB X CD ] 16 = [X ABX CD ] 16 = 0 since in order to send X AB andX AB to the origin requires the action of the complexified superconformal group rather than SU(2, 2|1) (in particular, sending X m = 1 2 X αβ (Γ m ) αβ to X m = (X + , 0, 0) requires a complex Poincare translation). Consequently, additional constraints that relate X AB andX AB are needed.
The correct additional constraint turns out to be that [X ABX CD ] 24 = 0, or explicitly transforms in the adjoint representation of SU(2, 2|1). Expanding in components, the constraints are is exactly the desired four-dimensional coset superspace 1 parameterized by the coordinates (x µ = (y +ȳ) µ /2, θ,θ), see e.g. [31]. 1 The need for the constraints [X AB X CD ] 16 = [X ABX CD ]1 6 = 0 and [X ABX CD ] 24 = 0 was first pointed out in [30]. Note that any further quadratic constraints, namely those involving [X AB X CD ] 46 and [X AB X CD ] 59 or [X ABX CD ] 96 are too constraining and do not yield the correct four-dimensional space.
Given the transformation properties of X AB worked out above, it is now straightforward to determine how the parameters (y µ , θ a ) transform under superconformal transformations.
The result is that a transformation with parameter induces the transformation which exactly coincides with the supersymmetry transformations of four-dimensional chiral N = 1 superspace (y µ , θ a ) (see e.g. ref. [31]). This establishes the equivalence between points X AB subject to the above covariant constraints and four-dimensional chiral superspace, at least near the origin. The full superspace, obtained by including the conjugate variablē X AB , is labeled by (x µ = (y +ȳ) µ /2, θ,θ). Analogous results have been obtained using the super-twistor formalism, see Refs. [27,29]. The relation between our fermionic generator Q α and the more conventional Poincare supercharge Q a and special-superconformal supercharge S a is then We conclude this section by noting that the point X AB (y, θ) defined by the action of U B A (x, θ,θ) on the "origin" can also be reached starting at "infinity", by applying a combination of special conformal and special superconformal transformations.
Such transformations yield a point that is projectively equivalent to X AB (x, θ,θ). This result indicates in particular that there exists a superconformal transformation that can simultaneously send any two points to the origin and to infinity, respectively.

IV. THE CHIRAL SECTOR OF SCFTS
As in the case of SO(4, 2) invariant field theory, the operator content of a four-dimensional SCFT has a covariant description in terms of fields living on the projective "superlightcone" structure introduced above. In this section we formulate the correspondence between superfields on the superlightcone and superconformal multiplets in four dimensions.
We start with superfields Φ(X,X) satisfying for arbitrary complex parameter λ. The superfield Φ can be assigned to any linear representation of SU(2, 2|1), although in this paper we will focus for simplicity on scalars with X ′ andX ′ given by the RHS of Eqs. (55) and (56). More general tensors, for instance the important case of the supercurrent multiplet, will be considered elsewhere.
The scalar superfield Φ(X,X) is reducible, as the holomorphy condition ∂Φ/∂X = 0 is SU(2, 2|1) invariant. For such a holomorphic field, the condition in Eq. (102) now becomes simply Φ(λX) = λ −∆ Φ(X). We will now focus on holomorphic fields and show that, upon projection to four dimensions, they correspond to SCFT multiplets whose lowest component is a chiral primary operator. Expanding in powers of θ, Φ(X) has the component field where the component fields are functions of (X αβ , ϕ). It follows from the scaling law Φ(λX AB ) = λ −∆ Φ(X AB ) that the component fields satisfy A(λX αβ , λϕ) = λ −∆ A(X αβ , ϕ), ψ(λX αβ , λϕ) = λ −∆−1 ψ(X αβ , ϕ), etc. We may therefore define a four-dimensional superfield, dependent only on the chiral coordinates (y µ , θ a ), by Restricted to the four-dimensional superspace, terms with more than two powers of θ α vanish, and using θ α θ β = −iθ 2 X + X αβ from Eq. (88) we obtain the explicit relations ψ a (y) = −(X + ) ∆+1 ψ α=a (X αβ , 0) + iy µ (σ µ )˙b a ψ α=ḃ (X αβ , 0) , The transformation of Φ(y µ , θ a ) under superconformal symmetries is fixed by the defi- Then δΦ(y, θ) ≡ Φ ′ (y, θ) − Φ(y, θ) is given by which implies that the components transform as In particular we note that at the origin δA(0) = 0, so that A(0) is annihilated by the special superconformal generators S a ,S˙a. This establishes that Φ(y, θ) is indeed a chiral field Thus, we recover the proportionality between the R-charge of the chiral multiplet and its scaling dimension ∆ at a superconformal fixed point [23]. To make contact with the standard normalization of R-charge in four-dimensional SCFTs, we may define R 4d = 4 3 R, in which case we have R 4d (θ a ) = 1 and R 4d (Φ) = 2 3 ∆. The R-charges of the component fields The property that R-charge is proportional to ∆ in the chiral sector follows from unitarity of the irreducible representations of SU(2, 2|1), and in superconformal field theories is reflected in the fact that the R-current lives in the same multiplet as the energy-momentum tensor. In the embedding space approach, this relation is a natural consequence of the scaling property Φ(λX) = λ −∆ Φ(X) of holomorphic superfields.

V. APPLICATIONS TO CORRELATORS AND TO THE OPE
Having established that the embedding space holomorphic operator Φ(X) gives a covariant representation of the chiral multiplet of four-dimensional N = 1 SCFTs, we can now determine the consequences of the formalism for SCFT correlation functions.
A correlator involving arbitrary insertions of holomorphic and anti-holomorphic fields is a function of SU(2, 2|1) invariants constructed from the supercoordinates. It follows from the analysis of SU(2, 2|1) Casimir invariants in Ref. [32] that an (over) complete set of invariants is given by supertraces of products of coordinate supermatrices. We will sometimes employ the notation 1234 · · · = str(X 1 λX 2 X 3 λX 4 · · · ) (115) for such supertraces.
For generic values of the scaling dimensions, correlators that only contain holomorphic field insertions must vanish, This follows because tensor products of any number of holomorphic supercoordinates X AB do not contain SU(2, 2|1) singlets. We note however, that for certain cases, non-zero results for purely chiral 2-point and 3-point [33][34][35][36][37] and 4-point [38,39] functions exist in the literature. While the formalism presented here does not appear to recover those cases, we note that the chiral 2-point function, which is only non-zero for dimension ∆ = 3/2 is purely local (i.e. delta function) and thus not relevant to long distance physics. The results for the 3-point function in [33][34][35][36][37] and the 4-point function in [38,39] cannot be expressed in a way that simultaneously exhibits manifest holomorphy or superconformal invariance, which is perhaps why they are not present in our formalism 2 .
We now consider correlators with both holomorphic and anti-holomorphic insertions. The simplest object is the two-point function Φ 1 (X 1 )Φ 2 (X 2 ) . In principle, it is a function of an 2 In addition, the results of [33][34][35][36][37] on purely chiral 3-and 4-point functions [38,39] apply only to the case in which the dimensions ∆ i of fields in the correlators obey the constraint i ∆ i = 3. However, in a unitary CFT, in which dimensions are bounded by ∆ ≥ 1, these results can only apply to free fields, for which ∆ = 1. In that case, these correlators must vanish identically despite being allowed by symmetry. Indeed, the vanishing of the three-point function has been checked explicitly within the context of a specific model in ref. [40].
infinite number of SU(2, 2|1) invariants of the form 1212 · · · . However, all these invariants reduce to simple powers of the basic invariant 12 = X 1 ·X 2 . This is most easily seen by applying a superconformal transformation that simultaneously sends X 1 to the origin andX 2 to conformal infinity. In that frame, the supertraces reduce to ordinary SU(2, 2) invariant traces of products of the matrix (X 1X2 ) α β , or equivalently, to the SO(4, 2) invariant dot product η mn X 1 mX n 2 . Using the scaling property Φ 1 (λX) = λ −∆ 1 Φ 1 (X), the only consistent possibility is therefore where, for the same reasons as in the ordinary SO(4, 2) case, c 12 = 0 unless ∆ 1 = ∆ 2 . For X 1 andX 2 on the superlightcone, so the four-dimensional two-point function is given by where ∆ = ∆ 1 = ∆ 2 . Expanding both sides in powers of θ 1 ,θ 2 gives component field twopoint functions. Note that for ∆ = 1 this is just the superpropagator of a massless chiral field. For general ∆ this result agrees with previous results [33,34,36,37].
Similar arguments can be applied to the more general case of correlators involving N holomorphic fields Φ i (X i ) of dimension ∆ i and one anti-holomorphic operatorΦ(X) of dimension ∆. The correlator is a function of invariant supertraces of the form, e.g., Working in a frame in which X is sent to the origin, one sees that the supertraces reduce to ordinary SU(2, 2) traces involving the matrices (X iX ) α β . Such traces are expressible in terms of SO(4, 2) invariant scalar products of the vectors X m i ,X m among each other. BecauseX mX m = 0, it follows that the invariants such as those in Eq. (120) are reducible to products of the basic SU(2, 2|1) invariants X i ·X, not just in the frame with X at the origin, but in an arbitrary frame. Using the scaling properties of the holomorphic fields Φ i , the only possible function of the invariants X i ·X is Finally, using the scaling behavior ofΦ(X), it follows that for non-zero c 12...N , the dimensions must obey the constraint ∆ = i ∆ i , ensuring R-charge neutrality of the correlator. For the special case of N = 2 holomorphic insertions, Eq. (121) is consistent with previous results obtained in refs. [33,34,36,37], but the expression for arbitrary N appears to be new.
The correlator of Eq. (121) has implications for the OPE of chiral operators. Multiplying the operator product Φ 1 (X 1 → X 2 )Φ(X 2 ) by an operatorΦ(X) of dimension ∆, and taking vacuum expectation values gives, using Eq. (117) and Eq. (121) with N = 2, where less singular terms have been omitted. In this equation, c 12,Φ is the numerical coefficient in the 3-point function Φ 1 (X 1 )Φ 2 (X 2 )Φ(X) , which is non-zero only for ∆ = ∆ 1 + ∆ 2 (we have chosen an operator basis in which the coefficients of the two-point functions are normalized to unity). We have recovered the fact that the OPE of two chiral fields must be non-singular. In particular, composite operators can be defined multiplicatively, without the need to renormalize UV divergences, and the dimension of the composite operator is the sum of the dimensions of its constituent chiral fields. On the other hand, the OPE of chiral and anti-chiral fields can have short distance singularities. If ∆ 1 > ∆ 2 there is a singularity of the form where Φ(X), if it exists, is an operator of dimension ∆ 1 − ∆ 2 . Meanwhile for ∆ 2 > ∆ 1 , the leading singularity is instead of the form where now the dimension of Φ(X) is ∆ 2 − ∆ 1 . For ∆ 1 = ∆ 2 , the only consistent possibility is that Φ is the identity operator. A more complete description of the subleading terms appearing in the OPE of chiral operators in N = 1 SCFTs, using four-dimensional language, has been developed in [4].
Coming back to the supersymmetric SU(2, 2|1) case, the four-point function is in principle a generic function of an infinite number of supertrace invariants such as that in Eq. (115). By going to the gauge in which X 1 is at the origin and X 2 is at infinity, these traces reduce to ordinary traces of the four matrices (X 1X3 ) α β , (X 1X4 ) α β , (X 2X3 ) α β , and (X 3X4 ) α β . As before, such traces can be expressed as linear combinations of scalar products of the SO(4, 2) vectors X m i=1,2 ,X m i=3,4 . The scalar products involving one X and oneX are simply the SU(2, 2|1) invariant scalar products X i=1,2 ·X j=3,4 expressed in our preferred coordinate system. The traces can also contain factors of the form (1 · 2)(3 ·4) (by R-invariance, and by the lightcone constraint Setting θ i=1,2 =θ j=3,4 = 0, one sees that u and v become equivalent to the SO(4, 2) invariant cross ratios that parametrize the four-point correlator of a generic CFT.
Given these results, the four-point function for the case of SU(2, 2|1) invariance is then of the form where consistency requires ∆ 1 + ∆ 2 = ∆ 3 + ∆ 4 . This expression is not unique as one can always re-define f (u, v) by a power of u, which changes the exponents. The function f (u, v) cannot be fixed by symmetry considerations, although its behavior near certain points is fixed by the OPE. For example in the limit X 1 → X 2 , corresponding to (u, v) → (1, 1/2), the The limit for ∆ 3 < ∆ 1 . Thus, In the limit X 1 → X 4 , corresponding to u, v → ∞, the asymptotic behavior is we have determined the number of independent superconformal cross ratios that parametrize the dynamics.
We hope that the new simplified kinematics introduced in this paper will lead to new results in superconformal field theory. Besides the natural extension to non-holomorphic operators and to higher-rank tensor fields, the formalism presented here could be developed in several directions: • Superconformal correlators in curved four-dimensional spaces: The SO(4, 2) invariant projective lightcone describes not only four-dimensional Minkowski space, but any curved spacetime that is conformally flat. For instance the intersection of the lightcone with a slice of constant 2X 6 = X + − X − describes AdS 4 , with a slice of constant 2X 4 = X + + X − describes dS 4 , and with a surface of constant (X 6 ) 2 + (X 0 ) 2 describes the space S 1 × S 3 . CFT observables written down in terms of the six-dimensional coordinates X m (restricted to the lightcone) then automatically describe not just Minkowski space physics but also physics in these other spacetimes (after applying suitable boundary conditions and/or iǫ prescription). Likewise, the superembedding formalism introduced in this paper can be used to study superconformal dynamics in curved spacetime (for some previous work on the case of AdS, which acts as an infrared cutoff [42] that preserves supersymmetry, see for instance [43]).
• Extended superconformal symmetry: While we have focused on the case of N = 1 superconformal invariance in this paper, it should be possible to extended this framework to study SCFTs with more supersymmetry. The starting point for this would be spinors of the extended superconformal group SU (2, 2|N ), and supercoordinates X AB ∼ V A V B containing the antisymmetric bosonic coordinates X αβ , the symmetric bosonic coordinates X IJ , and the fermionic coordinates θ I α . In fact, many of the results presented here already hold if the N = 1 coordinates introduced in Sec. III are naively replaced with these extended supercoordinates, although the expressions that result upon projection to four dimensions will obviously differ.
With these definitions the correspondence between the SO(4, 2) vector X m and the SU(2, 2) anti-symmetric bi-spinor X αβ is and The inner product on vectors is where SO(4, 2) vector indices are raised and lowered with the metric η mn introduced in the main text and its inverse η mn .
Note finally that the differential operators commute with the full set of superconformal generators. They are supersymmetric generalizations of the scaling operator X m ∂/∂X m that plays a role in the projective light cone formulation of SO(4, 2) invariant theories.