${\cal N}=(1,0)$ Anomaly Multiplet Relations in Six Dimensions

We consider conformal and 't Hooft anomalies in six-dimensional ${\cal N}=(1,0)$ superconformal field theories, focusing on those conformal anomalies that determine the two- and three-point functions of conserved flavor and $SU(2)_R$ currents, as well as stress tensors. By analyzing these correlators in superspace, we explain why the number of independent conformal anomalies is reduced in supersymmetric theories. For instance, non-supersymmetric CFTs in six dimensions have three independent conformal $c$-anomalies, which determine the stress-tensor two- and three-point functions, but in superconformal theories the three $c$-anomalies are subject to a linear constraint. We also describe anomaly multiplet relations, which express the conformal anomalies of a superconformal theory in terms of its 't Hooft anomalies. Following earlier work on the conformal $a$-anomaly, we argue for these relations by considering the supersymmetric dilaton effective action on the tensor branch of such a theory. We illustrate the utility of these anomaly multiplet relations by presenting exact results for conformal anomalies, and hence current and stress-tensor correlators, in several interacting examples.

Standard conformal anomalies can also be interpreted directly in terms of the underlying CFT, without background fields. Then T µ µ vanishes as an operator, and hence has trivial correlation functions at separated points, but it has non-vanishing contact terms dictated by the anomalies. Such contact terms are ultimately related to stress-tensor correlators at separated points, and hence they provide meaningful and useful information about the CFT.
In this paper we focus on conformal anomalies of CFTs in d = 6 dimensions. Coupling such a theory to a curved background metric leads to four possible anomalies, whose coefficients are conventionally called a and c 1 , c 2 , c 3 [1][2][3][4], (1.1) Here E is the Euler density in six dimensions, which is given in terms of the Riemann curvature two-form R ab as E ∼ ε a 1 a 2 a 3 a 4 a 5 a 6 R a 1 a 2 R a 3 a 4 R a 5 a 6 , (1.2) while the I i are expressions constructed out of the Weyl curvature tensor (for precise expressions see e.g. [4]), (1. 3) In terms of correlation functions at separated points, the coefficient c 3 determines the twopoint function of T µν , while c 1 , c 2 , c 3 fix the three-point function of T µν . The anomaly coefficient a first arises in a four-point function of stress tensors. We fix a convenient convention for SCFTs in d = 6 such that a single, free N = (2, 0) tensor multiplet has a = c 1 = c 2 = c 3 = 1. Table 1 lists the conformal anomalies of some six-dimensional CFTs in these conventions.
In CFTs with continuous flavor symmetries, there are additional conformal anomalies.
To lighten notation, we focus on a single factor G F of the full flavor symmetry; here G F could be either U(1) or simple. In the presence of background gauge fields (with field strength F µν ) that source the associated conserved flavor current J F µ , these flavor conformal anomalies take the following form, Here we omitted several additional terms that are fixed by conformal symmetry; we also dropped total derivative terms. We conclude that every U(1) or simple factor G F of the (2, 0) Theory with algebra g 16 7 h ∨ g d g + r g 4h ∨ g d g + r g 4h ∨ g d g + r g 4h ∨ g d g + r g Table 1: Conformal anomalies of some d = 6 CFTs. The first three lines are the minimal free field theories in six dimensions. The two subsequent lines are the free N = (1, 0) SCFTs. The final line summarizes the interacting N = (2, 0) SCFTs. These are labelled by an ADE Lie algebra g, and their conformal anomalies are determined in terms of the rank r g , the dimension d g , and the dual coxeter number h ∨ g of g. The large-N behavior of these conformal anomalies was established in [5]. The exact formula for a was computed in [6]. The exact formula for the c-anomalies of N = (2, 0) theories was conjectured in [7] and derived in [8].
flavor symmetry gives rise to three conformal anomaly coefficients: τ F 1 , τ F 2 , and ρ F . Note that the anomaly controlled by ρ F , which is cubic in the field strength F µν , is proportional to the structure constants f abc of G F and hence vanishes for abelian flavor symmetries.
As we will review below, these conformal anomalies encode basic data about correlation functions of conserved currents in the CFT at separated points and in the absence of background fields. Specifically, τ

Supersymmetry Constraints on Conformal Anomalies
In general, non-supersymmetric CFTs, the three c i anomalies are independent. Below we will prove that N = (1, 0) SCFTs only have two independent c i anomalies. In such theories, the three c i coefficients are related by a universal, linear relation dictated by supersymmetry, (1.5) As we will see, one consequence of this relation is that the two distinct unitary N = (1, 0) free field theories, i.e. the free hypermultiplet and the free tensor multiplet, span the space of stress-tensor supermultiplet three-point functions.
The relation (1.5) was conjectured in [9], where it was noted that it is satisfied by all free-field and holographic examples, and that it is analogous to a known relation obeyed by four-dimensional SCFTs in the context of conformal collider physics [10] (see below). A linear relation between the c i anomalies is also suggested by the more recent observation [11] that minimal conformal supergravity in d = 6 appears to only admit two independent candidate invariants that can supersymmetrize the I i invariants in (1.1) and (1.3).
In section 3 we establish (1.5) via a superspace analysis. We show that the superspace three-point function of stress-tensor supermultiplets T (x i , θ i ) takes the general form which depends on three real constants C T T T 1,2,3 that are linearly related to the c i anomalies. (Here Z = (X, Θ) is a superspace variable that is constructed using the three superspace coordinates (x i , θ i ). ) We then argue that the conservation equation obeyed by the stresstensor multiplet imposes one linear constraint on the coefficients C T T T 1,2,3 , which in turn leads to the relation (1.5) among the c i .
We also use superspace to analyze those supermultiplet three-point functions that can be obtained from (1.6) by replacing one, two, or all three stress-tensor multiplets by conserved flavor-current multiplets. In every case supersymmetry imposes one additional linear relation on the coefficients of the corresponding non-supersymmetric correlators. It follows that some of the conformal anomalies related to flavor symmetries, which were defined in (1.4), vanish in all N = (1, 0) SCFTs, To streamline the notation, we will often write for the unique non-vanishing conformal anomaly coefficient associated with the flavor symmetry G F . The relations above imply that τ F governs both the J F J F two-point function, as well as the T J F J F and J F J F J F three-point functions.
In terms of conformal collider observables [10], the fact that τ F 2 = 0 implies that the energy flux E( n) in a state created by the flavor current is independent of its polarization.
Similarly, the fact that ρ F = 0 implies that the charge flux Q( n) in such a state is also independent of the polarization. These properties of flavor currents in six-dimensional SCFTs are direct analogues of relations that hold for SCFTs in four dimensions [10].
Every N = (1, 0) SCFT has an SU(2) R symmetry. The associated conserved current J R µ resides in the stress-tensor supermultiplet, and consequently the conformal anomaly coefficients τ R 1,2 and ρ R it gives rise to need not satisfy the relations (1.7) that hold for flavor currents. Adapting the arguments outlined above to this case, we find that the conformal anomaly coefficients associated with the SU(2) R symmetry can be expressed using two linearly independent c i anomaly coefficients (with the third one given by (1.5)), Let us consider the special case of N = (2, 0) theories, which have an Sp(4) R symmetry. When viewed as N = (1, 0) theories, it is natural to focus on the maximal sub- Here SU(2) R and SU(2) F are R-and flavor symmetries that are distinguished by the choice of N = (1, 0) subalgebra, but they are exchanged by a Weyl reflection inside the full Sp(4) R symmetry. It follows that the vanishing conditions (1.7) that hold for the SU(2) F symmetry must also apply to SU(2) R ⊂ Sp(4) R , i.e. τ R 2 = ρ R = 0.

2
Comparing with (1.9) (and using (1.5)), we conclude that the c i anomalies of all N = (2, 0) theories necessarily coincide, in agreement with the explicit formulas in Table 1.

Anomaly Multiplets
So far we have discussed conformal anomalies (i.e. 't Hooft anomalies for conformal symmetry), which arose from various conserved currents such as the stress tensor or a flavor current. These currents may themselves have more conventional 't Hooft anomalies, which lead to violations of current conservation in the presence of background fields. Such anomalies are conveniently summarized by an anomaly 8-form, from which the anomalous transformation of the effective action follows via the standard descent procedure (see e.g. [14] for a detailed recent discussion with references). For instance, 't Hooft anomalies for the SU(2) R symmetry or diffeomorphisms are characterized by the following anomaly polynomial, The corresponding 't Hooft anomaly coefficients α, β, γ, δ are universal and independent observables in any six-dimensional N = (1, 0) SCFT. They are often exactly calculable, e.g. via string constructions and anomaly inflow arguments [15,16] or by analyzing RG flows [17][18][19][20].
We enumerate these anomaly coefficients for several N = (1, 0) theories in Table 2. (2, 0) Theory with algebra g h ∨ g d g + r g 1 2 r g 1 8 r g − 1 2 r g Table 2: SU(2) R and diffeomorphism 't Hooft anomaly coefficients of some N = (1, 0) theories. Note that the free vector multiplet is not conformal, but nevertheless enjoys an SU(2) R symmetry. The N = (2, 0) anomalies for g = su(N) were first computed in [15]. The general formulas for any ADE Lie algebra g were conjectured in [21] and verified in [22,17,6].
It is an important and general fact that supersymmetry relates conformal anomalies and 't Hooft anomalies. This means that the typically challenging and delicate conformal anomalies can be analyzed using the more accessible and robust 't Hooft anomalies. In the case of d = 4 SCFTs, such relations where established in [23] by analyzing the anomalous stress-tensor supermultiplet of the SCFT in the presence of background supergravity fieldsan object often referred to as the anomaly multiplet. In [24] we derived an anomaly multiplet relation between the conformal anomaly a and 't Hooft anomaly coefficients in (1.10), Rather than analyzing the anomalous stress-tensor supermultiplet in d = 6 SCFTs, this relation was derived by studying the dilaton effective action on the tensor branch of the SCFT -a technique we will also utilize in this paper. 4 A consequence of this derivation was a proof of the a-theorem for RG flows from the SCFT onto its tensor branch. 5 The behavior of a under Higgs branch RG flows was subsquently explored in [30], and the anomaly multiplet relation (1.11) has been verified holographically in [31].
In this paper, we likewise establish anomaly multiplet relations for the c i conformal anomalies in (1.1), by expressing them in terms of the 't Hooft anomaly coefficients α, β, γ, δ in (1.10) via the following formulas, Note that these formulas are compatible with the universal linear relation c 1 = 1 2 c 2 + c 3 in (1.5), even though α, β, γ, δ are independent.
In theories with flavor symmetries, there are additional 't Hooft anomaly coefficients that are visible in the presence of background flavor gauge fields. As above, we consider a single abelian or simple factor G F of the full flavor symmetry, and we focus on the mixed anomalies of G F with the SU(2) R symmetry or diffeomorphisms, We will argue that the 't Hooft anomaly coefficients α F 2 R 2 and α F 2 T 2 determine the conformal anomaly coefficient τ F in (1.8) as follows, (1.14) The formulas (1.12) have already appeared in the recent literature [32,33]. There linear relations between the c i conformal anomalies and the 't Hooft anomaly coefficients α, β, γ, δ were postulated, and the unknown coefficients in these relations were fixed by considering examples. This is complicated by the fact that each anomaly multiplet relation is specified by four coefficients, while there are only three independent classes of unitary SCFTs for 4 A direct analysis of anomalous stress-tensor multiplets is more challenging in d = 6 than in d = 4. For instance, non-conformal stress-tensor multiplets (of which the anomaly multiplet is a special case) and the associated supergravity theories have been thoroughly analyzed in d = 4 (see e.g. [25,26] and references therein), while their d = 6 counterparts are not nearly as well studied. Moreover, a direct investigation of d = 6 anomaly multiplets via anomalous supercurrents requires detailed knowledge of certain R 3 supergravity invariants, which is technically rather ominous. Some recent progress in this direction appears in [11]. 5 It was shown in [27][28][29] that the only supersymmetric RG flows in six dimensions that start at a SCFT fixed point are flows onto the moduli space of vacua of that SCFT. which all pertinent anomalies are reliably known (the free hyper-and tensor multiplets, and the N = (2, 0) theories). To circumvent this problem, the authors of [32,33] considered a non-unitary but superconformal free field theory constructed from an abelian vector multiplet with a higher-derivative kinetic term as a fourth data point. This theory had previously been shown to satisfy the anomaly multiplet relation (1.11) for the a conformal anomaly [34].
In section 4, we will instead follow [24] and argue for the anomaly multiplet relations (1.12) and (1.14) by studying conformal and 't Hooft anomaly matching along RG flows from an SCFT onto its tensor branch. As in [24], our main tool will be the supersymmetric dilaton effective action on the tensor branch.

Application to the Small E 8 Instanton SCFTs
The anomaly multiplet relations (1.12) and (1.14) have a wide variety of applications.
Nevertheless, as we illustrate in section 5, the conformal anomalies of these stronglycoupled SCFTs can often be determined using anomaly multiplet relations. For instance, in the N = (1, 0) SCFT described by N small E 8 instantons in string theory [37] we can use (1.14) to compute the two-point function coefficient τ E 8 of the E 8 flavor currents, This formula has already appeared in [49], where it was found to agree with bootstrap results.

Current and Stress-Tensor Two-and Three-Point Functions
In this section we review results from [50,51] about two-and three-point functions of conserved flavor currents J a µ associated with a global flavor symmetry G F , and the stresstensor T µν . We explain how these correlation functions are related to conformal anomaly coefficients, completing various discussions in the existing literature. In this section we consider general CFTs, without assuming supersymmetry. For each two-and three-point current correlator, we first present results for a general spacetime dimension d before specializing to d = 6. As above, we assume that the flavor symmetry G F is abelian or simple. Flavor Lie algebra indices will be denoted by a, b, c, . . . . Two-point functions are completely determined by conformal symmetry, up to an overall coefficient. For flavor currents and the stress tensor we have where In d = 6 dimensions, the two-point function coefficient C T is proportional to the conformal anomaly coefficient c 3 in (1.1). To determine the proportionality constant, it suffices to compare them for free-field CFTs. For a theory with n φ free real scalar fields and n ψ free fermion fields (with dim(ψ) complex spinor components), it was shown in [50] that ; in the last expression we have set d = 6 and taken n ψ to be the number of chiral fermions, with dim(ψ) = 4 complex components. We can now compare with our normalization for c 3 in Table 1 to conclude that Similarly, the flavor-current two-point function coefficient C F is proportional to the conformal anomaly coefficient τ F 1 in (1.4). We will chose a convention for the proportionality factor that is convenient for six-dimensional SCFTs. It was shown in [50] that in free scalar theories, with flavor current J a µ = φt a φ ∂ µ φ (here t a φ is real and antisymmetric), and in free fermion theories, with flavor current J a µ = ψt a ψ γ µ ψ (here t a ψ is complex and antihermitian) the coefficient C F is given by The first expression is valid for general d, while we have set d = 6 in the second expression.
The coefficients depend on the quadratic indices for the representations of the bosons or We choose our normalization convention for τ F 1 such that the Sp(4) R symmetry that acts on a free N = (2, 0) tensor multiplet has τ Sp(4) R 1 = 1. This is equivalent to the statement that the SU(2) F flavor symmetry that acts on a single, free N = (1, 0) hypermultiplet has τ F 1 = 1. Thinking of such a free hypermultiplet as a half-hyper that transforms as an SU(2) F doublet, we conclude that it has T φ = 2, T ψ = 1 2 , so that For generic d, it was shown in [50,51] that the flavor-current three-point function is fully determined by conformal symmetry and conservation laws, up to two coefficients A F F F and B F F F , where f abc are the structure constants and (2.8) A Ward identity implies that [50,51] Thus the JJJ three-point function introduces one additional, theory-dependent constant beyond the JJ two-point function coefficient C F .
The free-field values of the coefficients A F F F and B F F F were computed in [50], (2.10) Instead of using A F F F , B F F F to parameterize the JJJ three-point function, we can also express it in terms of free scalar or fermion correlators, The two coefficients n F F F φ and n F F F ψ are linearly related to A F F F and B F F F . The exact relation can be extracted from the free-field results summarized above.
The stress-tensor three-point function T µν (x)T ρσ (y)T κλ (z) is also determined by conformal symmetry and conservation laws up to three coefficients [50,51]. A Ward identity relates one linear combination of these three coefficients to the two-point function coefficient C T . In d = 6 dimensions, the three stress-tensor three-point function coefficients are linearly related to the three conformal anomalies c 1 , c 2 , c 3 in (1.1). We can span the three structures using the stress-tensor three-point functions of free fields: a free scalar φ, a free Weyl fermion ψ, and a free, chiral two-form B (with self-dual three-form field strength H), see [52] for details, In a free theory, the coefficients n T T T φ,ψ,B coincide with the number n φ , n ψ , n B of free scalars, Weyl fermions, or chiral two-forms, but in an interacting CFT they are defined by (2.12).
In general, these coefficients are constrained by unitarity and conformal collider inequalities (see section 3.4). Comparing to the known free-field conformal anomalies (see [53,52] and Table 1) we conclude that (2.13) We now consider the three-point function T JJ of one stress tensor and two flavor currents, which was also analyzed in [50,51] and shown to be determined by two coefficients.
One linear combination of these coefficients is proportional to the JJ two-point function coefficient C F (or equivalently to τ F 1 , see (2.6)) thanks to a Ward identity, while the remaining independent structure constant can be thought of as the OPE coefficient of the stress tensor in the fusion of two flavor currents.
In the notation of [50,51] (see for instance the discussion around equation (3.14) of [50]), the two coefficients that determine the T JJ three-point function are called c T F F and e T F F . Their free-field values can be found in equation (5.10) of [50]; setting d → 6 and dim (ψ) = 4 in these formulas, we find that 14) It follows that the two independent structures in the T JJ three-point function are spanned by free field theories in which the flavor current only arises from charged scalars or fermions, As reviewed in section 1.1, a U(1) or simple flavor symmetry G F gives rise to the conformal anomalies (1.4) in the presence of suitable background flavor and gravity fields, The anomaly coefficients τ F 1,2 and ρ F are theory dependent. In the absence of background fields, each conformal anomaly in (2.16) represents a contact term associated with a particular three-point function at separated points. To derive these relations, one can for instance follow [54,50] and work in position space using the method of differential regularization; alternatively, one can work in momentum space.
It follows that the conformal anomalies τ    [53,13,55,56]. Converting to our conventions, we find that free field theories satisfy By comparing this to the free-field formulas (2.10) and (2.14) above, we conclude that Together with (2.6), this completes the relations between two-and three-point function co-efficients of currents and stress tensors, and the conformal anomalies associated with flavor symmetries in (2.16). We summarize these relations here,

Current and Stress-Tensor Supercorrelators in Six-Dimensional SCFTs
In this section we examine the constraints of supersymmetry on two-and three-point functions of flavor currents and stress tensors in six-dimensional SCFTs. We will show that the conformal anomalies of all N = (1, 0) SCFTs satisfy the following universal relations, Here G F can be any U(1) or simple flavor symmetry. We also show that the conformal anomalies associated with the SU(2) R symmetry can be expressed in terms of the c i conformal anomalies as follows, Finally we briefly discuss conformal collider bounds on these anomaly coefficients.

Free SCFTs
We begin by considering a theory of n H free hypermultiplets and n T free tensor multiplets. As we will see below, some relations uncovered in this simple free-field context continue to hold for general N = (1, 0) SCFTs. In fact, the input from free field theories will be used in the general proof of these relations below.
Using (2.13) with n = n T , we find that Note that these formulas are consistent with (3.1), and that a free N = (2, 0) tensor multiplet with n H = n T = 1 indeed satisfies c 1 = c 2 = c 3 = 1.
In supersymmetric theories we distinguish between flavor symmetries, which commute with the supercharges, and R-symmetries, which do not: • Flavor Symmetries: Only hypermultiplets can carry flavor charge, with T H is the quadratic index of the hypermultiplet flavor representation. Thus, the free-field expression for the flavor-current two-point function coefficients is Note that a single free hypermultiplet has an SU(2) F flavor symmetry with T F H = 1 2 and hence τ F 1 = 1. This is easy to see by reformulating the theory as a half-hyper in the doublet representation of SU(2) F .
For the T J F J F three-point function, it follows from (2.14) that Substituting into the expressions (2.19) for the flavor conformal anomalies gives • SU(2) R Symmetry: The scalars in each hypermultiplet transform as a complex SU(2) R doublet, so that T R φ = 2n H . By contrast, the fermions in every tensor multiplet transform as half-doublets of SU(2) R , so T R ψ = 1 2 n T . It follows that the free-field expression for the SU(2) R current two-point function coefficient is given by The free-field expressions for the J R J R J R and T J R J R three-point function coefficitents take the following form, Substituting into (2.19) and comparing with (3.3), we find that

Supercorrelators for Conserved Supermultiplets: Overview
We will apply the superspace formalism of [57] to the two-and three-point functions of operators in conserved flavor-current and stess-tensor supermultiplets. Analogous considerations for four-dimensional SCFTs can be found in [58][59][60]. Here we will give a brief survey of the results that will follow from this analysis (see section 3.3 below for details): •   • All non-zero three-point functions of operators in the stress-tensor supermultiplet are completely determined by two coefficients, one of which is related to the two-point function coefficient c 3 by a Ward identity. In particular, this proves that the three conformal anomalies c 1 , c 2 , c 3 necessarily satisfy a linear relation in any six-dimensional SCFT, as originally conjectured in [9]. As before, the coefficients in this linear relation are fixed by the free-field formulas (3.3), Since the SU(2) R current J R also resides in the stress-tensor supermultiplet, it follows that its three-point functions are also linear combinations of c 1 and c 3 ; the coefficients are determined by the free-field formulas (3.9) and (3.10). It follows that all N = (1, 0) SCFTs satisfy This is compatible with the Ward identity relating the SU(2) R current three-and twopoint functions: π 3 ( 1 6 A RRR + B RRR ) = C R = 5c 3 /4π 6 , which fits (2.6) with τ R 1 = c 3 . Likewise, the coefficients e T RR and c T RR that determine the three-point functions of one stress tensor and two SU(2) R currents are universal linear combinations of c 1 and c 3 that can be fixed by free-field reasoning (see again (3.9) and (3.10)), The main goal of the superspace analysis below is to cut down the number of independent structures in the two-and three-point functions of flavor-current and stress-tensor supermultiplets by imposing the constraints of superconformal symmetry. Once the number of these structures is sufficiently small, their coefficients can be determined by free-field reasoning.
The constraints of conformal symmetry on two-and three-point functions are standard: two-point functions are always determined by one overall coefficient. By contrast, three-point functions are described by finitely many tensor structures, whose exact number can depend on the Lorentz representations of the operators participating in the three-point functions.
Both of these results follow from the fact that the bosonic conformal generators can be used to bring the spacetime coordinates of the operators appearing in these correlators to standard form (see for instance [61] and references therein for a recent discussion).
In superspace, the Q and S supercharges can be used to set the Grassmann coordinates As we will discuss below, the superspace three-point function T (z 1 )T (z 2 )T (z 3 ) of the stress-tensor multiplet can be expressed in terms of a homogeneous function H T T T (X, Θ) (here the bosonic variable X is the superpartner of Θ), which is completely determined up to three independent coefficients C T T T 1,2,3 . We then impose the shortening condition on T , which amounts to setting a level-three null state and its descendants to zero. As we will see, this imposes one linear relation on the coefficients C T T T 1,2,3 , so that the T (z 1 )T (z 2 )T (z 3 ) supermultiplet three-point function is in fact completely determined by two independent constants. Once this has been established, free-field reasoning is sufficient to deduce all facts about the stress-tensor supermultiplet three-point function that were summarized above.
When analyzing the three-point supercorrelator J flavor current multiplets, as well as supercorrelator J (i 1 j 1 ) (z 1 )J (i 2 j 2 ) (z 2 )T (z 3 ) of two flavorcurrent multiplets and one stress-tensor multiplet, we must impose the shortening condition satisfied by J (ij) in addition to that of T . This completely determines both correlators up to one overall coefficient, which in turn is related to the flavor-flavor two-point function coefficient τ F 1 by a Ward identity.
4.5 → 0 and its descendants. In superspace, Two-point functions can be written in terms of coordinates that are invariant under superspace translations [57], The superspace two-point function of the stress-tensor supermultiplet T (z) is completely determined up to an overall coefficient c T ∼ C T , giving the superspace generalization of (2.1), The two-point functions of all operators in the T -multiplet follow upon expanding in θ 1 , θ 2 .
In particular, the J  [65]. 8 Similarly, the two-point functions of flavor-current supermultiplets J a(ij) are completely determined up to an overall coefficient c J , (3.26) The factors of u i j (z 12 ) (defined in (3.24)) account for the SU(2) R representation of J a(ij) . The actual flavor-current two-point function arises from the θ 2 1 θ 2 2 component of (3.26). Finally, the two-point function of a flavor-current and a stress-tensor multiplet must vanish, For the bottom components, this follows from SU(2) R symmetry, while superconformal symmetry ensures that the same is true for all other operators in these multiplets.
We now turn to the three-point functions involving the T and J a(ij) multiplets. As in [50,57], they can be expressed in terms of a superspace variable Z ≡ (X µ , Θ αi ) ∈ R 6|8 that is formed from the three superspace coordinates z 1,2,3 as follows,   Θ). The function H can always be written as follows,  is not independent of the invariants that were already introduced above.
If we expand the three-point function (3.30) in components, the coefficient C determines the three-point function of the superconformal primaries. The coefficients C are associated with three-point functions of descendant operators. For long multiplets, the are independent, but for short multiplets they are satisfy additional constraints that follow from the requirement that the null-state multiplet vanish in superspace.
Let us apply the general reasoning above to the the three-point function of stress-tensor multiplets. Setting O i (z i ) = T (z i ) and ∆ i = 4 in (3.30), we obtain We must now impose the shortening condition (3.18), as well as exchange symmetry under z 1 ↔ z 2 ↔ z 3 , since the three supermultiplets are identical. In the present context, exchanging z 1 ↔ z 2 takes Z → −Z and does not constrain the coefficients in (3.31). 9 However, the shortening condition (3.18) can be shown to lead to the following constraint on the function H T T T that appears in (3.31), To analyze this constraint, we count monomials in X [αβ] and Θ i α that share its [1, 0, 0] Lorentz and R-symmetry quantum numbers and could therefore appear as terms in (3.32).
It follows that evaluating the differential operator on the left-hand side of (3.32) on the function H T T T in (3.31) can (schematically) only give terms of the form We can also use superspace to analyze three-point correlators that involve a flavor-current multiplet (3.21). This requires writing down general combinations of u's, χ's, X, and Θ with the correct symmetries and quantum numbers, and then imposing the shortening conditions The condition on the flavor current is particularly constraining, because the analogue of (3.33) now involves multiple independent structures on the right-hand side. The condition that all of these vanish separately determines the relative coefficients of all terms in the Θ-expansion.
The upshot is that all three-point functions involving a flavor-current multiplet are fully determined up to an overall coefficient. Moreover, Ward identities relate this overall coefficient to a two-point function coefficient c F ∼ τ and Here the canonical correlators on the right-hand side are specific, completely determined functions on superspace, which are fixed by exchange symmetries and null-state conditions.
This six-dimensional analysis is closely analogous to the four-dimensional analysis in [60].

Conformal Collider Inequalities
The average null energy condition [67,68] places unitarity bounds on the three-point function coefficients discussed above. These bounds are conveniently derived using the conformal collider setup of [10,9,69]. (See [70][71][72][73] for some related recent work.) As described in [9,55], the six-dimensional version of the collider bounds of [10] can be written as where t 2,4 are linear combinations of the c i conformal anomalies given in [9]. In terms of the parameterization in (2.12), these inequalities take the simple form It was pointed out in [10] that t 4 = 0 for d = 4 SCFTs. By analogy, it was conjectured in [9] that supersymmetry should impose the linear relation t 4 = 0 on the c i conformal anomalies of d = 6 SCFTs, and it was verified that this is indeed the case in free-field examples. In our conventions, the condition t 4 = 0 precisely coincides with the relation (3.13) derived above. The remaining inequalities in (3.37) then reduce to to [9], The lower bound is saturated by a free hypermultiplet. In fact, the upper bound in (3.39) can be strengthened to 10 9 < 31 27 in SCFTs, and this stronger upper bound is saturated by a free tensor multiplet. Note that unitarity of the two-point function implies c 3 > 0, so that (3.39) gives c 1 > 0. Similarly, using (3.13) gives (c 2 /c 3 ) = 2(c 1 /c 3 ) − 1 so that (3.39) implies and hence c 2 > 0.
To derive stronger constraints on the c i in the case of SCFTs we follow [10] and apply the conformal collider constraints to states created by the R-current. Let us first review the nonsupersymmetric conformal collider bounds on states created by a conserved flavor current J F µ in d dimensions. As reviewed below (2.13), the T J F J F correlator is parametrized by two coefficients, c T F F and e T F F , one linear combination of which is fixed by the flavor-current [50,51] for details). As shown in [70,74], the ratio c T JJ /C F satisfies the following d-dimensional collider bounds, where the first inequality is saturated for free scalars and the second one for free fermions.
In d = 6 we can rewrite these inequalities as where the order of the inequalities has been reversed, i.e. the first one is saturated for free fermions and the second one for free scalars.
We can now apply (3.42) to the T J R J R correlator in SCFTs. Substituting the free-field formulas for c T RR , e T RR in terms of c 1 , c 3 in (3.15) into these inequalities, we find that (3.43) The lower bound coincides with that in (3.39), but the upper bound is stronger; it is saturated by a free N = (1, 0) tensor multiplet. In terms of the free-field parameterization of the c i conformal anomalies in (3.3), the inequalities (3.42) reduce to the intuitive requirement that

Anomaly Multiplet Relations on the Tensor Branch
In this section we explore how the anomaly multiplet relations derived above are reflected on the tensor branch of N = (1, 0) SCFTs in six dimensions. This leads to a simple, intuitive argument for some of these relations.

Dilaton Effective Action on the Tensor Branch
Here we closely follow the discussion in [24], but also make some new observations. On the tensor branch, the SU(2) R symmetry is unbroken and the massless dilaton field ϕ (i.e. the Nambu-Goldstone boson associated with spontaneous conformal symmetry breaking) resides in a tensor multiplet. The constraints of conformal symmetry on the effective action of ϕ with and without background gravity fields were analyzed in [75]. The additional terms that arise in the presence of background flavor gauge fields were described in [55]. When all background gauge fields are set to zero and the background metric is taken to be the flat Minkowski metric, the minimal low-energy effective action for ϕ schematically takes the form Here, ∆a = a UV − a IR is the change in the conformal a-anomaly along the RG flow from the SCFT at the origin to the low-energy effective theory on the tensor branch. A basic fact that will play an important role below is that the coefficient b in (4.1) is subject to a dispersion relation that implies its positivity [76], This inequality is saturated if and only if ϕ is a free field with trivial scattering S-matrix.
The constraints of N = (1, 0) supersymmetry on the dilaton effective action were analyzed in [24]. In the absence of non-trivial background fields, the N = (1, 0) dilaton effective action on the tensor branch is given by the supersymmetrization of (4.1), with ϕ residing in a tensor multiplet together with its superpartners ψ i α and B. Here ψ i α is a Majorana-Weyl fermion, and B is a two-form gauge field, whose field strength H = dB is self dual (i.e. H = * H). The upshot of this analysis is two-fold [24]: the first conclusion is that the supersymmetric completion of the leading 4-derivative term in (4.1) is of the form where the ellipisis denotes higher-order terms in the expansion of ϕ = ϕ +δϕ in fluctuations around its vev. The second conclusion is that the 6-derivative term in (4.1) proportional to ∆a is in fact related to the term in (4.3) by supersymmetry, which implies the following universal quadratic relation, ∆a = 98304π This immediately implies positivity of ∆a, and hence the a-theorem, for this class of flows [24].
In the presence of background fields, the dilaton effective action (4.1) gets extended in various ways. Among these extensions there are certain additional terms that are needed to compensate any apparent mismatch between the 't Hooft anomalies of the SCFT at the origin and the low-energy theory on the tensor branch. These terms are Green-Schwarz (GS) like couplings that involve the dynamical two-form gauge field B residing in the tensor multiplet and various background field strengths or background curvatures [17,18], and their supersymmetric completions. Given, for instance, a background flavor field strength F µν , the corresponding GS term takes the form It follows that c 2 (F ) (the background flavor instanton density) acts as a source for the dynamical B-field, whose field-strength is H, This in turn implies that n F is an integer. The supersymmetric completion of (4.5) contains a dilaton couplling ∼ n f ϕ tr F µν F µν . Together with (4.5), this Lagrangian is similar to the interacting gauge-tensor Lagrangians used to described N = (1, 0) SCFTs in [36], except that here the gauge fields are fixed backgrounds associated with global flavor symmetries while only the tensor multiplet is dynamics.
In addition to the flavor background fields discussed above, the tensor multiplet containing B and ϕ also couples to supergravity background fields -in particular a background metric and background gauge fields for the SU(2) R symmetry. Here we will briefly review these terms, following [24]. The GS terms involving the dynamical B-field and supergravity background fields are where x, y are real coefficients. These terms account for the following mismatches ∆α = α UV − α IR etc. in the SU(2) R and diffeomorphism 't Hooft anomaly coefficients (1.10), It was shown in [24] that the GS terms (4.7) are related to certain R 2 supergravity terms [77], . (4.9) Specializing to a conformally flat background and using results from [75], this shows that the leading four-dilaton interaction (4.1) is determined by the GS coefficeints x and y, The inequality (4.2) then shows that y ≥ x. This inequality can only be saturated if the dilaton is a free tensor multiplet, in which case there is no RG flow to begin with. In particular, ∆a ∼ b 2 ∼ (y − x) 2 vanishes in this case [24].

The c i Conformal Anomalies on the Tensor Branch
We will now use the tensor branch to argue for the anomaly multiplet formulas (1.12) that determine the c i conformal anomalies in terms of the 't Hooft anomalies α, β, γ, δ.
We will not provide a complete analysis of these anomaly multiplet relations on the tensor branch, which would require the dilaton effective action coupled to background supergravity fields up to six-derivative order. Instead, we will make the plausible assumption that all sixderivative terms responsible for anomaly matching in the dilaton effective action arise from the completion of four-derivative GS terms and their superpartners. (This assumption was explicitly demonstrated in [24] for the six-derivative terms associated with the a conformal anomaly.) We also postulate that the c i anomalies obey linear anomaly multiplet relations of the form Using (4.8), it follows that the change in these anomalies under the RG flow onto the tensor branch is At the special locus x = y, the dilaton is free and there is no non-trivial RG flow, so that all ∆c i must vanish. It follows that the quadratic polynomials in (4.12) must factor as ∆c i = (y − x)(u i x + v i y) (recall that this is the case for ∆a ∼ (y − x) 2 ), so that

Flavor Conformal Anomalies on the Tensor Branch
In four dimensional SCFTs, the flavor-current two-point function coefficient τ F is determined by the 't Hooft triangle anomaly of two flavor currents and one R-current. This follows from the d = 4 multiplet of anomalies in the presence of background supergravity and flavor gauge fields [23]. We anticipate that a similar relation exists for six-dimensional SCFTs, which should relate the flavor-current two-point function coefficient τ F to 't Hooft anomalies involving background supergravity and flavor gauge fields. Since τ F scales quadratically with the charges (e.g. in free field theory), such a putative relation must involve a mixed 't Hooft anomaly with two flavor background gauge fields. The remaining two background fields can either be SU(2) R or gravity backgrounds, i.e. we consider mixed flavor-SU(2) R and flavordiffeomorphism anomalies. We therefore postulate a linear relation between τ F and the 't Hooft anomaly coefficients α F 2 R 2 and α F 2 T 2 in the anomaly 8-form polynomial, (4.14) Recall from the discussion above (2.6) that we normalize τ F so that a free N = (2, 0) tensor multiplet contributes has τ Sp(4) R 2 = 1. In this normalization, the τ 2 conformal anomaly and the 't Hooft anomalies in (4.14) for some known examples are summarized in Table 3.
This is the relation in (1.14).
We can now subject the proposed formula (4.15) to a stringent consistency check by considering an RG flow onto the tensor branch, under which The anomaly mismatches ∆α F 2 T 2 and ∆α F 2 R 2 are accounted for by the GS terms for supergravity background fields in (4.7), in conjunction with the flavor GS term (4.5), This leads to 't Hooft anomaly matching contributions beyond (4.8), with the same overall proportionality factor as in (4.8). Substituting into (4.16), we find that This is indeed proportional to b ∼ y − x, and trivializes as expected when x = y.

N = (2, 0) Theories
These theories have an Sp(4) R symmetry. Thinking of them in N = (1, 0) language, it is natural to focus on its SU(2) F × SU(2) R subgroup, with SU(2) F a flavor symmetry. In our normalization, the conformal anomalies of the interacting N = (2, 0) theory based on an ADE lie algebra g are given by Consider a rank-one tensor branch of the type g theory, associated with the adjoint breaking pattern g → h + u(1). The apparent change in the anomaly polynomial is [21] 4!∆I 8 = ∆k p 2 (Sp(4) R ) −→ ∆k c 2 (L) − c 2 (R) 2 , ∆k ≡ k(g)−k(h) , k(g) ≡ h This requires GS anomaly matching terms (4.17), with y = 0 and x = n F ∼ − ∆k/6.

10
The simplest example in this class occurs for g = a 1 and h trivial, so that ∆k/6 = 1. The interacting a 1 SCFT at the origin has c i = τ F = τ R 1 = 25 and a = 103 7 . This theory was explored using numerical bootstrap techniques in [78].

Small E 8 Instanton SCFTs
We can apply our general formulas to determine the conformal anomalies of the N = (1, 0) SCFTs E[N] that describe N small E 8 instantons in string theory, or alternatively N M5 branes probing an end-of-the-world M9 brane in M-theory. These theories have an SU(2) F × E 8 flavor symmetry. Aspects of the corresponding flavor-current correlators were discussed in [79,49]. Note that our definition of the E[N] theory includes the free, decoupled hypermultiplet describing overall translations of the N M5 branes in the four transverse directions.
We will choose N > 0, so that SU(2) R is the N = (1, 0) R-symmetry, while SU(2) F is a flavor symmetry.
To O(N 3 ), these anomalies agree with those obtained from a naive Z 2 orbifold of the a N −1 N = (2, 0) theory, as expected from the M5 brane construction of the E[N] theory. Note however that the c i anomalies in (5.7) already differ at O(N 2 ).
As was already stated above, the E[N] SCFTs have an SU(2) F × E 8 flavor symmetry.
The associated 't Hooft anomalies can be extracted from the anomaly polynomial (5.3), e.g.
Substituting into (1.14) gives the SU(2) F two-point function coefficient, The fact that τ F and τ R 1 coincide at leading O(N 3 ) is again consistent with a naive Z 2 orbifold of the a N −1 N = (2, 0) theory, and again the two expressions differ at O(N 2 ). Note that (5.9) and (5.10) are not simply related by the N → −N transformation mentioned above. This is not a contradiction, since it need not be the case that replacing N → −N exchanges SU(2) F and SU(2) R for all purposes. 11 As a check of (5.9), note that when N = 1 the SU(2) F flavor symmetry only acts on the free, decoupled hypermultiplet associated with M5 brane motion in the transverse directions. Indeed substituting N = 1 in (5.9) gives τ F = 1, as expected for a free hypermultiplet.
The anomaly coefficients in (5.3) that involve the E 8 background gauge field are Substituting into (1.14) then gives the E 8 two-point function coefficient, This result has already appeared in [49].
Finally, let us consider a tensor-branch deformation corresponding to moving one M5 brane away from the M9 brane and the other N −1 M5 branes, so that E[N] → E[N −1]+E [1].
The anomaly mismatch is accounted for by a GS mechanism [18], Since it follows from (5.14) that N + 1

Holographic Examples
Consider d = 6 SCFTs with AdS 7 holographic duals; see e.g. [81] and references therein for examples. The AdS d+1 /CFT d dictionary relates the flavor-current two-point function [82], where L is the AdS d+1 length scale, and g Moreover, the relation (1.14) implies that the g −2 F F Yang-Mills coefficient of the AdS 7 supergravity theory is related to the bulk Chern-Simons terms responsible for the α F 2 T 2 and α F 2 R 2 't Hooft anomalies on the boundary. See [83] for some related comments.