Deformations of Superconformal Theories

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 admit relevant or marginal deformations, and some admit neither. The classification is complicated by the fact that short superconformal multiplets display a rich variety of sporadic phenomena, including supersymmetric deformations that reside in the middle of a multiplet. We illustrate our results with examples in diverse dimensions. In particular, we explain how the classification of irrelevant supersymmetric deformations can be used to derive known and new constraints on moduli-space effective actions.


Introduction
In this paper we consider unitary superconformal field theories (SCFTs) in 3 ≤ d ≤ 6 spacetime dimensions. 1 Our main result is a classification of their possible relevant, irrelevant, and marginal operator deformations that preserve the non-conformal Poincaré supersymmetries and Lorentz invariance, but not necessarily conformal symmetry. These deformations are tabulated in section 3, which is self-contained. The classification utilizes the fact that the deforming operators reside in unitary representations of the superconformal symmetry, which are much more constrained that representations of Poincaré supersymmetry. 2 Since we only rely on general properties of these representations, our results are model independent and do not require a Lagrangian.

Deformations of Conformal Field Theories
Quantum field theories can be thought of as renormalization group (RG) flows from short distances in the UV to long distances in the IR. The endpoints of such flows are RG fixed points, and hence scale invariant. In relativistic theories, it is common to further assume that the fixed-point theory is a conformal field theory (CFT), whose spacetime symmetry is enhanced to the conformal algebra so (d, 2). 3 In addition to free CFTs, which exist in every dimension, there is compelling evidence for a vast landscape of interacting CFTs in diverse dimensions. 4 Many of these theories are non-Lagrangian, i.e. they do not possess a known presentation in terms of fields and a Lagrangian, and in some cases the existence of such a presentation is believed to be unlikely. 1 SCFTs also exist in d = 1, 2. They are particularly well-studied in d = 2, where the superconformal algebra is typically enhanced to a super-Virasoro algebra. 2 As we will discuss below, one consequence of this fact is that deformations of SCFTs constitute a proper subset of the deformations that can arise in more general supersymmetric theories. 3 In d = 2 spacetime dimensions, this enhancement follows from unitarity and Poincaré invariance [1,2]. See [3] for a review of what is known in other spacetime dimensions, and [4][5][6][7][8][9] for developments in d = 4. 4 There are no known interacting CFTs in d > 6.
Given a CFT, we would like to analyze the nearby quantum field theories that can be obtained by deforming it, i.e. we would like to analyze the possible RG flows in the vicinity of the corresponding fixed point. Broadly speaking, such deformations fall into three classes: 1.) Adding local operators to the Lagrangian: This is the most common way to modify the dynamics of a theory, where the Lagrangian L is deformed as follows, Here g is a (typically running) coupling constant and O is a local operator in the original, undeformed CFT at g = 0. Note that the deformation δL in (1.1) can always be defined using conformal perturbation theory, 5 even if the original CFT is non-Lagrangian and L is not known, or perhaps does not exist.

2.) Gauging a global symmetry:
In the most familiar case, the symmetry is a continuous flavor symmetry with a conserved one-form current j µ , but it could also be discrete or a higher-form global symmetry (see for instance [11] and references therein). The gauging procedure involves projecting out some degrees of freedom from the original theory (those that are not gauge invariant) and adding new ones, which arise from the gauge fields, and it typically involves a choice of continuous or discrete coupling constants. Gauging a global symmetry may be obstructed by anomalies or lead to a theory with a Landau pole that is not UV complete. Note that gauging cannot be understood as an operator deformation (1.1). A similar comment applies to Chern-Simons terms, which are not gauge-invariant local operators.
3.) Moving onto a moduli space of vacua: In d > 2 non-compact spacetime dimensions, a CFT may possess a non-trivial moduli space of vacua, which is continuously connected to the conformal vacuum at the origin. This is the case for many superconformal theories. Deforming away from the conformal vacuum involves tuning the boundary conditions at spatial infinity and leads to vacuum expectation values for some fields, which generate a scale and break conformal symmetry spontaneously. Unlike the deformations in 1.) and 2.), which modify the dynamics of the theory at short distances, moving along a moduli space of vacua represents a modification in the deep IR, via boundary conditions. Nevertheless, one can consider an RG flow that interpolates be-tween the UV physics of the CFT at the origin and the IR physics on the moduli space of vacua.
In this paper, we will almost exclusively focus on deformations of type 1.), i.e. adding local operators to the Lagrangian as in (1.1).
The local operators must reside in representations of the conformal algebra so(d, 2). They can be labeled by their weights under the maximal compact subalgebra so(d) × so (2). Here the so(d) weight specifies the (Wick-rotated) Lorentz representation, and the so(2) eigenvalue is related to the scaling dimension ∆, see e.g. [12][13][14] for more detail. There is a natural inner product on all CFT operators, which is defined by their twopoint functions in flat space, or equivalently by the the inner product on the Hilbert space of states in radial quantization. In unitary theories, all primary and descendant operators must nave non-negative norms with respect to this inner product. This leads to unitarity bounds for the scaling dimension ∆ O in terms of the Lorentz representation L O , see e.g. [15,12,14] ∆ When the bound is saturated, the representation has null states, i.e. zero-norm descendants that can be consistently removed from the representation.
The possible operator deformations (1.1) can be understood using the structure of unitary so(d, 2) representations. First note that the deformation O should be a conformal primary. Descendants are total derivatives of well-defined operators, and adding them to the Lagrangian leads to boundary terms that do not modify the bulk dynamics. Similarly, we do not consider deformations by the identity operator 1, since these only modify the vacuum energy, but not the dynamics. If we further restrict O to be an so(d) scalar, as we will do throughout most of this paper, then the deformation preserves Lorentz symmetry. In this case the strongest unitarity bound (1.2) comes from demanding that the norm of the descendant O be non-negative, The bound is saturated if O is a free scalar field satisfying O = 0.
The qualitative properties of the deformation depend on the value of ∆ O relative to the spacetime dimension d. This is the standard distinction between relevant, irrelevant, and marginal operators: Here the CFT at g = 0 is the UV fixed point of an RG flow that is initiated by turning on the deformation. The relevant coupling g grows in the IR, and conformal perturbation theory in g is expected to break down eventually. actually exist in a given CFT, and when they lead to well-behaved RG flows, cannot be answered using only representation theory.

Superconformal Theories
We will use the structure of unitary superconformal multiplets to analyze the possible supersymmetric deformations of unitary superconformal theories in 3 ≤ d ≤ 6 spacetime dimensions, generalizing the analysis of [16,17]. By this we mean deformations of the form (1.1) that preserve all Poincaré Q-supersymmetries, but not necessarily the superconformal Ssupersymmetries. From now on, we will always use the term deformations to refer to such supersymmetric operator deformations.
Together, the Q-and the S-supersymmetries anticommute to the superconformal algebra, whose bosonic subalgebra contains the conformal algebra so(d, 2), as well as an Rsymmetry algebra. Unlike the Poincaré supersymmetry algebra, which exists in all dimensions, superconformal algebras are highly constrained: they do not exist in d ≥ 7 dimensions, and in 3 ≤ d ≤ 6 dimensions the only consistent superconformal algebras are given by [18] (see also [12] for a nice discussion), In every case, we have indicated the maximal bosonic subalgebra, which factorizes into the superconformal algebra so(d, 2) and the R-symmetry. As usual, N ∈ Z ≥1 is a positive integer that indicates the number of supercharges in units of the minimal amount of supersymmetry that is possible in a given dimension. We use N Q to denote the total number of independent supercharges. In d = 3, 4, 5, 6 dimensions, minimal N = 1 supersymmetry corresponds to N Q = 2, 4, 8, 8 supercharges, respectively. For d = 5, there is a unique superconformal algebra, with N = 1 supersymmetry; theories with more supersymmetry (e.g. N = 2 maximally supersymmetric Yang-Mills theory) exist, but cannot be superconformal. By contrast, the superconformal algebras in d = 3, 4, 6 come in infinite families, labeled by a positive integer N . However, it can be shown [19] that interacting superconformal field theories only exist for N ≤ 8, 4, 2 in d = 3, 4, 6 dimensions, respectively, and hence we will only discuss these values of N . Note that SCFTs in six dimensions are often referred to as (N , 0) theories.
There is compelling evidence for the existence of many interacting SCFTs in these allowed ranges of d and N .
We will make extensive use of known facts about unitary representations of the superconformal algebras in (1.4), especially results from [20,12,21,22,13,23,19], 6 which we briefly review. Each unitary irreducible representation of a superconformal algebra decomposes into a finite number of irreducible representations of the bosonic subalgebra, which consists of the conformal algebra so(d, 2) and the R-symmetry. In other words, the superconformal multiplet decomposes into a supermultiplet of conformal representations. Since we are interested 6 In addition to references dedicated to the representation theory of the superconformal algebras, there are also many supergravity papers that considered these representations from the perspective of what was later understood to be the holographic AdS duals. See for instance [24][25][26] for a discussion of 1 2 -BPS multiplets in maximally supersymmetric SCFTs in d = 4, 3, 6 dimensions, respectively. We will not attempt the challenging task of assembling a complete set of references.
in deformations of the form (1.1), we will only consider conformal primaries O, which are labeled by their Lorentz-and R-symmetry representations L O and R O , as well as their scaling (1.5) Throughout the paper, we will use integer-valued Dynkin labels to specify the representa- the S-supersymmetries, with dimension − 1 2 , and the special conformal generators K µ , with dimension −1). The other operators in the superconformal multiplet (the other conformal primaries, and all conformal descendants) are superconformal descendants of V, i.e. they are obtained by acting on V with any number of Q-supersymmetries, whose scaling dimension is + 1 2 . Demanding that all of these operators have non-negative norm leads to unitarity bounds for the superconformal primary. These bounds are are stronger than the bosonic unitarity bounds (1.2) because there are more Q-descendants than P µ -descendants, all of whose norms must be non-negative. Schematically, Whenever such a bound is saturated, the representation has null states. These must themselves form a superconformal representation (though not necessarily a unitary one) that can be consistently removed from the multiplet. We will refer to all representations with null states as short, and those without null states as long. A superconformal representation is completely determined by the quantum numbers of its superconformal primary V. As a result, the multiplet is typically described by specifying the quantum numbers (1.5) for V.
We are interested in supersymmetry-preserving deformations (1.1), i.e. conformal primaries O that are annihilated by the action of all Q-supersymmetries, up to a total derivative.
Schematically, We will heavily draw on the results of [19], where a solution to 1.) is presented for all unitary superconformal multiplets in 3 ≤ d ≤ 6 dimensions, generalizing the results of [13] for d = 4.
However, the methods of [19], do not immediately solve 2.) as well. The problem is in principle straightforward: it can be solved by explicitly expressing all conformal primaries as Q-descendants of the superconformal primary and, if the multiplet is short, imposing the vanishing of all null states. This head-on approach was used in [16,17] to analyze the deformations of four-dimensional N = 1, 2 SCFTs, but it is prohibitively tedious in many other cases. Here we will carry out a classification of supersymmetric deformations while largely sidestepping this problem. As a result, the completeness of our classification depends on some assumptions that are spelled out in section 2.

Supersymmetric Deformations: Generic and Sporadic Phenomena
There are several familiar classes of top components, and hence supersymmetric deformations, that can be described in a uniform manner. Given a superconformal primary V, its descendants are obtained by acting with the Q-supersymmetries. Operators of the form Q ℓ V, which are obtained by acting with ℓ supercharges on V are said to reside at level ℓ. The expression Q ℓ V should always be understood as ℓ nested (anti-) commutators. Since we are only interested in conformal primaries, we can drop all spacetime derivatives, ∂ µ ∼ P µ ∼ 0, so that the Q-supercharges effectively anticommute, The only exception is the identity operator 1, which has already been excluded as a deformation.
Here N Q is the total number of supercharges. By Fermi statistics, conformal primaries only occur at levels 0 ≤ ℓ ≤ ℓ max , where ℓ max must satisfy the bound ℓ max ≤ N Q . This bound is saturated for long multiplets, without null states, for which Q N Q V is the unique top component. Its quantum numbers are the same as those of V, because Q N Q is a Lorentz and R-symmetry singlet. This leads to the generalized supersymmetric D-term deformation, which is a Lorentz singlet if the superconformal primary V is a Lorentz singlet; its dimension When a superspace formulation is available, the generalized Dterm (1.9) can be written as an integral over all of superspace. A typical example is the Kähler potential in four-dimensional theories with N = 1 supersymmetry and N Q = 4 supercharges.
Another common type of deformation is a generalized F -term. It is constructed using a short, 1 2 -BPS multiplet, whose superconformal primary V BPS is annihilated by half of the supercharges. Then the F -term deformation is given by the action of the other 1 2 N Q supercharges on V BPS , (1.10) When a superspace formulation is available, a generalized F -term can be written as an integral over half of superspace. A typical example is the chiral superpotential W in fourdimensional N = 1 theories, which satisfies QαW = 0 and leads to the F -term deformation L F = Q 2 W . 8 (The Hermitian conjugate deformation Q 2 W is also an F -term.) The detailed structure of 1 2 -BPS multiplets changes for different d and N , but they often lead to generalized F -term deformations. Also, we will see below that some theories admit different types of F -term deformations that reside in distinct 1 2 -BPS multiplets. The D-and F -term deformations are generic: they are constructed using multiplets that exist for all (or most) values of d and N , and for a variety of quantum numbers. By contrast, there are deformations that reside in special, typically very short multiplets and only occur sporadically, i.e. only for certain values of d and N , and only when the quantum numbers of the superconformal primary take certain small values.
• The Lorentz scalar [0] (0;0) 2 at ℓ = 2 is also a top component, even though it occurs in the middle of the multiplet. Acting on it with the Q-supercharges leads to an operator with quantum numbers [1] (1;1) 3/2 , but there is no such conformal primary at ℓ = 3.
The scalar top component at ℓ = 2 gives rise to a relevant deformation of the theory with scaling dimension ∆ = 2, just like a fermion mass term. Since it occurs in the stresstensor multiplet, this relevant deformation exists for all three-dimensional N = 4 SCFTs, and we will refer to it as a universal mass. Its existence invalidates the standard lore that supersymmetric deformations necessarily reside at the highest level of a multiplet. (As we will explain in section 2, this lore is correct for suitably generic multiplets.) Similar universal mass deformations, which reside in the middle of stress-tensor multiplets, exist in threedimensional theories with N ≥ 5 supersymmetry. Such deformations are further discussed in section 4.3. As we review there, they lead to exotic deformations of the non-conformal supersymmetry algebra that includes the R-symmetry generators, even though they do not commute with the supercharges.
The main result of this paper is a classification of all Lorentz-invariant, supersymmetric deformations that can arise for SCFTs in 3 ≤ d ≤ 6 dimensions. The full classification is tabulated in section 3, and a brief summary appears in table 1. Even at this level of detail, two unifying themes emerge: 1.) Many theories possess special deformations that reside in multiplets together with conserved currents. We have already mentioned the universal mass deformations for N ≥ 4 9 The standard half-integral su(2) spins are given by  theories were classified in [16,17]. The fact that N = 4 theories in three dimensions and N = (1, 0) theories in six dimensions do not possess marginal deformations was independently found in [27,28], while the absence of relevant or marginal deformations in genuine N = 3 theories in four dimensions was observed in [29].

Outline
In section 2, we explore aspects of long and short superconformal representations, and In section 4, we discuss Lorentz-invariant deformations that reside in superconformal multiplets together with conserved currents, focusing on flavor currents and the stress tensor.
Such deformations can lead to a modified supersymmetry algebra, which may contain central or non-central charges. The latter are particularly interesting, since they naively contradict the supersymmetric extension [30] of the Coleman-Mandula theorem [31]. We also use flavor mass deformations to illustrate the fact that deformations which preserve supersymmetry at leading order need not do so at higher order.

Superconformal Multiplets and Supersymmetric Deformations
As discussed in the introduction, the problem of classifying supersymmetric deformations amounts to identifying top components of superconformal multiplets, i.e. conformal primaries that are annihilated by all Q-supersymmetries up to a total derivative, as in (1.7). Since total spacetime derivatives play no role in this discussion they can be dropped without repercussion so that the Q-supersymmetries anticommute, as in (1.8), As discussed around (1.9), it follows from (2.1) that a superconformal multiplet can only contain a finite number of conformal primaries, which must occur at levels 0 ≤ ℓ ≤ ℓ max , where ℓ max ≤ N Q by Fermi statistics. Thus, every multiplet contains at least one top component, which resides at level ℓ max . In this section we will explore supermultiplets with a unique top component, as well as others that possess multiple top components. This will enable us to precisely formulate our classification scheme for supersymmetric deformations.

Long Multiplets and the Racah-Speiser Algorithm
Long multiplets do not possess any null states, i.e. the supercharges Q i do not satisfy any relations other than Fermi statistics (2.1) when acting on the superconformal primary V. The primary V transforms irreducibly under the Lorentz-and R-symmetry, and the independent conformal primaries at level ℓ of a long multiplet transform in the reducible representation Here R Q is the Lorentz-and R-symmetry representation of the supercharges and ∧ ℓ R Q denotes its ℓ-fold totally antisymmetric wedge power. 10 It follows from the antisymmetry of the wedge power that this multiplet has a unique top component at level Since the maximal wedge power of R Q transforms as a Lorentz and R-symmetry singlet, the top component Q N Q V has the same Lorentz and R-symmetry quantum numbers as the superconformal primary V, but its dimension is ∆ V + 1 2 N Q . The structure of long multiplets is conceptually straightforward. The short multiplets are more complicated. It is useful to use the Racah-Speiser (RS) algorithm for decomposing tensor products of Lie-algebra representations, which was applied to superconformal multiplets in [13,32,23] and plays a crucial role in [19]. Here we will only briefly sketch the algorithm and use it to illustrate various general features of superconformal multiplets. In broad strokes, the RS construction of a long multiplet proceeds as follows: • Select the highest-weight state V h.w. ∈ V of the superconformal primary with respect to both the Lorentz and the R-symmetry.
• At each level ℓ, consider all sequences of ℓ supercharges acting on the highest weight state V h.w. , which are distinct up to rearrangements using (2.1), Adding the Lorentz and R-symmetry weights of the supercharges in (2.4) to those of V h.w. for all such sequences leads to a set of RS trial weights W (ℓ) RS at level ℓ. As long as the representation V is sufficiently large, the RS algorithm states that the highest weights of all irreducible representations that occur in (2.2) are in one-to-one correspondence with the RS trial weights W • When the representation V is too small, the bijection between irreducible subrepresen-tations of (2.2) and RS trial weights in W (ℓ) RS can fail. This happens when one or several trial weights cannot be highest weights of an irreducible representation, because some of their Dynkin labels are negative. In this case the RS algorithm states that these states should be removed, possibly at the expense of also removing other weights from W (ℓ) RS , or adding new ones, according to a precise set of group-theoretic rules.
As a simple example, consider long multiplets in three-dimensional N = 1 SCFTs.
According to (1.4), the R-symmetry is trivial and the Lorentz symmetry is su (2). The supercharges Q α (α = ±) transform in a Lorentz doublet, which we denote as R Q = [1].
(As in (1.11), we use integer-valued su(2) Dynkin labels.) If the Lorentz representation of the superconformal primary V is [n], it can be represented by an n-index symmetric   representation at ℓ = 1 in (2.5) is In order to illustrate this phenomenon, we consider a long multiplet in three-dimensional N = 4 theories. As in the discussion around (1.11), the supercharges Q i,i ′ α transform in the trifundamental [1] (1;1) 1/2 of the su(2) R × su(2) ′ R symmetry and the su(2) Lorentz symmetry. Here i, i ′ , α = ± are doublet indices for the respective su(2)'s. For simplicity, we take the superconformal primary V ∈ [0] (0;0) to be a singlet. We examine the true highest weight states S, O, O ′ of three conformal primaries that occur at levels ℓ = 2, 3, 4 in the multiplet, ℓ = 2 : The fact that some transitions do not occur even through they are consistent with all quantum numbers raises the possibility of accidental top components, which cannot be inferred from the decomposition of a superconformal multiplet into conformal primaries. As we argued around (2.2) above, this does not occur in long multiplets, which have unique top components. We will now examine the short multiplets.

Short Multiplets and Manifest Top Components
Short multiplets possess null states, which must be removed from the representation. In some cases, this can be done by simply dropping some of the supercharges and constructing a long multiplet using the remaining ones. (See for instance [13] for a discussion in four dimensions.) Since the resulting multiplets are essentially long multiplets constructed using a reduced set of Q-supersymmetries, they have unique top components. More generally, the null states lead to (potentially very complicated) relations, which must be solved explicitly. 11 Such multiplets may possess additional top components, which can be categorized as follows: 2.) Accidental Top Components: As discussed after (2.8), these are hypothetical conformal primaries at level ℓ that are mapped into descendants, even though there are conformal primaries at level ℓ + 1 whose quantum numbers occur in the tensor product R Q ⊗ O.
We do not know any examples of such accidental top components, and we suspect they do not exist, but we have not ruled them out systematically. 12 Note that representations with multiple top components do not respect the reflection symmetry of the Clifford algebra (2.1) between occupied and unoccupied levels (i.e. particles and holes), which exchanges the Q-and S-supersymmetries.
In this paper we will only discuss manifest top components, which can be analyzed using the decomposition of a superconformal multiplet into conformal primaries. In the remainder of this section, we illustrate various properties of manifest top components in simple examples.
The vast majority of manifest top components reside at the highest level ℓ max of a multiplet.
To our knowledge, the only Lorentz-invariant deformations that reside in the middle of a multiplet are the universal mass deformations in three-dimensional theories with N ≥ 4 supersymmetry (see the discussion around (1.11) and in section 4.3).
Suitably generic multiplets have a single (generally not Lorentz-invariant) operator at level ℓ max . For long multiplets this was discussed around (2.2) above. Here we consider an example of a generic short multiplet in three-dimensional N = 2 theories, where the supercharges Q α and Q α carry u(1) R charges −1 and +1, i.e. they transform reducibly as [1] ∆ denotes an operator of Lorentz spin 1 2 j ∈ 1 2 Z ≥0 , R-charge r, and scaling dimension ∆. Consider an (0) 1 2 j+1 multiplet, which obeys a shortening condition of type A 1 with respect to Q α (see table 4) and a shortening condition of type A 1 with respect to Q α (see table 5), with generic Lorentz spin and vanishing R-charge, r = 0. The superconformal multiplet decomposes into the following conformal primaries: Note that there is a unique operator at ℓ max = 2. This multiplet exists for any j ≥ 1.
It only contains conserved currents (generally with high spin). Taking into account the conservation laws leads to 4 + 4 independent operators, independent of j. The case j = 2 is the superconformal stress-tensor multiplet.
As the Lorentz and R-symmetry quantum numbers of a short multiplet are specialized to small values, we encounter a host of sporadic phenomena that can result in additional top components. We have analyzed these phenomena on a case-by-case basis, by relying on the explicit construction of unitary superconformal multiplets in [19]. As an example, consider 1 flavor current multiplet in three-dimensional N = 2 theories, which obeys a Qshortening condition of type A 2 (see table 4) and a Q-shortening condition of type A 2 (see table 5). It can be viewed as the specialization of the A 1 A 1 [j] (0) 1 2 j+1 multiplets discussed above to j = 0. As indicated by the subscripts on A and A, the primary null states jump from ℓ = 1 There are now two manifest top components at ℓ max = 2. The conserved flavor current [2] (0  . (For generic R, the multiplet is tabulated in equation (4.37) of [13].) However, when R = 0, 1 this top component disappears, and the multiplet undergoes further shortening, i.e. ℓ max decreases. The case R = 0 is the stress-tensor multiplet, which also contains the conserved R-symmetry and supersymmetry currents; the stress tensor is the unique top component at ℓ max = 4. However, for R = 1 the multiplet has two manifest top components, at ℓ max = 5, as seen in its explicit decomposition into conformal primaries: Using bosonic conformal unitarity bounds (see for instance section 2.5 of [12]), it can be checked that this multiplet does not contain any conserved currents.

Tables of Supersymmetric Deformations
In this section we tabulate all Lorentz-invariant supersymmetric deformations of interacting SCFTs in 3 ≤ d ≤ 6 dimensions. The subsections describing the results for different values of d and N are largely self-contained and can be read independently. In each case we briefly summarize our conventions and review the Lorentz and R-symmetry transformation properties of the supercharges. As was already stated in the introduction, we always use integer-valued Dynkin labels to denote Lie-algebra representations. 13 For each d and N , we summarize the possible unitarity superconformal multiplets, relying on the results of [20,12,21,22,13,23]. We use a streamlined labeling scheme for superconformal representations that uniformly covers all values of d and N . (See [19] for a detailed discussion.) Multiplets are denoted by capital letters that indicate whether they satisfy any shortening conditions. Long multiplets are always denoted by L, while the letters A, B, C, D indicate short multiplets. A-type multiplets exist for all values of d and N . They reside at the threshold to the continuum of long multiplets, and their Lorentz or R-symmetry quantum numbers are not restricted. By contrast, the letters B, C, D denote families of short multiplets that are isolated from the continuum and whose Lorentz or R-symmetry quantum numbers are restricted. The notation is chosen such that A, B, C, D-type multiplets with the same Lorentz and R-symmetry quantum numbers are ordered according to their scaling Short multiplets have null states, which descend from a primary null state whose quantum numbers are uniquely fixed by those of the superconformal primary. We will use a subscript ℓ to denote the level of the primary null state, e.g. A ℓ denotes an A-type shortening condition whose primary null state resides at level ℓ. In d = 4 and in three-dimensional N = 2 theories there are independent Q and Q supercharges, both of which give rise to shortening conditions. In these theories, we denote multiplets by a pair of capital letters (one unbarred and one barred) to indicate the Q, Q null states, e.g. LB 1 or A 1 A 1 .
For every value of d and N , we list the superconformal shortening conditions allowed by unitarity, the possible Lorentz and R-symmetry quantum numbers of the superconformal primary, the restrictions on its scaling dimension imposed by unitarity, and the quantum numbers of the primary null state. In theories with Q and Q supercharges, we independently list the corresponding shortening conditions, which must be combined in a consistent fashion to obtain a sensible superconformal multiplet.
In each case, we then tabulate (and briefly comment on) all Lorentz-invariant supersym- 13 For the rank-r odd and even orthogonal algebras so(2r + 1) and so(2r), the relation between Dynkin labels R i ∈ Z and orthogonal labels h i ∈ 1 2 Z (which are, for instance, used in [23,12]) is given by metric deformations. In these tables, we indicate both the superconformal primary of the multiplet containing the deformation, as well as the deformation itself. Here we would like to make some general comments, which apply for all values of d and N .
• In this section, we only discuss Lorentz-invariant deformations. 14 As can be seen from the tables below, the superconformal primaries of the multiplets that harbor such deformations are also always Lorentz scalars. In order to streamline the presentation, we will therefore omit the (trivial) Lorentz quantum numbers from the deformation tables.
• We shift the quantum numbers of the superconformal primaries by constant offsets, to make the quantum numbers of the deformations as uniform as possible. This facilitates the comparison of deformations that reside in different multiplets.
• The deformations are ordered according to the level at which they reside in their respective multiplets. Every table starts with deformations that reside in the shortest possible multiplets and ends with generalized D-term deformations, which reside in long multiplets.
• For some values of d and N , we find deformations residing in multiplets that also contain additional supersymmetry currents. We will not include such deformations in our tables, since they can be thought of as deformations of a theory with enhanced supersymmetry. Similarly, we will not tabulate deformations residing in multiplets that also contain higher-spin currents, since such theories are expected to be free [34]. See [19] for a systematic discussion of superconformal multiplets with conserved currents.
• Some deformations are related by Hermitian conjugation. We indicate conjugate pairs by including a common symbol, e.g. ( * ) or (⋆), in the 'comments' column of the deformation tables. We similarly indicate deformations that are related by mirror symmetry or so(8) R triality in three-dimensional N = 4 or N = 8 theories.

Three Dimensions
In this subsection we list all Lorentz-invariant deformations of three-dimensional SCFTs with 1 ≤ N ≤ 6 and N = 8 supersymmetry. Unitarity SCFTs with N ≥ 9 exist, but are necessarily free, because the stress-tensor multiplet also contains higher-spin currents [19].
Genuine theories with N = 7 supersymmetry do not exist: they always enhance to N = 8, because the N = 7 stress-tensor multiplet contains eight, rather than seven, supersymmetry currents [35,19]. The pertinent superconformal algebras and their unitary representations are briefly summarized below. (See for instance [12,23,19] and references therein for additional details.) Throughout, representations of the so(3) = su(2) Lorentz algebra are denoted by Here j is an integer-valued su(2) Dynkin label, so that the [j]-representation is (j + 1)dimensional. (The conventional half-integral su(2) spin is j 2 .) We write [j] ∆ whenever we wish to indicate the scaling dimension ∆.
The Q-supersymmetries transform as The superconformal unitarity bounds and shortening conditions are summarized in table 2.

Name Primary
Unitarity Bound Null State  Table 3: Deformations of three-dimensional N = 1 SCFTs.

D-Term
Operators of R-charge r ∈ R are denoted by (r). There are independent Q and Q supersymmetries, which transform as (1/2) 1/2 multiplet, which is annihilated by Q 2 as well as all Q supercharges. Conserved flavor currents reside in an A 2 A 2 [0] (0) 1 multiplet, while the stresstensor multiplet is given by 2 multiplet containing the marginal deformation (and its complex conjugate) can pair up with an Table 6: Deformations of three-dimensional N = 2 SCFTs. Here r ∈ R denotes the u(1) R charge of the deformation.

Primary O Deformation δL Comments
The N = 3 superconformal algebra is osp(3|4), so that there is a so(3) R ≃ su(2) R symmetry. The R-charges are denoted by (R), where R ∈ Z ≥0 is an su(2) R Dynkin label.
The Q-supersymmetries transform in the vector representation 3 of so(3) R , The superconformal unitarity bounds and shortening conditions are summarized in table 7.

Name Primary
Unitarity Bound Null State  Primary O Deformation δL Comments The N = 4 superconformal algebra is osp(4|4), hence the R-symmetry is so(4) R ≃ su(2) R × su(2) ′ R . Its representations are denoted by (R ; R ′ ), where R, R ′ ∈ Z ≥0 are Dynkin labels for su(2) R and su(2) ′ R , respectively. For example, (1; 0) and (0; 1) are the left-and right-handed spinors 2 and 2 ′ of so(4) R , while (1; 1) is its vector representation 4. Note that the su(2) R and su(2) ′ R factors of the R-symmetry algebra are inert under complex conjugation. However, they are exchanged by the action of mirror symmetry M, which is an outer automorphism of the N = 4 superconformal algebra. (It need not be a symmetry of the field theory, although it can be.) The Q-supersymmetries transform as The superconformal unitarity bounds and shortening conditions are summarized in table 9.
1/2 is a free twisted hypermultiplet. The two multiplets are exchanged by the mirror automorphism M. By contrast, the stress-  Primary O  The N = 5 superconformal algebra is osp(5|4) and therefore the R-symmetry is so(5) R . Its representations are denoted by (R 1 , R 2 ), where R 1 , R 2 ∈ Z ≥0 are so(5) R Dynkin labels.

Deformation δL Comments
For example, (1, 0) is the vector representation 5, while (0, 1) is the spinor representation 4. 15 The Q-supersymmetries transform as The superconformal unitarity bounds and shortening conditions are summarized in   Primary O Deformation δL Comments The N = 6 superconformal algebra is osp(6|4) and thus the R-symmetry is so(6) R .

Deformation δL
Comments    , which are also exchanged by T . An irreducible quantum field theory, without locally decoupled sectors, is expected to possess a unique stress tensor (see for instance [34]), and hence only one stress-tensor multiplet. Specifying the stress-tensor multiplet therefore completely fixes the triality frame.

Name Primary
Unitarity Bound Null State  (0,0,1,1) 2 multiplet is distinguished by the fact that it contains an extra supersymmetry current, which enhances N = 8 to N = 9. This is consistent with the fact that N ≥ 9 theories exist, but are necessarily free [19].

Primary O
Deformation δL Comments

Four Dimensions
In this subsection we list all Lorentz-invariant supersymmetric deformations of fourdimensional 1 ≤ N ≤ 4 SCFTs. (Unitary SCFTs with N ≥ 5 do not exist, because they do not possess a stress tensor [19].) The corresponding superconformal algebras and their unitary representations are briefly summarized below. (See for instance [20,12,13,19] and references therein for more detail.) Throughout, representations of the so(4) = su(2) × su (2) Lorentz algebra are denoted by Here j, j are integer-valued su(2) Dynkin labels, so that the representation in (3.9) has dimension (j + 1)(j + 1). We use [ j; j ] ∆ to indicate the Lorentz quantum numbers of an operator with scaling dimension ∆.
Operators of R-charge r ∈ R are denoted by (r). The Q-supersymmetries transform as

Name Primary
Unitarity Bound Q Null State   Primary O The N = 2 superconformal algebra is su(2, 2|2), so that there is a su(2) R × u(1) R symmetry. The R-charges of an operator are denoted by (R ; r), where R ∈ Z ≥0 is an su(2) R Dynkin label, while r ∈ R is the u(1) R charge. The Q-supersymmetries transform as is a multiplet that contains a conserved flavor current. 17 17 In brief, the relation between our labeling scheme (which is somewhat similar to that used in [32]) and

Deformation δL Comments
the labeling scheme of [13] is as follows (see [19] for more detail): is a long multiplet, and the short multiplets are given by (note that some of them are referred to as semi-short in [13]), Analogous relations for the multiplets C R,r(j,) , B R,r(j,0) , E r(j,0) , and D R(j,0) can be obtained by complex conjugation.   up with other multiplets into a long multiplet (see for instance [13,32]). There are also irrelevant 1 4 -BPS deformations that reside in generic LB 1 and B 1 L multiplets. (As indicated by the symbol ( †) in table 22, they are related by complex conjugation.)

Deformation δL
Comments Table 22: Deformations of four-dimensional N = 2 SCFTs. The su(2) R Dynkin label R ∈ Z ≥0 and the u(1) R charge r ∈ R denote the R-symmetry representation of the deformation.
The u(1) R charge is given by r ∈ R. The Q-supersymmetries transform as (3.12) 18 A standard argument (based on the single-particle representations of the N -extended super-Poincaré algebras and the CPT theorem, see for instance [36]) shows that weakly coupled N = 3 SCFTs must actually be N = 4 theories. However, no known argument rules out the existence of strongly-coupled N = 3 SCFTs that are not N = 4 theories. Aspects of such theories were recently discussed in [29,37]. is the stress-tensor multiplet.

Name
Primary  as was also observed in [29]. However, there is a rich variety of irrelevant supersymmetric deformations, many of which reside in short multiplets. Pairs of multiplets that share one of the symbols ( * ), (⋆), ( †), ( ‡) are related by complex conjugation.

Primary O
Deformation δL Comments Term   Table 25: Deformations of four-dimensional N = 3 SCFTs. The su(3) R Dynkin labels R 1 , R 2 ∈ Z ≥0 and the u(1) R charge r ∈ R denote the R-symmetry representation of the deformation.
The R-charges are denoted by su(4) R Dynkin labels (R 1 , For instance, the stress-tensor multiplet is given by

Deformation δL
Comments Stress Tensor

Five Dimensions
In this subsection we list all Lorentz-invariant supersymmetric deformations of fivedimensional SCFTs. The unique superconformal algebra in five dimensions is f(4) and corresponds to N = 1 supersymmetry. The Lorentz algebra is so(5) = sp(4) and the R-symmetry is su(2) R . Lorentz representations are denoted by sp(4) Dynkin labels j 1 , j 2 ∈ Z ≥0 , e.g. [1,0] and [0, 1] are the spinor 4 and the vector 5 representations of so (5). The R-charges are denoted by (R), where R ∈ Z ≥0 is an su(2) R Dynkin label. The quantum numbers of an operator with scaling dimension ∆ are indicated as follows, 14) The Q-supersymmetries transform as The superconformal unitarity bounds and shortening conditions are summarized in table 29.

Name
Primary Unitarity Bound Null State

Primary O
Deformation δL Comments

Six Dimensions
In this subsection we list all Lorentz-invariant deformations of six-dimensional SCFTs with (N , 0) supersymmetry for N = 1, 2. (Unitarity SCFTs with N ≥ 3 do not exist, because they do not admit a stress tensor [19].) The corresponding superconformal algebras and their unitary representations are briefly summarized below. (See for instance [12,[21][22][23]19] and references therein for additional details.) Representations of the so(6) = su(4) Lorentz algebra are denoted using su(4) Dynkin labels, The N = (1, 0) superconformal algebra is osp(8|2), hence the R-symmetry is sp(2) R ≃ su(2) R . Its representations are denoted by (R), where R ∈ Z ≥0 is an su(2) R Dynkin label. 19 As representations of su(4), the 4 ′ is typically denoted by 4, which is related to the 4 by complex conjugation. However, in six-dimensional Minkowski space, chiral spinors are not related by complex conjugation.

Name
Primary Unitarity Bound Null State Therefore, the only possible supersymmetric RG flows out of these fixed points are triggered by moving onto a moduli space of vacua [38]. The fact that there are no marginal deformations was also discussed in [28].
Primary O  is the stress-tensor multiplet.

Name
Primary Unitarity Bound Null State  Primary O Deformation δL Comments D-Term Table 34: Deformations of six-dimensional N = (2, 0) SCFTs. The R-symmetry representation of the deformation is denoted by the sp(4) R Dynkin labels R 1 , R 2 ∈ Z ≥0 .

Deformations Related To Conserved Currents
In this section we discuss Lorentz-invariant supersymmetric deformations which reside in supermultiplets that also contain conserved currents, focusing on flavor currents and the stress tensor. (As in section 3, we will not discuss multiplets containing additional supersymmetry currents or higher spin currents. A detailed analysis of all superconformal multiplets that contain conserved currents can be found in [19].) Such deformations can lead to the appearance of additional bosonic charges in the supersymmetry algebra, which arise from the currents that reside in the same multiplet as the deformation. We also comment on the fact that some of these deformations can fail to be supersymmetric at higher order, even though they preserve supersymmetry at leading order.

Flavor Current Multiplets
Using 3 flavor current multiplet. All of these deformations arise in myriad well-studied examples. In addition to standard mass terms for scalars and fermions in matter multiplets, which are possible in all cases, we also mention the following two interesting possibilities in three and five dimensions: • In three dimensions, FI-terms for abelian gauge fields can be interpreted as flavor masses for the topological current ⋆F , where F the abelian field strength. (By contrast, deformations of SCFTs in 4 ≤ d ≤ 6 dimensions do not include FI-terms, see section 5.2.) Abstractly, such FI flavor masses cannot be distinguished from conventional flavor masses for matter fields, as reflected by the fact that the two can be exchanged by duality (see for instance [39,40]).
• In five dimensions, many SCFTs possess flavor symmetries. Upon activating the corresponding flavor masses, such theories often flow to weakly-coupled, generally nonabelian, gauge theories in the IR. Each gauge group gives rise to a topological current ⋆ Tr(F ∧ F ), which descends from a particular flavor current of the SCFT in the UV. The corresponding Yang-Mills kinetic term descends from the flavor mass deformation related to that UV flavor current (see for instance [41][42][43][44]).
In all cases discussed above, the Lorentz-invariant flavor mass deformation resides at level two in a flavor current multiplet whose superconformal primary J a,I is also a Lorentz scalar. (Here I collectively denotes all R-symmetry labels.) Schematically, we can therefore write the deformation as follows,

Stress Tensor Multiplets
We . They reside at the top of the multiplet, at level ℓ = 4, together with the stress tensor T µν .
• Three-dimensional SCFTs with N ≥ 4 supersymmetry have universal relevant deformations M univ. of dimension ∆ = 2. In analogy with the flavor masses M a flav. discussed in section 4.1, we will refer to them as universal mass deformations. They reside in the middle of the stress-tensor multiplet, at level ℓ = 2, while T µν resides at level ℓ = 4.
The M univ. are generally charged under the R-symmetry (see below), but since they reside in the same multiplet as T µν , they are neutral under any flavor symmetries.
We will now comment on these two kinds of universal deformations in turn.
Since the marginal deformation O in four-dimensional N = 4 theories resides in the same multiplet as T µν , many of its properties follow directly from superconformal Ward identities (see for instance [47][48][49] and references therein). First, the deformation is necessarily exactly marginal, because its dimension is tethered to that of the stress tensor. 20 It therefore gives rise to a conformal manifold labeled by one complex parameter τ . The local geometry of this manifold is fixed by Ward identities, which imply that its Zamolodchikov metric has constant negative curvature, Here C is proportional to the stress tensor two-point function, which is determined by the As a simple example, we write down the universal mass deformation for a free N = 4 (1;0) 1/2 multiplet. The scalars H i and the fermions ψ i ′ α are doublets under the first and second factors of the su(2) R × su(2) ′ R symmetry. The action of the supercharges on the scalars is given by where the ∼ indicates that we are not keeping track of convention-dependent proportionality factors. The universal mass deformation resides at level two of the A 2 [0] (0;0) 1 stress tensor multiplet, whose bottom component is ε ij H i H j . According to (4.3), the universal mass is R-symmetry (as well as Lorentz) invariant. Explicitly, where m is a real mass parameter. Note that the two supercharges are contracted to a Lorentz and R-symmetry singlet. As was the case for flavor masses (see (4 flavor current multiplets, which are exchanged by mirror symmetry, the universal mass is inert under the mirror automorphism. In this example, the universal mass (4.4) leads to a fully gapped theory.
As we will review below, this is a general, model-independent property of all universal mass deformations.

Deformed Stress-Tensor Multiplets and Supersymmetry Algebras
In an SCFT, the stress tensor T µν resides in a short superconformal multiplet (whose structure essentially only depends on d and N ), together with the N Q supersymmetry currents S i µα and the R-symmetry currents. The multiplet is typically completed by other operators that need not be conserved currents. At the conformal point, both the stress tensor and the supersymmetry currents are traceless. Schematically, where we have used four-dimensional notation to indicate the spin-1 2 projection of the supersymmetry currents in all dimensions. The vanishing traces in (4.5) allow the definition of dilatation and special conformal generators, as well as the superconformal S-supersymmetries.
See [19] for a unified discussion of superconformal stress-tensor multiplets for all d and N .
When a CFT is deformed by a scalar operator O of dimension ∆, i.e. δL = λO, then the trace of the stress tensor is deformed to T µ µ ∼ λ (∆ − d) O, at leading order in λ. (More generally, the coefficient of O in T µ µ is proportional to the β-function of the coupling λ.) If the deformation O is marginal, the deformation preserves conformal symmetry at leading order and T µ µ remains zero at that order. (A non-zero trace, i.e. a beta function, may be generated at higher order.) In all other cases, conformal symmetry is broken and T µν acquires a trace. For the case of an SCFT deformed by a Lorentz-scalar operator O that preserves the Q-supersymmetries but breaks conformal symmetry (and hence the S-supersymmetries), the currents T µν , S i µα remain conserved, but both of the traces that were set to zero in (4.5) are activated by the deformation. Together with their superpartners, they deform the conformal stress-tensor multiplet into a multiplet of Poincaré supersymmetry that contains the currents T µν , S i µα . Typically, it contains more operators than the conformal stress-tensor multiplet, which are supplied by the multiplet of the deformation O. (However, operators from other multiplets can also participate, see below.) As an example, consider four-dimensional N = 1 theories. At the superconformal point, the stress-tensor resides in an A 1 A 1 [0; 0] (0) 2 multiplet (see tables 17 and 18), whose primary J µ = σ αα µ J αα is the u(1) R current. The superconformal shortening conditions take the form After using these to eliminate all null states (including conservation laws), the multiplet has 8 + 8 bosonic and fermionic operators: the conserved R-current J µ , as well as the conserved and traceless supersymmetry currents S µα , S µα and stress-tensor T µν .
Unlike the superconformal case, the representation theory of the Poincaré supersymmetry algebra admits several different non-conformal stress-tensor multiplets. (See [57,58] and references therein for a detailed discussion.) The most common such multiplet is Ferrara-Zumino (FZ) multiplet. In addition to J αα , it also contains a chiral submultiplet X (and its conjugate anti-chiral multiplet X). The superconformal shortening condition (4.6) is deformed to The 4 + 4 bosonic and fermionic operators in X combine with the 8 + 8 operators in the superconformal multiplet to make a 12 + 12 multiplet. The bottom component J µ = σ αα µ J αα is no longer a conserved current; its non-zero divergence is one of the additional operators residing in X. Therefore, the FZ-multiplet only contains a conserved R-current if X vanishes and the theory is superconformal. If a non-conformal N = 1 theory has a u(1) R symmetry, the associated R-current resides in a different 12 + 12 stress-tensor multiplet, which is related to the FZ-multiplet by an improvement transformation.
The vast majority of N = 1 theories admit an FZ-multiplet. The only known exceptions are models with FI-terms, or theories that can be obtained from such models by RG flow, e.g. sigma models whose target spaces contain compact two-cycles [59,57,58]. Since supersymmetry deformations of N = 1 superconformal theories never give rise to FI-terms (see the more detailed discussion in section 5.2), we expect that all deformed N = 1 SCFTs possess an FZ stress-tensor multiplet, whose X submultiplet is determined by the deformation. 21 We will now consider the effect of flavor mass deformations (see section 4.1) on the stresstensor multiplet. For simplicity, we will focus on theories with N Q = 8 supercharges in d = 5, 4, 3 dimensions, where the discussion is fairly uniform. 22 As explained below (4.1), flavor currents in these theories are conveniently understood in terms of dimensional reduction, staring with a D 1 [0, 0, 0]  (see table 9). Explicitly, the primaries and shortening conditions of these flavor current multiplets are In three dimensions, the symmetrization of the su(2) R doublet indices i, j and the su(2) ′ R 21 In a weakly-coupled theory of chiral superfields with superpotential W and Kähler potential K, the chiral operator X in the FZ multiplet is a linear combination of W and Q 2 K. Using conformal perturbation theory, it should be possible to find an analogous expression for X in any N = 1 SCFT that has been deformed by general F -and D-terms (see table 19), but we will not pursue it here. 22 See [58] for a discussion of stress-tensor multiplets in three-dimensional N = 2 theories, including the effects of flavor masses. 23 In five dimensions, spinors are contracted using the sp(4)-invariant symplectic matrix Ω αβ .
doublet indices i ′ , j ′ denotes a projection onto the representation with Dynkin labels (2 ; 2). This perspective immediately suggests the form of the non-conformal stress-tensor multiplet in the presence of flavor mass deformations (4.1), since they can be viewed as Wilson lines that wrap the reduced directions (see the discussion below (4.1)), and hence contribute to the momentum in those directions. We simply interpret a suitable linear combination of the flavor currents J ij a as the KK current, which no longer decouples in the presence of flavor mass deformations. This logic leads to a non-conformal stress-tensor multiplet in d = 5, 4, 3 that was first described in [60] for N = 2 theories in four dimensions, (4.10c) Here the m a are real and R-neutral in d = 5, while they are complex (with complex conjugates m a ) and charged under U(1) R in d = 4. In three dimensions, both m i ′ j ′ a and (m ′ ) ij a ′ are real and transform as triplets of su(2) ′ R and su(2) R , respectively. The conserved flavor currents on the right-hand sides of (4.10) integrate to Lorentz-scalar central charges in the supersymmetry algebra, which is therefore deformed. For instance, in five dimensions they lead to a term Ω αβ ε ij Z ⊂ {Q i α , Q j β }, where the real central charge Z = m a F a is determined by the masses m a and the flavor charges F a corresponding to the currents J ij a . This is consistent with the interpretation of m a J ij a as the KK current, since the central charge Z can be identified with the momentum in the reduced direction.
An even more drastic modification of the supersymmetry algebra takes place in the presence of universal mass deformations in three-dimensional theories with N ≥ 4 supersymmetry. As discussed in section 4.3, the universal mass deformation resides in the stress-tensor multiplet. This leads to an unusual deformation of the supersymmetry algebra by non-central terms proportional to the unbroken R-symmetry generators, see [53,54,56]. For instance, in N = 4 theories the universal mass m preserves the entire su(2) R × su(2) ′ R symmetry, with generators R ij , (R ′ ) i ′ j ′ , so that the deformed algebra takes the form In interacting theories, the appearance of the non-central R-symmetry generators on the right-hand side explicitly contradicts the supersymmetric extension [30] of the Coleman-Mandula theorem [31], which follows from certain analyticity assumptions on the S-matrix.
(In free theories, such as (4.4), the S-matrix is trivial and there is no contradiction.) As we will review below, the deformed theory is necessarily gapped. However, it may contain massive anyons, which can lead to an S-matrix with non-standard analytic properties (see for instance [61] for a recent discussion). In Lagrangian theories based on hypermultiplets interacting with gauge fields, the universal mass deformation triggers same-sign real mass terms for all hypermultiplet fermions, as in (4.4), which leads to a gapped theory with induced Chern-Simons terms in the deep IR.
It follows directly from the algebra (4.11) that N = 4 theories with a universal mass deformation are gapped, i.e. the low-energy effective theory must be empty or topological and does not contain massless particles. To see this, we follow [54] and consider a massless particle with lightcone momentum P + = E and P − = P 3 = 0, on which (4.11) reduces to In Lorentzian signature, the supercharges Q ii ′ − are Hermitian, so (4.12b) implies that they are represented trivially, Q ii ′ − = 0. It then follows from (4.12c) that the R-charges R ij , (R ′ ) i ′ j ′ must also act trivially. Since Q ii ′ + transforms as a bifundamental under these generators, this is only consistent if Q ii ′ + = 0, so that the entire representation is trivial, and in particular E = 0. Therefore, the only massless single-particle states are vacua, and hence the theory is gapped.
The argument straightforwardly generalizes to all N ≥ 5 theories with universal masses.

Comments on Preserving Supersymmetry at Higher Order
Throughout this paper, we have focused on deformations δL = λO (1) that preserve supersymmetry at leading order in the deformation parameter λ, i.e. the operator O (1) is annihilated by the Q-supercharges of the undeformed theory at λ = 0, up to a total derivative. However, once the deformation has been activated, it typically does not preserve supersym-metry at O(λ 2 ). Resorting to weakly-coupled, Lagrangian intuition, this is due to the fact that we used the equations of motion of the undeformed theory to show that O (1) is annihilated by the undeformed supercharges. Therefore the supercharges themselves are corrected at O(λ) and may no longer annihilate O (1) , leading to a remainder term at O(λ 2 ). If this term can be cancelled by the supersymmetry variation of a local operator O (2) , then the following Lagrangian is supersymmetric up to and including O(λ 2 ), (4.13) If such an operator O (2) does not exist, then δL breaks supersymmetry at O(λ 2 ), even though the leading-order deformation O (1) was supersymmetric. (An explicit example where this happens will be discussed below.) Even if O (2) exists, the deformation δL may fail to be supersymmetric at O(λ 3 ). We must then repeat the procedure and look for a local operator O (3) that can be added to δL to preserve supersymmetry at this order. This procedure may terminate after a finite number of steps, or it can continue indefinitely. A simple example in the former category arises by deforming the free SCFT consisting of a single chiral superfield Φ = (φ, ψ α ) by a superpotential λW (Φ). 24 The O(λ) leading deformation takes the form (see table 19), where the ∼ means that we are omitting convention-dependent numerical factors. We see from (4.14) that the O(λ) deformation O (1) only contains the Yukawa couplings, but not the scalar potential. The latter must be added to restore supersymmetry at O(λ 2 ), i.e. we must choose O (2) ∼ |W ′ (φ)| 2 , after which the procedure terminates. A deformation that requires corrections at all orders in λ is the Born-Infeld-like higher-derivative F -term that arises on the Coulomb branch of four-dimensional N = 4 Yang-Mills theories (see section 5.1.2).
We will not attempt to systematically determine which deformations preserve supersymmetry beyond leading order. 25 Rather, we will use the flavor mass deformations discussed in sections 4.1 and 4.3 above to give a simple example of an allowed leading-order deformation that breaks supersymmetry at second order. Following the discussion around (4.1), we con- 24 Here we are using an on-shell formalism without auxiliary fields, since it more closely resembles the situation encountered when deforming an abstract SCFT. 25 One possible approach, which was mentioned in [62], is to examine the OPE of two or more leading-order deformations O (1) . The supersymmetry Ward identities may require the presence of certain operator-valued contact terms that can be identified with the higher-order corrections O (n≥2) to the deformation. where f abc are the totally antisymmetric structure constants of the flavor Lie algebra g.
Therefore, supersymmetry requires all of the m a,I to reside in a Cartan subalgebra of g. This requirement is trivial in d = 5, but not so in d = 4, 3. For instance, if f abc m a,− m +,b = 0 in four dimensions, then N = 2 supersymmetry is explicitly broken to N = 1 [46]. Note that this is a quadratic constraint, which occurs at second order in the deformation parameters m a,I , all of which preserve supersymmetry at leading order.

Constraints on Moduli-Space Effective Actions
Many supersymmetric theories in d > 2 dimensions have a moduli space of vacua M that is parametrized by the expectation values of massless, scalar moduli fields Φ I , which are nearly free in the deep IR. The low-energy effective Lagrangian L M on the moduli space includes a sigma model for the Φ I with target space M, whose kinetic terms determine the metric g IJ (Φ) on the moduli space. 27 The moduli-space Lagrangian L M is constrained by supersymmetry, and the constraints depend on the supermultiplets in which the scalars Φ I reside. (Such constraints are often referred to as non-renormalization theorems.) In this subsection we will explain how our results on irrelevant supersymmetric deformations can be used to understand the supersymmetry constraints on L M , including the moduli-space metric g IJ (Φ) (section 5.1.1), as well as higher-derivative terms (section 5.1.2). We will follow the standard scaling rules for moduli-space effective actions, which assign scaling weight 0 to the moduli Φ I and weight 1 2 to the Q-supercharges, so that a derivative ∂ µ has weight 1. This fixes the scaling weights of all superpartners of the Φ I .
Our starting point is an expression for L M as a sum of terms with moduli-dependent coefficient functions f i Φ multiplying operators O i (constructed from Φ I and its superpartners) that can be organized according to their scaling weight, 28 Here we have explicitly indicated the kinetic terms of the Φ I , which determine the moduli space metric g IJ (Φ). They carry scaling weight 2 and are often the terms of lowest scaling weight in (5.1). 29 We would like to understand how the functions f i (Φ) and the possible O i are constrained by supersymmetry. Our strategy will be to focus on a neighborhood of a point Φ I on M and to consider the fluctuations δΦ I around that point. In the deep IR, the δΦ I are free fields with canonical kinetic terms, and we will reorganize the modulispace Lagrangian in terms of higher-derivative, irrelevant corrections to this free theory. We therefore expand 2) and substitute into (5.1). This leads to a series of irrelevant operators in the fluctuations δΦ I and their superpartners, such as If such a term cannot be interpreted as an irrelevant supersymmetric deformation of the free IR theory consisting of the δΦ I and their superpartners, it must be absent, which leads to a differential constraint on the coefficient functions f i (Φ). Some higher-order terms that should be absent according to this rule can in fact be generated by supersymmetrically completing a lower-order deformation, as in the discussion around (4.13). In this case the higher-order deformation is not independent: its coefficient is completely determined by the lower-order deformation that induces it. Some examples of this phenomenon are described in [62,38]. 28 We can also contemplate couplings of the moduli fields to other degrees of freedom that may be present at low energy, but we will not do it here. 29 Some supersymmetric theories allow scalar potentials (e.g. superpotentials in four-dimensional N = 1 theories), which carry weight 0. Note that such terms can lift all or part of the moduli space.
The preceding discussion offers an alternative perspective on known moduli-space nonrenormalization theorems, and can be used to derive new ones. We will discuss several examples below, relying on our understanding of irrelevant supersymmetric deformations in free SCFTs. In some cases, the free IR theory of the fluctuations δΦ I is not an SCFT. For instance, this happens on the Coulomb branches of maximally supersymmetric Yang-Mills theories in d ≥ 5, because free vector fields in these dimensions are not conformally invariant (see for instance [63]). In such cases, our classification of irrelevant deformations does not apply and must be worked out separately. 30

Kinetic Terms and the Moduli-Space Metric
We will now apply the general procedure outlined above to constrain the weight-2 kinetic terms of the moduli-space sigma model in (5.1), and hence the moduli-space metric g IJ (Φ).
Expanding in Riemann normal coordinates δΦ I around a point Φ I , we can express where R IKJL is the Riemann curvature tensor, evaluated at the point Φ I , and the ellipsis denotes terms with five or more powers of δΦ I . The term proportional to R IKJL contains four powers of δΦ I and two derivatives, i.e. it has weight 2. It must therefore be accounted for by a weight-2 deformation tabulated in section 3 that involves a product of four fields.
(In a free theory, the number of fields is preserved by the action of the supercharges.) In the examples discussed below, the number of fields is simply related to the R-symmetry quantum numbers of the superconformal primary that gives rise to the deformation.
By examining the tables in section 3, we see that a large class of theories does not admit any supersymmetric deformations that satisfy these requirements: • In three-dimensional N = 5 theories (see table 12) all deformations involving four or more fields take the form Q n O, with n ≥ 6, and therefore have weight ≥ 3. They can therefore not account for the term proportional to the Riemann tensor in (5.4), and hence the moduli-space metric must be flat.
• Repeating the argument for N = 6, 8 theories in three dimensions (see tables 14 and 15), and N = 3 theories in four dimensions (see table 25) immediately shows that these theories must also have flat moduli-space metrics. 30 For maximally supersymmetric Yang-Mills theories in all dimensions, this was done in [64][65][66].
• Four-dimensional N = 4 theories have a flat metric, because there are no weight-2 terms with four fields, but they do admit weight-2 terms with two fields (see table 28). These are the exactly marginal deformations that change the gauge coupling multiplying the kinetic terms. The latter have been canonically normalized in (5.4).
• The metric on the tensor branch of six-dimensional N = (1, 0) theories must also be flat. Although there is a candidate F -term deformation of weight-2 that involves four fields (see table 32), it requires fields that carry su(2) R charge, e.g. hypermultiplets (see below). By contrast, the tensor-multiplet scalars are R-symmetry neutral, because they reside in a C 2 [0, 0, 0] 2 multiplet (see table 31).
• In six-dimensional N = (2, 0) theories, the entire moduli space must be flat, because there are no candidate weight-2 deformations (see table 34).
Intuitively, tensor branches in six dimensions must have flat metrics because self-dual twoform gauge fields do not admit continuous (and hence also not moduli-dependent) couplings.
Theories with less supersymmetry allow richer possibilities for the moduli-space metric.
For instance, it is a classic result [67] that g IJ (Φ) must be Kähler in theories with N Q = 4 supercharges. In four-dimensional N = 2 theories, the Coulomb branch is parametrized by complex scalars Φ I residing in A 2 B 1 [0; 0] and their complex conjugates. Supersymmetry requires the Coulomb-branch metric to obey the constraints of rigid special Kähler geometry (see for instance [68,69] and references therein). This can be understood in terms of the fact that the only weight-2 deformations that can be constructed out of the su(2) R neutral vector multiplets are chiral and antichiral F -terms residing in LB 1 and B 1 L multiplets (see table 22). The bottom components of these multiplets furnish the holomorphic prepotential and its anti-holomorphic complex conjugate that give rise to special geometry.
Finally, we will discuss the geometry of Higgs branches parametrized by hypermultiplets, which exist in all theories with N Q = 8 supercharges. 31 It is well-known that supersymmetry requires the Higgs-branch metric to be hyperkähler [70,71]. A hyperkähler manifold of quaternionic dimension n has real dimension 4n, and the Riemannian holonomy of its metric must lie in Sp(2n) ⊂ SO(4n). Here we will briefly outline how the hyperkähler constraint arises from the perspective of the free hypermultiplets that constitute the deep IR of the Higgs-branch effective theory. 31 In d = 6, 5, 4, 3 they reside in D 1 [0, 0, 0] (1;0) 1/2 multiplets (see tables 31,29,20,21,9). In three-dimensional N = 4 theories, there are also B 1 [0] (0;1) 1/2 twisted hypermultiplets, which are related to conventional hypermultiplets by mirror symmetry.
For simplicity, we will focus on Higgs branches of rank one, which are described by a single hypermultiplet H i and its complex conjugate H i = H i . Here i = 1, 2 is an su(2) R doublet index, which is raised and lowered with the su(2) R invariant ε-symbol. The real target space index I which appears in (5.4) is now replaced by a pair i, i of complex indices.
Here i is also an su(2) R doublet index, which refers to the components of H i . The barred and unbarred indices transform in the same way under su(2) R , so that ε ij is also an invariant symbol. For instance, the kinetic terms in (5.4) are now proportional to ε ij ∂ µ H i ∂ µ H j . Note that the indices i, i are not standard holomorphic indices on the Higgs branch, because the corresponding Kähler form is proportional to ε ij and hence su(2) R invariant. By contrast, the usual su(2) R triplet of hyperkähler forms is proportional to the Pauli matrices σ a ij . By examining tables 32, 30, 22, 10, which list the supersymmetric deformations of theories with N Q = 8 supercharges in d = 6, 5, 4, 3 dimensions, we see that these theories admit a unique irrelevant deformation of weight 2 constructed out of four hypermultiplet fields. In each case, the deformation is obtained by acting with four Q-supercharges on four hypermultiplet scalars in a totally symmetric representation of the su(2) R symmetry with Dynkin label (4), while the deformation itself is an su(2) R singlet. 32 Comparing with the normal coordinate expansion in (5.4), we see that the Riemann tensor must be constructed using combinations of su(2) R invariant ε-symbols, in accord with its usual algebraic symmetry properties. This leads to (here ∼ means that we are not keeping track of numerical coefficients) together with similar expressions for other components, which are related to those in (5.5) by complex conjugation or algebraic symmetries. The highly constrained form of the Riemann tensor in (5.5) implies that the local holonomy of the metric lies in su(2) ≃ sp(2) ⊂ so(4).
This is reflected by the fact that the Riemann tensor -viewed as a map from two-forms to two-forms -annihilates the su(2) R triplet of hyperkähler forms proportional to σ a ij (see for instance [72]). Therefore, the metric is locally hyperkähler. Note that our local analysis does not allow us to conclude that the Higgs-branch metric should be globally hyperkähler, as is in fact required by supersymmetry [70,71]. 32 In d = 6, 5, 4, 3 dimensions, these deformations are the unique top components of D 1 [0, 0, 0]

Higher Derivative Terms
The approach to moduli-space effective actions described above can also be used to constrain higher-derivative terms. We will focus on a representative example: the four-derivative Born-Infeld-like deformation that exists on the Coulomb branch of four-dimensional N = 4 theories. 33 Perhaps surprisingly, these terms are strongly constrained by supersymmetry, as was first observed in the context of BFSS matrix quantum mechanics [73].
For simplicity, we will consider a Coulomb branch of rank one, which is described by a free abelian N = 4 vector multiplet, which constitutes a B 1 B 1 [0; 0] (0,1,0) 1 representation of the superconformal algebra (see tables 26 and 27). In particular, the moduli space is parametrized by six real scalars Φ I , which transform in the vector representation 6 of the so(6) R ≃ su(4) R symmetry. Schematically, the term of interest takes the following form, where F denotes the abelian field strength and the ellipsis indicates terms with fermions. For our purposes, it will be sufficient to know that we are looking for a term of weight 4 (there are four derivatives and four abelian field-strengths) that involves four fields. By comparing with  In order to determine the function f (ϕ), we expand (5.6) in fluctuations δΦ I around a fixed expectation value Φ I . This leads to an infinite number of terms of the schematic form ∂ I 1 ∂ I 2 · · · ∂ I n f | Φ δΦ I 1 δΦ I 2 · · · δΦ I n F 4 + (∂Φ) 4 + · · · . (5.7) All of these terms have weight 4 and involve n + 4 fields. It follows from table 28 that the only such deformation is an F -term that arises by acting with Q 4 Q 4 on a B 1 B 1 [0; 0] (0,n+4,0) n+4 multiplet constructed out of n + 4 symmetrized vector-multiplet scalars. The deformation itself has R-symmetry quantum numbers (0, n, 0), i.e. it is a totally symmetric, traceless tensor of so(6) R . This immediately implies that the coefficients ∂ I 1 ∂ I 2 · · · ∂ I n f | Φ must also be traceless. Taking n = 2, we obtain δ IJ ∂ I ∂ J f (Φ) = 0 . (5.8) Therefore f (Φ) is a harmonic function, as was first pointed out in a quantum mechanical context [73]. Since f (Φ) only depends on the radial variable ϕ = δ IJ Φ I Φ J , it is fixed in terms of two constants f (ϕ) = A + B ϕ 4 . (5.9) Note that the constant A is dimensionful and must vanish on the moduli space of a conformal theory. However, our discussion did not assume that the theory whose Coulomb branch we are discussing is conformal. 34

Fayet-Iliopoulos Terms
In this section, we use the classification of supersymmetric deformations in section 3 to comment on the status of Fayet-Iliopoulos (FI) terms in different dimensions. As we will see, FI-terms cannot arise as deformations of SCFTs unless d = 3. These results complement the restrictions on field-theoretic FI-terms discussed in [59,57,58]. We will only consider theories The R-symmetry representation of the FI-parameter ξ is conjugate to that of the D-term, e.g. it is an R-symmetry singlet in four-dimensional N = 1 theories and an su(2) R triplet in four-dimensional N = 2 theories.
Since the D-term resides in a vector multiplet, and at the same level as the abelian field strength F , its scaling dimension is ∆ = 2 in every spacetime dimension. By examining tables 19, 30, 32, we see that SCFTs in d = 4, 5, 6 do not admit relevant deformations of scaling dimension ∆ = 2. Thus, despite their common appearance in supersymmetric theories with abelian gauge fields, FI-terms cannot arise as deformations of UV-complete SCFTs. Conversely, an abelian gauge theory with FI-terms cannot have a UV fixed point. In d = 4, these statements can also be understood from the following alternative point of view: interacting CFTs with abelian gauge fields necessarily require electrically and magnetically charged de-grees of freedom [79]. However, it was shown in [58] that the magnetic current identically vanishes in theories with FI-terms. This leaves only the free abelian vector multiplet, for which the deformation (5.10) vanishes on-shell and can be removed by a field redefinition, up to an innocuous shift of the vacuum energy.
In three dimensions, the situation is different: as can be seen from tables 3, 6, 8, 10, 12, 14, 16, all SCFTs in d = 3 admit relevant deformations of scaling dimension ∆ = 2, and hence the FI-term is not ruled out. On the contrary, FI-terms exist and can be interpreted as flavor mass deformations associated with the topological current ⋆F , as was already discussed in section 4.1.

Lorentz Non-Invariant Deformations
Throughout this paper, we have focused on deformations that preserve the full super-Poincaré algebra. We can use the same techniques to enumerate deformations that preserve supersymmetry but break Lorentz invariance. These are much less restricted, e.g. all operators at the highest level ℓ = ℓ max of any multiplet furnish such deformations. Here we briefly mention a well-studied example, which arises in the context of non-commutative gauge theories (see [80] and references therein). Using the Seiberg-Witten map [81], gauge theories on a non-commutative geometry with coordinates x µ , x ν = iθ µν can be described as ordinary gauge theories deformed by a series of irrelevant operators weighted by powers of θ µν . These operators break Lorentz-invariance, but they may preserve supersymmetry. For instance, the leading non-commutative deformation of a four-dimensional N = 4 gauge theory is (see [51]), where θ αβ and θαβ are the self-dual and anti-self-dual components of θ µν , and O is a Poincaré supersymmetries.