The \( \mathcal{N}=2 \) superconformal bootstrap

Journal of High Energy Physics, Mar 2016

In this work we initiate the conformal bootstrap program for \( \mathcal{N}=2 \) super-conformal field theories in four dimensions. We promote an abstract operator-algebraic viewpoint in order to unify the description of Lagrangian and non-Lagrangian theories, and formulate various conjectures concerning the landscape of theories. We analyze in detail the four-point functions of flavor symmetry current multiplets and of \( \mathcal{N}=2 \) chiral operators. For both correlation functions we review the solution of the superconformal Ward identities and describe their superconformal block decompositions. This provides the foundation for an extensive numerical analysis discussed in the second half of the paper. We find a large number of constraints for operator dimensions, OPE coefficients, and central charges that must hold for any \( \mathcal{N}=2 \) superconformal field theory.

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The \( \mathcal{N}=2 \) superconformal bootstrap

Accepted: March = 2 superconformal bootstrap Christopher Beem 0 1 3 6 7 Madalena Lemos 0 1 3 4 7 Pedro Liendo 0 1 3 5 7 Leonardo Rastelli 0 1 3 4 7 Balt C. van Rees 0 1 2 3 7 Extended Supersymmetry, Supersymmetric gauge theory 0 Zum Gro en Windkanal 6 , 12489 Berlin , Germany 1 Stony Brook , NY 11794-3840 , U.S.A 2 Theory Group, Physics Department , CERN 3 Princeton , NJ 08540 , U.S.A 4 C. N. Yang Institute for Theoretical Physics, Stony Brook University 5 IMIP, Humboldt-Universitat zu Berlin, IRIS Adlershof 6 Institute for Advanced Study , Einstein Drive 7 CH-1211 Geneva 23 , Switzerland In this work we initiate the conformal bootstrap program for N = 2 superconformal eld theories in four dimensions. We promote an abstract operator-algebraic viewpoint in order to unify the description of Lagrangian and non-Lagrangian theories, and formulate various conjectures concerning the landscape of theories. We analyze in detail the four-point functions of avor symmetry current multiplets and of N = 2 chiral operators. For both correlation functions we review the solution of the superconformal Ward identities and describe their superconformal block decompositions. This provides the foundation for an extensive numerical analysis discussed in the second half of the paper. We nd a large number of constraints for operator dimensions, OPE coe cients, and central charges that must hold for any N = 2 superconformal eld theory. Conformal and W Symmetry; Conformal Field Models in String Theory 1 Introduction 2 The N = 2 superconformal bootstrap program 2.1 2.2 2.3 2.4 The insu ciency of Lagrangians The bootstrap philosophy Operator algebras of N = 2 SCFTs A rst look at the landscape: theories of low rank 3 The moment map four-point function 3.1 Structure of the four-point function 3.1.1 3.1.2 Constraints of crossing symmetry Fixing the meromorphic functions 3.2 Superconformal partial wave expansion 3.2.1 Fixing the short multiplets 3.3 su(2) global symmetry 3.4 e6 global symmetry 4 The Er four-point function 4.1 Structure of the four-point function 4.2 Crossing symmetry 4.1.1 4.1.2 The r0 (x1) The r0 (x1) r0 (x2) channel r0 (x2) OPE 4.2.1 Free theory expansion 5 6 Operator bounds from crossing symmetry Results for the moment map four-point function 6.1 su(2) global symmetry 6.2 e6 global symmetry 6.1.1 6.1.2 6.1.3 6.1.4 6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 Constraints on c and k Dimension bounds for su(2) Bounds for theories of interest Bounds for defect SCFTs Constraints on c and k Dimension bounds in the singlet channel Bounds for theories of interest The rank one theory Bounds for defect SCFTs 7 Results for the Er four-point function 7.1 7.2 7.3 Central charge bounds Dimension bounds for non-chiral channel E2r OPE coe cient bounds { i { Conclusions A Unitary representations of the N = 2 superconformal algebra B Superconformal block decompositions B.1 Superconformal blocks for the B^1 four-point function B.2 Superconformal blocks for the Er four-point function B.2.1 B.2.2 B.2.3 Selection rules in the non-chiral channel Selection rules in the chiral channel Superconformal blocks in the non-chiral channel B.2.4 Superconformal blocks in the chiral channel C Semide nite programming and polynomial inequalities C.1 A toy model for polynomial inequalities C.1.1 C.1.2 The primal problem: ruling out solutions The dual problem: constructing solutions C.2 Notes on implementation D Polynomial approximations and conformal blocks E Exact OPE coe cients for the N = 2 chiral ring 70 72 72 77 77 78 79 80 81 83 83 85 86 88 90 towards this ambitious goal. In this work we initiate the conformal bootstrap program for four-dimensional conformal eld theories with N = 2 supersymmetry. These theories are extraordinarily rich, both physically and mathematically, and have been studied intensively from many viewpoints. Nevertheless, we feel that a coherent picture is still missing. We hope that the generality of the conformal bootstrap framework will allow such a picture to be developed. We also feel the time is ripe for such an investigation | the recent explosion of results for N = 2 superconformal eld theories (SCFTs) calls out for a more systematic approach, while the methods rst introduced in [1] have reinvigorated the conformal bootstrap [2{8] with a powerful and exible toolkit for studying conformal eld theories. The rst examples of N = 2 superconformal eld theories (SCFTs) were relatively simple gauge theories with matter representations chosen so that the beta functions for all gauge couplings would vanish. Since then, the library of known theories has grown in size, with the new additions including many Lagrangian models [9], but remarkably also many theories that appear to admit no such description. In particular, the class S construction of [ 10, 11 ] gives rise to an enormous landscape of theories, most of which resist description by conventional Lagrangian eld theoretic techniques. Despite this abundance, the current catalog seems fairly structured, and one may reasonably suspect that a complete classi cation of N = 2 superconformal eld theories (SCFTs) will ultimately be possible. = 2 superconformal bootstrap seems an indispensable step { 1 { Our rst task is to introduce an abstract operator-algebraic language for N = 2 SCFTs. In this reformulation, we retain only the vector space of local operators (organized into representations of the superconformal algebra), and the algebraic structure on this vector space de ned by the operator product expansion. From this viewpoint, we can see that a theory is free (or contains a free factor) if its operator spectrum includes higher spin currents; we can see that a theory has a Higgs branch of vacua if its operator algebra includes an appropriate chiral ring that is the coordinate ring of an a ne algebraic variety; and so on and so forth. Representation theory of the N = 2 superconformal algebra proves an invaluable tool, as its shortened representations neatly encode di erent facets of the physics. This algebraic viewpoint is remarkably rich, and we have dedicated the next section to its extensive presentation. Once equipped with the proper language, we can make an informed decision on where and how to employ numerical bootstrap methods. We explain that there are three classes of four-point functions that should be the starting point for any systematic exploration of this type: the stress-tensor four-point function; the moment map four-point function; and the four-point function of N = 2 chiral operators. In the present work, we report on numerical investigations into speci c examples of the latter two classes. The requisite superconformal block expansion for the rst correlator, which is the most universal, is not yet available, so this case is left for future work. The moment map four-point function is related to the avor symmetry of the theory, and we focus on the cases of su(2) and e6. The su(2) case is clearly the simplest and is a natural starting point, while e6 case is interesting because exceptional avor symmetries cannot appear in any Lagrangian eld theory, and e6 is (among others) the simplest case to bootstrap after su(2). On the other hand, the four point function of N = 2 chiral operators gives us access to a very di erent aspect of the physics, namely the Coulomb branch chiral ring. There are two broad types of questions that we can hope to address by bootstrap methods. First of all, we can constrain the space of consistent N = 2 SCFTs. There are a number of universal structures that appear throughout the N = 2 catalog that cannot be satisfactorily explained in the abstract bootstrap language. Are Coulomb branch chiral rings always freely generated? Are central charges bounded from below by those of free theories, or are there exotic theories with even lower central charges? Is every N = 2 conformal manifold parametrized by gauge couplings? As we will see, these questions can sometimes be connected with the constraints of crossing symmetry, and then numerical analysis can provide (partial) answers. Our second motivation is to learn more about speci c N = 2 SCFTs. There are many cases where supersymmetry can tell us a lot about an N = 2 SCFT even when we have no Lagrangian description. In many examples we know, e.g., the central charges (including avor central charges), the spectrum of protected operators, and some OPE coe cients associated with protected operators. This partial knowledge can be used as input for a numerical bootstrap analysis. Optimistically, we may hope that this protected data and the constraints of crossing symmetry are enough to determine the theory uniquely. The bootstrap may then allow us to e ectively solve the theory along the lines of what has been done for the three dimensional Ising CFT [12{14]. Because the bootstrap is completely { 2 { nonperturbative in nature, it is a natural tool for studying intrinsically strongly coupled (non-Lagrangian) theories. In fact, when it comes to studying unprotected operators in a non-Lagrangian theory, the bootstrap is really the only game in town. The detailed organization of the paper can be found in the table of contents. In the rst part (sections 2-4) we develop the algebraic viewpoint and the details of the superconformal block expansion for the two classes of correlators that we consider, while in the second part (sections 5-8) we present our numerical investigations. Several appendices complement the main text with technical and reference material. 2 The N = 2 superconformal bootstrap program In the bootstrap approach to conformal eld theories, one adopts an abstract viewpoint that takes the algebra of local operators as the primary object. On the other hand, the majority of conventional wisdom and communal intuition about N = 2 eld theories arises from a Lagrangian | or at least quasi-Lagrangian | perspective. This leads to something of a disconnect. The bootstrap perspective is likely to be unfamiliar to many experts in supersymmetric eld theory, while amongst readers with a background in the conformal bootstrap the additional structure that follows from N = 2 supersymmetry may not be well known. In this section we will try to bridge this divide. 2.1 The insu ciency of Lagrangians Let us recall some aspects of Lagrangian N = 2 eld theories, which provide a historical foundation of the subject and help to guide our thinking even for the non-Lagrangian theories discussed below. The building blocks of an N = 2 four-dimensional Lagrangian are vector multiplets, transforming in the adjoint representation of a gauge group G, and hypermultiplets (the matter content ), transforming in some representation R of G.1 For the theory to be microscopically well-de ned, the gauge group should contain no abelian factors,2 so we can take G to be semi-simple, G = G1 G2 Gn : (2.1) To each simple factor Gi is associated a complexi ed gauge coupling i 2 C, Im i > 0, and for each choice of (G; R; f ig) there is a unique, classically conformally invariant N = 2 Lagrangian. For the quantum theory to be conformally invariant, the matter content must be chosen so that the one loop beta functions for the gauge couplings vanish. Thanks to N = 2 supersymmetry, this is also a su cient condition at the full quantum level. The classi cation of the pairs (G; R) that lead to N = 2 SCFTs can therefore be reduced to a purely combinatorial problem, whose complete solution has been described recently in [9]. The simplest examples are N = 2 superconformal QCD, which has gauge 1More generally, for appropriate choices of gauge group one can allow for \half-hypermultiplets", i.e., N = 1 chiral multiplets, transforming in pseudo-real representations of G. See, e.g., [9] for a recent discussion. 2An exception is when no hypermultiplet is charged under the abelian factors, in which case there are decoupled copies of the free vector multiplet SCFT in the theory. { 3 { HJEP03(216)8 group G = SU(Nc) and Nf = 2Nc hypermultiplets in the fundamental representation, and N = 4 super Yang-Mills theory (which can be regarded as an N = 2 SCFT), for which G is any simple group and the hypermultiplets transform in the adjoint representation. The conformal manifold of a CFT is the space of theories that can be realized by deforming a given CFT by exactly marginal operators. In a slight abuse of terminology we often refer to the conformal manifold of an N = 2 SCFT as the (not necessarily proper) submanifold of the full conformal manifold where in addition the full N = 2 supersymmetry is preserved. For a Lagrangian theory this submanifold coincides with the space of gauge couplings f ig, up to the discrete identi cations induced by generalized S-dualities.3 The conformal manifold comes endowed with a metric | the Zamolodchikov metric | which is Kahler and with respect to which the weak coupling points (where some i ! 1 in some S-duality frame) are at in nite distance as measured from the interior. Thus the conformal manifold of any N = 2 Lagrangian SCFT is non-compact with boundaries where gauge couplings are turned o . Lagrangian theories also always possess nontrivial moduli spaces of supersymmetric vacua. The simplest parts of the moduli space are the Coulomb branch and the Higgs branch. The Coulomb branch consists of vacua where the complex scalar elds 'i in the vector multiplets acquire nonzero vacuum expectation values (vevs), while the complex scalars (q; q~) in the hypermultiplets are set to zero | this branch is characterized by the fact that SU(2)R is unbroken, while U(1)r is broken. Alternatively, on the Higgs branch only the hypermultiplet scalars get nonzero vevs, and this branch is characterized by SU(2)R breaking with U(1)r preserved. There can also be mixed branches where the entire R-symmetry is broken, though we will not have much to say about mixed branches in this paper. The best way to parametrize these moduli spaces is by the vevs of gauge-invariant combinations of the elementary elds. The Coulomb branch is parametrized by the vevs of operators of the form fTr 'kg. These operators form a freely generated ring, called the Coulomb branch chiral ring, with generators in one-to-one correspondence with the Casimir invariants of the gauge group. Similarly, the Higgs branch can be parametrized by the vevs of gauge invariant composites of the hypermultiplet scalars. These operators also form a nitely generated ring, the Higgs branch chiral ring. The Higgs branch chiral ring is generally not freely generated, but rather has relations so that the Higgs branch acquires a description as an a ne complex algebraic variety. Alternatively, the Higgs branch can be expressed as a Hyperkahler quotient [15]. Isolated SCFTs and quasi-Lagrangian theories. Lagrangian SCFTs make up only small subset of all SCFTs. A wealth of strongly coupled N = 2 SCFTs with no marginal deformations are known to exist | by virtue of being isolated, they cannot have a conventional Lagrangian description. One particularly elegant way to nd such isolated theories is through generalized S-dualities of the kind discussed in [16]. By taking a Lagrangian theory and dialing a marginal coupling all the way to in nite strength, one may recover a 3Because the action of S-duality can have xed points in the space of gauge couplings, the conformal manifold may have orbifold points, so it may not really be a manifold. { 4 { weakly gauged dual description which involves one or more isolated SCFTs and a set of vector multiplets to accomplish the gauging. In this dual description the gauging procedure is described in what we may call a quasi-Lagrangian fashion: the isolated SCFT is treated as a non-Lagrangian black box with a certain avor symmetry, which is allowed to talk to the vector multiplets through minimal coupling of the conserved avor current of the isolated SCFT to the gauge eld. The one-loop beta function for each simple gauge group factor is given by = h_ + 4k ; (2.2) where h_ is the dual Coxeter number of the group and k the avor central charge, de ned from the two-point function of the conserved avor current. (Of course, this expression for applies also to the Lagrangian case, where the avor current is a composite operator made of the hypermultiplet elds.) The web of generalized S-dualities for large classes of theories can be elegantly described through the class S constructions of [ 10, 11 ]. These theories arise from twisted compacti cations of the six-dimensional (2; 0) theories on a punctured Riemann surface, with additional discrete data speci ed at each puncture. The marginal deformations of the four-dimensional theory correspond to the moduli of the Riemann surface, and weakly gauged theories arise if the Riemann surface degenerates. In this picture the isolated theories correspond to three-punctured spheres which have no continuous moduli. They do, however, depend on the discrete data at the three punctures as well as on a choice of g 2 fAn; Dn; Eng for the six-dimensional ancestor theory. In this way several in nite classes of isolated theories can be constructed. A few of these theories turn out to be equal to theories of free hypermultiplets, but most cases do not admit a Lagrangian description. Another large class of isolated theories are the Argyres-Douglas xed points [17] which describe the infrared physics at special points on the Coulomb branch of another N = 2 theory. At these distinguished points several BPS particles with mutually non-local charges become simultaneously massless, which precludes any Lagrangian description of the infrared theory. Alternatively, many Argyres-Douglas xed points can be constructed in class S by allowing for irregular singularities on the UV curve [18]. Argyres-Douglas theories have also recently been used as building blocks in a quasi-Lagrangian set-up [19]. In order to describe the currently known landscape of N = 2 SCFTs, then, it is clearly not su cient to only consider Lagrangians with hypermultiplets and vector multiplets. We can certainly accommodate any theory in a framework which takes as fundamental the spectrum and algebra of local operators. This is the basic starting point for the bootstrap approach that we take in this paper. The remainder of this section is dedicated to the development of such a framework. 2.2 The bootstrap philosophy In the bootstrap approach, we take a (super)conformal eld theory to be characterized by its local operator algebra.4 The aim is then to understand the constraints imposed upon 4In adopting this perspective, we are therefore willfully ignoring the complications associated with indates back to the foundational papers of [2{8]. See, e.g., [1, 20] for modern expositions. We will brie y recall the general logic, while placing particular emphasis on the role played by short representations of the conformal algebra. In the next subsection we describe the special features that arise in the N = 2 superconformal case. The local operators fOi(x)g of a CFT form a vector space that is endowed with a product that gives it something like an associative algebra structure. The product for local operators is known as the Operator Product Expansion (OPE), and takes the schematic form Any correlation function of separated local operators in at spacetime Rd can be evaluated by successive applications of the OPE, which is an absolutely convergent expansion. The OPE follows as a straightforward consequence of the state/operator correspondence.5 To each local operator is associated a state, obtained by acting on the vacuum with the operator inserted at the origin, O1(x)O2(y) = X c12k(x k y)Ok(y) : (2.3) and conversely each state de nes a unique local operator, O(x) ! jOi := O(0)j0i ; j i ! O (x) : (2.4) (2.5) As customary, we will use the language of operators or states interchangeably. To completely specify a CFT at the level of correlators of local operators, it is therefore su cient to list the set of local operators (that is, the set of their quantum numbers) and the structure constants appearing in their OPEs. Conformal invariance streamlines the presentation of this information. First, it allows the local operators to be assembled into conformal families, each of which transforms as a highest weight representation of the conformal algebra so(d; 2). The highest weight state, known as the conformal primary, is annihilated by all raising operators in the conformal algebra, notably the special conformal generators K . Specializing to the four-dimensional case, a representation R[ ; j1; j2] of so(4; 2) = su(2; 2) is labeled by the quantum numbers of the primary, namely its conformal dimension and its Lorentz spins (j1; j2). If the theory enjoys an additional global symmetry GF , then the local operators can be further organized into GF representations, labeled by some avor symmetry quantum numbers f , and the full representations are then denoted as R[ ; j1; j2; f ]. Conformal symmetry also restricts the spacetime dependence of the functions cijk(x) appearing in the OPE (2.3). In particular, the functions cijk(x) are uniquely determined in terms of the quantum numbers of the representations Ri, Rj , and Rk and the coe cients ijk that parametrize their three-point functions.6 All told, the s spacetime geometries. 5See [21] for a recent discussion. 6In the simplest case of three spacetime scalars (with no additional avor charges), the three-point function is completely xed up to a single overall coe cient ijk. In general there are multiple parameters isjk, s = 1; : : : mult(ijk), where the ( nite) multiplicity mult(ijk) is given by the number of independent conformally covariant tensor structures that can be built from the three reps Ri;j;k. { 6 { data that fully specify the local theory amount to a countably in nite list These data are constrained by the requirements that the theory be unitary and that the OPE be associative. The hypothesis underlying the conformal bootstrap is that these constraints are so powerful that they can completely determine the local data given some minimal physical input. In practice, one expects that the input will include the global symmetry of the theory and some simple spectral assumptions such as the number of relevant operators. ity bounds, Unitarity and shortening. We rst recall the constraints imposed by unitarity. Nontrivial7 unitary representations of so(4; 2) are required to satisfy the following unitarGeneric representations are denoted as A ;j1;j2 . Non-generic, or short, representations occur when the norm of a conformal descendant state in the Verma module built over some conformal primary is rendered null by a conspiracy of quantum numbers. This happens precisely when the unitarity bounds are saturated, leading to the following list of short representations: All of these representations have null states at level one with the exception of B, which has a null state at level two. The presence of short representations in the spectrum of a CFT is connected to the existence of free elds and symmetries in the theory. In particular, the primaries of Btype representations are decoupled free elds, and as such are not of much interest when studying interacting CFTs. For example, the primary of a B representation is a free scalar eld (x). Modding out by the null state at level two imposes the operator constraint P P = which is nothing but the free scalar equation of motion. Similarly, B?1 multiplets have as their primaries free Weyl fermions; the null state at level one imposes the free equation 2 7We use the quali cation \non-trivial" to exclude the vacuum representation, which consists of a single state with (x) = 0 ; 2j1 _ 1 _ 2j2 (x) = 0 : (2.10) (2.11) On the other hand, C-type representations have various conserved currents as their primaries; their level-one null state is the consequence of a conservation equation, Conserved currents with spin j1 + j2 > 2 are higher-spin currents, which are a hallmark of free CFTs [22, 23]. For the purposes of the bootstrap, we will usually impose by hand that no such multiplets appear. Conserved currents with (j1; j2) = (1; 12 ) and (j1; j2) = ( 12 ; 1) give rise to an enhancement of the conformal algebra to a super conformal algebra | when these operators are present one should therefore be taking full advantage of the power of superconformal symmetry. Thus, amongst the short representations of so(4; 2), those which may be present in an interacting non-supersymmetric CFT are C1;1 and C 1 ; 1 . In the former case, the con2 2 formal primary is the stress tensor T . In the latter case, the conformal primary is a conserved current J , so the presence of such multiplets portend the existence of continuous global symmetries. Locality in the operator algebra. An important remark is in order. When characterizing CFTs by their local operator algebra, certain ingredients which are usually automatically present in a Lagrangian context are no longer necessarily compulsory. For example, one need not assume that the local algebra includes a stress tensor at all. Indeed, there are interesting local algebras, such as the algebra of local operators supported on conformal defects in a higher-dimensional CFT, in which the stress tensor is not present. The presence of a stress tensor is clearly connected with the notion of locality in the CFT, and we will take the existence of a unique stress tensor (that is, the existence of a unique conformal representation of type C1;1) as part of the de nition of a local CFT. Similarly, in the Lagrangian context a continuous global symmetry implies the existence of a conserved current in the operator spectrum. We will assume the validity of this claim even in the non-Lagrangian context: Conjecture 1 (CFT Noether \theorem") In a local CFT, to any continuous global symmetry is associated a conserved current in the operator algebra that generates the symmetry. Clarifying the conceptual status of this \theorem" is an important open problem. On one hand, one may take it as part of the de nition of what it means for a CFT to be local, in which case this is a tautology. Alternatively, it is possible that the theorem may be { 8 { derived from general principles in a suitable axiomatic framework.8 Whatever the case may be, the proof of such a statement is of interest in part due to its reinterpretation via AdS/CFT, which is the statement that there are no continuous global symmetries in AdS quantum gravity. Canonical data. The data associated to short representations of the conformal algebra carries particular physical signi cance. The three-point function of the stress tensor depends on three parameters, two of which can be identi ed with the two coe cients appearing in the conformal anomaly, conventionally denoted by a and c. The a coe cient gives a measure of the degrees of freedom of the theory and serves as a height function in ow, aUV > aIR [24, 25]. However, since a can only be extracted from the stress tensor three-point function, it is rather di cult to access by bootstrap methods | one would generally need to consider correlation functions involving external stress tensors, which are very complicated [26]. By contrast, if one uses the canonical normalization for the stress tensor, its two-point function is proportional to c. The c coe cient will then appear in any four-point function containing an intermediate stress tensor, making its presence ubiquitous in the bootstrap literature. Using \conformal collider" observables, it was argued in [27] that in a general unitarity CFT the ratio of conformal anomaly coe cients must obey the bounds9 1 3 a c 31 18 : (2.12) The lower bound is saturated by the free scalar CFT, the upper bound by the free vector CFT. There is strong evidence that these free CFTs are the only theories saturating the bounds [29]. Similarly, the two-point function of canonically normalized currents depends on a parameter k often called the avor central charge that can be identi ed with an 't Hooft anomaly for the corresponding global symmetry [30, 31]. This parameter appears in the OPE of conserved currents as follows, J A(x)J B(0) 3k AB x2g Like the c central charge, the avor central charge makes frequent appearances in the bootstrap because it controls the contribution of the conserved current in a correlation function of charged operators. In a sense, the data associated to the spectrum of conserved currents and stress tensors and their associated anomaly coe cients is the most basic data associated to a conformal 8It is unclear whether the axioms for the algebra of local operators should be su cient for this purpose. It is possible that the existence of a conserved current could follow from the assumptions that the operator algebra is invariant under a continuous symmetry and that there is a stress tensor. Alternatively, the framework may need to be enlarged, perhaps allowing for correlation functions in non-trivial geometries, subject to suitable locality assumptions. 9The argument uses positivity of energy correlators in a unitarity theory, which is a reasonable physical assumption (see also [28]). It would be interesting to recover the HM bounds by conformal bootstrap methods. This will likely have to wait for the complete conformal block analysis of the stress tensor fourpoint function, a challenging technical problem. { 9 { eld theory. We designate this data as the canonical data for the CFT. It is natural to organize an exploration of the space of conformal eld theories in terms of these parameters, and if one wants to study a particular theory in detail this data is an obvious starting point. This has not always been the approach in the existing bootstrap literature thus far, but that is at least in part because the natural observables through which to pursue such a strategy would be the four point functions of conserved currents and stress tensors. At a technical level, these are much more complex observables than the correlators of spacetime scalars. The numerical bootstrap approach. Intuitively, associativity of the operator algebra is a tremendous constraint. However, aside from the case of two-dimensional CFTs where the global conformal symmetry algebra enhances to two copies of the in nite-dimensional Virasoro algebra, it seems very di cult to extract useful information from these conditions. The way forward was shown in [1], where the focus was shifted away from trying to solve the associativity problem and towards obtaining constraints for, e.g., the spectrum of local operators or their OPE coe cients in a unitary CFT. The prototypical bounds that can be obtained in this way are upper bounds for the dimension of the lowest-lying operator of a given spin, or a lower bound on the c central charge of a theory, all given some input about the spectrum of scalar operators. In order to test associativity it su ces to investigate four-point functions in a given CFT, where the OPE can be taken in three essentially inequivalent ways by fusing di erent pairs of operators together. For each choice one nds a representation of the four-point function as a sum over conformal blocks [20], with one block for each conformal multiplet that appears in both OPEs. The statement that these three decompositions have to sum to exactly the same result is known as crossing symmetry. It was shown in [1] that useful bounds can be extracted already from the requirement of crossing symmetry for a single four-point function involving four identical scalar operators. Such an analysis is conspicuously tractable | as opposed to trying to solve all of the in nitely many crossing symmetry constraints simultaneously, we simply nd the conditions that follow from a nite subset of those constraints. The structure of four-point functions and their OPE decompositions are severely constrained by conformal symmetry | see, e.g., [20] for an introductory exposition. The work of [1] has been extended in numerous directions, and bounds have been obtained in theories with and without supersymmetry and in various spacetime dimensions. Further numerical bootstrap results can be found for example in [12{14, 32{56]. An essential ingredient in the numerical analysis is the (super)conformal block decomposition of a four-point functions. These structures have been investigated in various cases in, e.g., [21, 26, 57{69]. In related work, [70{74] obtained nontrivial constraints for the operator spectrum by considering in particular the OPE in the limit where operators become lightlike separated. 2.3 Operator algebras of N = 2 SCFTs The superconformal case follows largely the same conceptual blueprint as the nonsupersymmetric case, where we replace the conformal algebra so(4; 2) with the superconformal algebra su(2; 2j2). The maximal bosonic subalgebra is just the conformal algebra so(4; 2) su(2; 2) times the R-symmetry algebra SU(2)R U(1)r. Additionally there are sixteen fermionic generators | eight Poincare supercharges and eight conformal supersu(2)1, and su(2)2 indices, respectively. charges | denoted as fQI ; QeI _ ; SJ ; SeJ _ g where I = 1; 2, = , and _ = _ are SU(2)R, The spectrum of local operators can be organized in highest weight representations of su(2; 2j2) whose highest weight states, known as superconformal primaries, are annihilated by all lowering operators of the superconformal algebra | in particular, by all the conformal supercharges S. These representations are labeled by the quantum numbers [ ; j1; j2; R; r] of the superconformal primary; the additional labels R and r that extend the ordinary conformal case are the eigenvalues of the Cartan generators of SU(2)R and U(1)r. We will also consider theories that are invariant under additional avor symmetry gF (a semisimple Lie algebra commuting with su(2; 2j2)), which introduces additional avor quantum numbers f . In summary, the local data for an N = 2 SCFT are HJEP03(216)8 with the conformal case, the coe cients encode the information needed to completely reconstruct the superspace three-point functions10 s ijk hRi(x1; 1) Rj (x2; 2) Rk(x3; 3)i. Unitarity and shortening. The unitary representation theory of the N = 2 superconformal algebra is more elaborate than that of the ordinary conformal algebra. The unitarity bounds are now given by where we have de ned i ; = i 2 or i ; ji 6= 0 ; ji = 0 ; 1 := 2 + 2j1 + 2R + r ; 2 := 2 + 2j2 + 2R r : The unitary representations of su(2; 2j2) have been classi ed in [75{77]. Short representations occur when one or more of these bounds are saturated, and the di erent ways in which this can happen correspond to di erent combinations of Poincare supercharges that can annihilate the highest weight state of the representation. There are again two types of shortening conditions, the B type and the C type. Each type now has four incarnations corresponding to the choice of chirality (left or right-moving) and the choice of 10In the conformal case, the isjk can be extracted from the three-point function of the conformal primaries, because descendant operators are simply derivatives of the primaries and their three-point functions contain no extra information. In general this is no longer the case with superconformal symmetry: knowledge of the three-point functions of the superconformal primaries does not always su ce. But at an abstract level there is no di erence: what matters are superconformally covariant structures that can be built from the three representations. (2.14) (2.15) (2.16) SU(2)R component: BI : BI : CI : CI : Q j i = 0 ; I QeI _ j i = 0 ; Some authors refer to B-type conditions as shortening conditions, and to C-type conditions as semi -shortening conditions, to highlight the fact that a B-type condition is twice as strong. We refer to appendix A for a tabulation of all allowed combinations of (semi)shortening conditions and for naming conventions for the resulting representations. Because of the proliferation of short representations in the N = 2 context, there is potentially much more \canonical data" than in the non-supersymmetric case. Indeed, these many short representations are closely related to various nice features theories with N = 2 supersymmetry. Here we focus primarily on three classes of short representations that have particularly straightforward connections to familiar physical characteristics of = 2 theories. These representations have the distinction of obeying the maximum number of shortening or semi-shortening conditions that can simultaneously be imposed (two and four, respectively). In the notations of [76], they are: multiplets, annihilated by the action of all left-handed supercharges.11 Er: half-BPS multiplets \of Coulomb type". These obey two B-type shortening conditions of the same chirality: B1 \ B2. In other terms, they are N = 2 chiral B^R: half-BPS multiplets \of Higgs type". These obey two B-type shortening condi tions of opposite chirality: B1 \ B2. These types of operators are sometimes called \Grassmann-analytic". C^0(j1;j2): the stress tensor multiplet (the special case j1 = j2 = 0) and its higher spin generalizations. These obey the maximal set of semi-shortening conditions: C1 \ C2 \ C1 \ C2. The CFT data associated to these representations encodes some of the most basic physical information about an N = 2 SCFT. We now look at each in more detail, starting from the third and most universal class, which contains the stress tensor multiplet. 11We are focusing on the scalar Er multiplets | Er := Er(0;0) in the notations of table 4. Representation theory allows for N = 2 chiral multiplets Er(0;j2) with j2 6= 0, but such exotic multiplets do not occur in any known N = 2 SCFT. See [78] for a recent discussion. Stress tensor data. The maximally semi-short multiplets C^0(j1;j2) contain conserved tensors of spin 2 + j1 + j2. For j1 + j2 > 0, such multiplets are not allowed in an interacting CFT, and we will always impose their absence from the double OPE of the four-point functions under consideration. The C^0(0;0) representation includes a conserved tensor of spin two, which we identify as the stress tensor of the theory. By de nition, a local N = 2 SCFT will contain exactly one C^0(0;0) multiplet.12 We will usually assume that the theories that we study are local, but we will also brie y explore non-local theories, which have no stress tensor and thus no C^0(0;0) multiplet. The superconformal primary of C^0(0;0) is a scalar operator of dimension two that is invariant under all R-symmetry transformations. The other bosonic primaries in the multiplet are the conserved currents for SU(2)R U(1)r and the stress tensor itself. An analysis in N = 2 superspace [79] reveals that the three-point function of C^0(0;0) multiplets involves two independent structures, whose coe cients can be parametrized in terms of the a and c anomalies. The N = 2 version of the Hofman-Maldacena bounds reads 1 2 a c 5 4 : (2.21) The lower bound is saturated by the free hypermultiplet theory, and the upper bound by the free vector multiplet theory. By a generalization of the analysis of [29], one should be able to argue that these are the only N = 2 SCFTs saturating the bounds. In this paper we will not study the four-point function of the stress tensor multiplet, because the requisite superconformal block expansion has not yet been worked out. We will, however, have indirect access to the c anomaly coe cient. As in the non-supersymmetric case, if one chooses the canonical normalization for the stress tensor then the two-point function of C^0(0;0) multiplets will depend on c only. The c coe cient will make an appearance in all four-point functions that we study, since C^0(0;0) appears in their double OPE. Coulomb and Higgs branches. As indicated by our choice of terminology, the two types of half-BPS multiplets | Er and B^R | are closely related to the Coulomb and Higgs branches of the moduli space of vacua, respectively. In Lagrangian theories, the superconformal primaries in the Er multiplets are the gauge-invariant composites of vector multiplet scalars that parametrize the Coulomb branch, and the superconformal primaries in the B^R multiplets are the gauge-invariant composites of hypermultiplet scalars that parametrize the Higgs branch. We should call attention to the fact that a satisfactory understanding of the phenomenon of spontaneous conformal symmetry breaking has not yet been developed in the language of CFT operator algebras. In principle, the local data should contain all necessary information to describe the phases of the theory where conformal symmetry is spontaneously broken. A method to extract this information is, however, presently not known. Even the basic question of whether a given CFT possesses nontrivial vacua remains out of 12A caveat to this de nition of locality is that in the tensor product of two local theories there will be two stress tensor multiplets. For the purposes of the conceptual discussion here we restrict our attention to theories that are not factorizable in this manner | we might call such theories simple. HJEP03(216)8 reach. Since all known examples of vacuum manifolds in CFTs occur in supersymmetric theories, one might speculate that supersymmetry is a necessary condition for spontaneous conformal symmetry breaking. of BPS multiplets. We are now ready to look in more detail at the CFT data encoded in the two classes Coulomb branch data. We will refer to the data associated to Er multiplets as Coulomb branch data. By passing to the cohomology of the left-handed Poincare supercharges, one nds a commutative ring of operators known as the Coulomb branch chiral ring, the elements of which can be identi ed with the superconformal primaries of Er multiplets. In all known examples, this ring is exceedingly simple, and it is natural to formulate a conjecture that the ring is always as simple as it is in the examples:13 Conjecture 2 (Free generation of the Coulomb chiral ring) In any N = 2 SCFT, the Coulomb branch chiral ring is freely generated. This conjecture can in principle be translated into a statement about the OPE coe cients of the Er multiplets. For instance, a simple consequence is that no Er superconformal primary can square to zero in the chiral ring, so an E2r operator must appear with nonzero coe cient in the OPE of the Er with itself. Precisely this kind of statement can be tested by numerical bootstrap methods, as we will describe in section 7. The number of generators of the Coulomb branch chiral ring is usually referred to as the rank of the theory. The set fr1; : : : rrankg of U(1)r charges of these chiral ring generators is one of the most basic invariants of an N = 2 SCFT. Unitarity implies r 1, with r = 1 only in the case of the free vector multiplet, so we will always assume r > 1. In Lagrangian SCFTs, the ri are all integers, but there are several non-Lagrangian models that possess Er multiplets with interesting fractional values of r. We are not aware of any examples where U(1)r charges take irrational values. It is widely believed that the Coulomb branch of the moduli space of any N = 2 SCFT is parametrized by assigning independent vevs to each of the Coulomb branch chiral ring generators. We will generally operate under the assumption that this statement is true, which amounts to assuming the validity of the following conjecture. Conjecture 3 (Geometrization of the Coulomb chiral ring) The Coulomb chiral ring is isomorphic to the holomorphic coordinate ring on the Coulomb branch. We note that the union of Conjecture 2 and Conjecture 3 implies that the Coulomb branch of any N = 2 SCFT just Cr, with r the rank of the theory. At present we are not sure how one might establish Conjecture 3 using bootstrap methods due to the obstacle of spontaneous conformal symmetry breaking discussed above. However, once one has found their way onto the Coulomb branch, the powerful technology of Seiberg-Witten (SW) theory becomes applicable. The e ective action for the low-energy U(1)rank gauge theory on the Coulomb branch is characterized by geometric data (in the 13To the best of our knowledge, this conjecture was rst explicitly stated in the literature by Yuji Tachikawa in [80]. HJEP03(216)8 simplest cases, this is the SW curve, more generally it is some abelian variety). There are well-developed techniques to determine the SW geometry, which apply to most Lagrangian examples and to several non-Lagrangian cases as well. In turn, the SW geometry determines a wealth of physical information, such as the spectrum of massive BPS states. Unfortunately, how to translate this information into CFT data remains an unsolved problem.14 In [82], Shapere and Tachikawa (ST) proved a remarkable formula that relates the a and c central charges to the generating r-charges fr1; : : : rrankg, 2a c = The ST sum rule holds in all known examples, and it is tempting to conjecture that it is a general property of all N = 2 SCFTs. The derivation of [82] requires that the SCFTs in question be realized at a point on the moduli space of a Lagrangian theory. The result can then be extended to all SCFTs connected to that class of theories by generalized S-dualities. In particular, this includes a large subset of theories of class S. According to the ST sum rule, a theory with zero rank necessarily has a=c = 1=2, which is the value saturating the lower HM bound. As remarked above, there are strong reasons to believe that the only SCFT saturating this bound is the free hypermultiplet theory. However, since the whole logic of [82] relies on the existence of a Coulomb branch, this reasoning is circular. An interacting SCFT of zero rank would be rather exotic, but we do not know how to rule it out with present methods. marginal operator O4 the complexi ed gauge coupling. The special case of the E2 multiplet is particularly signi cant. The top component of the multiplet, obtained by acting with four right-moving supercharges on the superconformal primary,15 O4 Q~4E2 is a scalar operator of dimension four. This operator provides an exactly marginal deformation of the SCFT that preserves the full N = 2 supersymmetry. (By CPT symmetry, there is also a complex conjugate operator O4 Q4E 2.) The converse is also true: any N = 2 supersymmetric exactly marginal operator O4 must be the top component of an E2 multiplet. It follows that the number of E2 multiplets is equal to the (complex) dimension of the conformal manifold of the theory. In a Lagrangian theory, there is an E2 multiplet for each simple factor of the gauge group, and the exactly Tr(F 2 + iF~2) (where F is the Yang-Mills eld strength) is dual to Another true feature of all Lagrangian SCFTs (and many non-Lagrangian ones in class S) is that they can be constructed by taking isolated building blocks with no marginal deformations (such as hypermultiplets in the Lagrangian case, or TN theories in the class S case) and gauging global symmetry groups for which the beta function will vanish. A natural conjecture is that this feature is indeed universal: 14See however [81] for a relation between the spectrum of BPS states on the Coulomb branch and a certain partition function (evaluated at the conformal point), which appears to be closely related to the superconformal index. represents the whole multiplet. 15In an abuse of notation, we are denoting the superconformal primary with the same symbol E2 that Conjecture 4 (Decomposability) Any N = 2 SCFT with an n-dimensional conformal manifold can be constructed by gauging n simple factors in the global symmetry group of a collection of isolated N = 2 SCFTs. Of course such a decomposition need not be unique | the existence of inequivalent decompositions of the same theory is what is often called \generalized S-duality". Note that the validity of this conjecture would imply the absence of compact conformal manifolds for N = 2 SCFTs.16 Higgs branch data. In a similar vein, the B^R multiplets are expected to encode the information about the Higgs branch of the theory. The B^R superconformal primaries, which are also SU(2)R highest weights, form the Higgs branch chiral ring. In all known examples this ring is described by a nite set of generators obeying polynomial relations. The algebraic variety de ned by this ring is then expected to coincide with the Higgs branch of vacua. This expectation can be formalized as follows: Conjecture 5 (Geometrization of the Higgs chiral ring) In any N = 2 SCFT, the Higgs branch chiral ring is isomorphic to the holomorphic coordinate ring on the Higgs branch of vacua. The Higgs branch of vacua is hyperkahler, so there are actually a CP1 worth of holomorphic coordinate rings on it depending on the choice of complex structure. The choice of complex structure corresponds to a choice of Cartan element in SU(2)R, so we have implicitly made the choice already. In this paper we will focus on the simplest non-trivial17 case of these multiplets, the B^1 multiplet. This multiplet plays a distinguished role, because it encodes the information about the continuous global symmetries of the theory. Indeed, the multiplet contains a conserved current, J _ = J KQI QeJ _ IK ; (2.23) where IJ is the operator of lowest dimension in the B^1 multiplet. It is an SU(2)R triplet and is often referred to as the moment map operator. (The superconformal primary is the highest SU(2)R weight 11.) The current J _ generates a continuous global symmetry, and is thus necessarily in the adjoint representation of some Lie group GF . Vice versa, if the theory enjoys a continuous global symmetry, it follows from Conjecture 1 that the CFT contains an associated conserved current J _ , and one can show that in an interacting N = 2 SCFT such a current must necessarily belong to a B^1 multiplet. Indeed, one can survey the list of superconformal representations and identify all the ones that contain conserved spin one currents that are also SU(2)R U(1)r singlets. The list is very short: B^1 and C^0( 12 ; 1 ). The latter multiplet has a conserved current as its superconformal primary, 2 but also contains conserved a spin three conserved current among its descendants, so by our usual criterion it is not allowed in an interacting SCFT. What's more, B^1 representations 16In the N = 1 case the existence of compact conformal manifolds has recently been established in [83]. The methods used there cannot easily be generalized to the N = 2 case. HJEP03(216)8 cannot combine with other short representations to form long representations, so the B^1 content of a theory is an invariant on the conformal manifold. To reiterate, a SCFT may have a avor symmetry enhancement only in a singular limit where some free subsector decouples (such as the zero coupling limit of a gauge theory) and C^0( 12 ; 12 ) multiplets split o from long multiplets hitting the unitarity bound. In the \bulk" of the conformal manifold, avor symmetries are always associated to B^1 multiplets. As we have already mentioned in the context of exactly marginal gauging of SCFTs, to each simple non-abelian factor of the global symmetry group is associated a avor central charge k, de ned from the OPE coe cient of the conserved current with itself (2.13). Thus the most basic data associated to the B^1 representations in an SCFT are the global symmetry group GF = G1 : : : Gk and the corresponding avor central charges. Chiral algebra data. It was recognized in [84] (see also [85, 86]) that the local operator algebra of any N = 2 SCFT admits a closed subsector isomorphic to a two-dimensional chiral algebra. The operators that play a role in the chiral algebra are the so-called Schur operators, which (by de nition) obey the conditions18 (j1 + j2) 2R = 0 ; j2 j1 r = 0 : Schur operators are found in the following short representations, B^R ; DR(0;j2) ; DR(j1;0) ; C^R(j1;j2) : (2.24) (2.25) One should in particular note the absence of the Er multiplets from this structure. Each supermultiplet in this list contains precisely one Schur operator: for the B^R multiplets, the Schur operator is the superconformal primary itself, while for the other multiplets in (2.25) it is a superconformal descendant.19 When inserted on a xed plane R 2 R4, parametrized by the complex coordinate z and its conjugate z, and appropriately twisted (the twist identi es the right-moving global conformal algebra sl(2) acting on z with the complexi cation of the su(2)R algebra), Schur operators have meromorphic correlation functions. The rationale behind this construction is that twisted Schur operators are closed under the action of a certain nilpotent supercharge, Q := Q 1 + Se1_ , and they have wellde ned meromorphic OPEs at the level of Q cohomology. This is precisely the structure that de nes a two-dimensional chiral algebra. We refer the reader to [84] for a comprehensive explanation of this construction. Here we mainly wish to emphasize that the chiral algebra data (i.e., the Schur operators and their three-point functions) are a very natural generalization of the Higgs data. Since they are subject to associativity conditions expressed by meromorphic equations, the chiral algebra data can be often determined exactly given some minimum physical input. The simplest example, and the one that will play a role in this paper, is the case of moment maps. Moment maps transform in the adjoint representation of the avor 18In fact one can show that the rst condition implies the second in a unitary theory. 19For example, the Schur operator in a C^0(0;0) multiplet is a single component of the SU(2)R conserved current. symmetry group, and in the associated chiral algebra they correspond to a ne Kac-Moody currents, where the level k2d of the a ne current algebra is related to the four-dimensional avor central charge k by the universal relation The four-point function of a ne currents is completely determined by meromorphy and crossing symmetry. In the present context, it admits a reinterpretation as a certain meromorphic piece of the full moment map four-point function. Crucially, this meromorphic piece contains the complete information about the contribution of short representations to the double OPE of the four-point function.20 All in all, combining the constraints of four-dimensional unitarity with the ability to solve exactly for the contributions of short representations leads to novel unitarity bounds for the level k and the trace anomaly coe cient c that are valid in any interacting N = 2 SCFT. These bounds will play a signi cant role in the analysis of section 6. rst look at the landscape: theories of low rank The ultimate triumph of the N = 2 bootstrap program would be the classi cation of N = 2 SCFTs. If the decomposability conjecture of section 2.3 holds true, then this problem is reduced to the enumeration of elementary building block theories with no conformal manifold. Still, this is completely out of reach at present, and any attempt at a direct attack on the classi cation problem would be premature. We are still very much in an exploratory phase. To organize our explorations we may characterize theories by their rank | i.e., the dimension of their Coulomb branch or the number of generators in the Coulomb branch chiral ring. Theories with low rank by and large have smaller values for their central charges than their higher-rank counterparts, so this may be a reasonable measure of the complexity of a theory. From the bootstrap point of view, theories with small central charges are attractive as targets for numerical study. The rank zero case is probably trivial. The simplest conjecture is that the only N = 2 SCFT with no Coulomb branch is the free hypermultiplet theory. This would be compatible with the universal validity of the Shapere-Tachikawa bound. For rank one, we can start by reviewing the list of established theories. This survey will prove useful in our e orts to interpret the numerical bootstrap results reported in later sections. The classic rank one theories are the SCFTs that arise on a single D3 brane probing an F -theory singularity with constant dilaton [87{92]. There are seven such singularities, denoted by H0, H1, H2, D4, E6, E7, E8. With the exception of the theory associated to the D4 singularity, which is an SU(2) gauge theory with Nf = 4 fundamental avors, these theories are all isolated non-Lagrangian SCFTs. They have an alternative realization in class S, where they are associated to punctured spheres with certain special punctures | see, e.g., [10, 18, 93{95]. 20To be able to uniquely reconstruct the contribution of the short representations from the meromorphic function, one must make the now-familiar assumption that the theory does not contain higher-spin conserved currents. a superconformal primary, then some algebraic manipulations lead to the following form of the Ward identity,38 (x3)h (x1) (x2) nQI ; hQeJ ; _ ; O0K11::;::::2Kjn_ 1::: _ 2j io (x3)i = I jh (x1) (x2)O0K11::;::::2Kjn_( _ 1::: _ 2j 1 J (x3)i __2j) + j + r 2 n + __ J(K1 h (x1) (x2)O0K12::;::::2Kjn_)1;:I:: _ 2j (x3)i : _ h (x1) (x2)O0K11::;::::2Kjn_ 1::: _ 2j (x3)i _ (B.14) are the U(1)r charge and dimension of O0. It follows from this identity that the three-point function including the superconformal descendant conformal primary. QI ; QeJ ; _ ; O0K1;:::Kn Similar results can be derived for all higher descendants of O0(x) using (B.13) plus the corresponding relation involving the conjugate supercharges. All told, we are left with the selection rule given above in (B.12). Given these selection rules, the possible superconformal representations that may appear in the OPE are severely restricted. Namely, only representations for which the superconformal primary has R = r = 0 and j := j1 = j2 may appear. A brief survey of the representations in appendix A leads to the following list, is xed in terms of the three point function of the superEr0(0;0) E r0(0;0) 1 + C^0(j;j) + A 0;0(j;j) : (B.15) We should note that this selection rule has only been derived here for the superconformal primaries of the Er0(0;0) and E r0(0;0) multiplets. B.2.2 Selection rules in the chiral channel The selection rules for the chiral OPE can be determined by a generalization of arguments of [35], where the analogous problem for N operator O(x) appears in the r0 = 1 SCFTs was considered. Suppose an r0 OPE. Ordinary non-supersymmetric selection rules imply that O must be an SU(2)R singlet with r O = 2r0 and j := j1 = j2 2 Z. There are then additional constraints that come from the supersymmetry properties of the chiral operators that are being multiplied. Namely, we observe that for any x, we have [QI ; r0 (x)] = 0 ; [SeI; _ ; r0 (x)] = 0 : The rst condition is simply a part of the de nition of the Er multiplet. The latter is automatic when x = 0 because r0 is the superconformal primary in its representation. For x 6= 0, we note the following relation from the N = 2 superconformal algebra, 38In this calculation we have assumed that O0(x3) is bosonic. A similar calculation leading to the same conclusion holds in the fermionic case. (B.16) (B.17) [ P _ ; SeI; _ ] = __ QI : It follows that when r0 is translated away from the origin, its variation under the action of SeI; _ is proportional to its variation under the action of a chiral supercharge, which vanishes. Thus we see that r0 (x1) r0 (x2) itself is invariant under the action of QI and SeI; _ , and so must be any operator appearing in the corresponding OPE, The only superconformal primary operator that can appear in the chiral OPE is therefore that of an E2r multiplet, and its superconformal descendants are excluded from appearing. Any other operator that appears must be a superconformal descendant obtained by acting on a given superconformal primary with all possible supercharges Q I that do not annihilate it. Thus only one conformal family per superconformal multiplet can contribute, and the superconformal blocks in this channel will be equal to the conventional conformal blocks for that family. Upon consulting the catalog of N = 2 superconformal multiplets reviewed in appendix A, it is straightforward to identify the multiplets that t the bill. (For simplicity, we temporarily assume that r0 > 1.) To illustrate the procedure, let us consider the case of long multiplets. The above argument implies that a long multiplet may only contribute to this OPE via a descendant of the schematic form O = Q4O0, where O0 is a superconformal primary. This descendant must be an SU(2)R singlet with r O = rO0 + 2 = 2r0 and spin ` O = 2j = `O0 . The relevant long multiplet is therefore of type A0;2r0 2(j;j). Unitarity requires that the dimension of the superconformal primary satis es O0 2r0 + `, so the contributing descendant will have O 2r0 + ` + 2. Similar reasoning leads to the complete list of short multiplets that may contribute to the OPE, with the nal selection rule taking the form E2r0(0;0) + C0;2r0 1(j 1;j) + B1;2r0 1(0;0) + C 12 ;2r0 23 (j 12 ;j) + A0;2r0 2(j;j): We note that again, this derivation applies only to the OPE for superconformal primaries of the Er0(0;0) multiplets. For r0 = 1 we can nd additional short multiplets of types D1(0;0); C^12 (j 12 ;j); C^0(j 1;j) : (B.19) (B.20) The last of these multiplets contains higher spin conserved currents, as is to be expected since the chiral operator with r0 = 1 is a free scalar eld. B.2.3 Superconformal blocks in the non-chiral channel The superconformal blocks for the various representations appearing in the non-chiral channel have been determined in [66]. In the language of section 4, these are the superconformal blocks in the 1^ channel. They are as follows, G^1Id(z; z) := 1 ; G^1 C^;`(z; z) := zz z z z ` 2 z 2F1 (` + 1; ` + 3; 2` + 4; z)) HJEP03(216)8 64( 16( 256( ` 2)2 `+2 G(`)+2(u; v) `+4 G(`+22)(u; v) + ` + 2) u 2 `+4 G(`+13)(u; v) u 2 `+2 G(`++13)(u; v) ` 2)2( + `)2 ` + 1)( + ` + 1)( + ` + 3) u 2 `+4 G(`)+4(u; v) : (B.22) G^1 ;`(z; z):= (zz) 2 z 2F1 1 2 + ` + 2; z) ` 2) ; ( ` + 2) ; `; z) Note that the superconformal block for the C^0(j;j) representation is just the specialization of the superconformal block for a long multiplet to the case = ` + 2. This is to be expected based on the recombination rules of appendix A. The superconformal block for a long multiplet can be decomposed into ordinary conformal blocks, which makes manifest the collection of conformal families from this multiplet that contribute to the four-point function: Gi=;^1`(z; z) = u 2 ` G(`)(u; v) + + `) + ` + 2) 1 2( u 2 ` G(`++11)(u; v) + u 2 `+2 G(`+11)(u; v) ( + `)2 4( + ` + 1)( + ` + 3) u 2 ` G(`++22)(u; v) The same multiplets contributing to the non-chiral channel also contribute to the 3^ channel via the \braided" version of the above superconformal blocks. The braided version is obtained by replacing each G(`) by ( 1)` G(`) in (B.22). B.2.4 Superconformal blocks in the chiral channel Because the supermultiplets appearing in the chiral channel contribute a single conformal family to the four point function, the superconformal blocks in the chiral channel (or 2^ channel in the language of section 4) are just the conventional conformal blocks appropriate to those conformal families. Table 7 displays the corresponding block for each allowed supermultiplet. The fourth and fth lines in table 7 correspond to short representations that lie at the unitarity bound for long multiplets. Accordingly, their superconformal blocks are simply the specializations of the long multiplet block to appropriate values of and `. On the other hand, the rst two classes of short representations are separated from the continuous spectrum of long multiplets by a gap. The last three representations are only present when we relax our assumption that there are no higher spin conserved currents or free elds in the theory. Contribution to G^i=^2(u; v) Restrictions Semide nite programming and polynomial inequalities This appendix is devoted to a review of the methods of [39], whereby the search for a linear functional of the type described in section 5 can be recast as a semide nite program. The principal observation that leads to this reformulation is that, up to a universal prefactor, any derivative of a conformal block for xed ` can be arbitrarily well approximated by a polynomial in the conformal dimension , that is ( ; `)Pm(`;)n( ) : Here ( ; `) may be complicated, but it is positive for all physical values of is independent of the choice of derivative. On the other hand, Pm(`;)n( ) is a and ` and nite order . For the superconformal blocks appearing in this paper, the details of this polynomial approximation are explained below in appendix D. With the aid of this approximation, we consider the action of a linear functional on smooth functions of z and z of the form [F (z; z)] = X m;n=0 Up to the positive prefactor described above, the action of this functional on a conformal block is now given by a nite order polynomial in the conformal dimension, [G(`)(z; z)] = ( ; `) am;nPm(`;)n( ) =: ( ; `) P`( ) : X m;n=0 The numerical problem in question (see section 5) is thus transformed into a search in the space of am;n 2 R such that the polynomial P`( ) 0 for `? for each `. Note that the range of values of for which the polynomial must be positive is always bounded from below, either by the unitarity bound or by the chosen value A polynomial in that is positive for all ? can always be decomposed as follows, ` . P( ) = P ( ) + ( )Q( ) ; where P ( ) and Q( ) are polynomials that are positive for all real . Furthermore, in terms of the monomial vector ~ := (1; ; 2; : : : ; N ), such non-negative polynomials can always be written as P ( ) = ~ tP ~ ; Q( ) = ~ tQ ~ ; where P and Q are positive semide nite matrices, which is notated as P; Q emphasize that the matrices P and Q are not completely xed in terms of P ( ) and Q( ). There is a redundancy to which we will return shortly. The action of the functional on conformal blocks will therefore be non-negative above `? in the spin ` channel if and only if there exist two positive semide nite where ci is a xed cost vector that de nes the objective function, and F i and F 0 are some xed square matrices. This semide nite program has a dual problem that is de ned as the search for a positive semi-de nite matrix Y that maximizes an appropriate objective function and satis es certain linear constraints, am;nP m(`;)n( ) = ~ tP (`) ~ + ( ` ) ~ tQ(`) ~ : In words, we are demanding that the left- and right-hand sides of (C.6) be the same , which amounts to linear relations between the coe cients of P (`) and Q(`) and the am;n. Such an equation must hold for each ` appearing in the crossing symmetry equation, and if there are multiple avor symmetry channels then there will be such an equation for each channel. The problem is thus reduced to the search for a set of positive semide nite matrices whose entries satisfy certain linear constraints. This is a prototypical instance of a semide nite program, the basic theory of which we review next. Semide nite programming. A semide nite program (SDP) is an optimization problem wherein the goal is to minimize a linear objective function over the intersection of the cone of positive semide nite matrices with an a ne space. Such a problem can be described in terms of a vector of real variables xi as follows, xi such that minimize (xici) X := xiF i F 0 0 ; Y maximize Tr(F 0 Y ) such that Y 0 ; Tr(F i Y ) = ci : (C.4) (C.5) (C.6) (C.7) (C.8) The original problem written in (C.7) | called the primal problem | and the dual problem of (C.8) are not generally guaranteed to be equivalent. Indeed, given a solution xi to the primal problem and a solution Y to the dual problem, a measure of the inequivalence of the solutions is the duality gap: xici Tr(F 0 Y ) = xiTr(F i Y ) Tr(F 0 Y ) = Tr(X Y ) 0 ; (C.9) where the last line holds because both matrices are positive semide nite. The absence of a duality gap, and the existence of an optimal solution to the primal (dual) problem, is guaranteed if the dual (primal) problem is bounded from above (below) and has a strictly feasible solution, i.e., there exists a matrix Y 0 (X 0) satisfying the relevant constraints. This is called Slater's condition. C.1 A toy model for polynomial inequalities To demonstrate the application of semide nite programming techniques to the type of crossing symmetry problem being considered in this paper, let us consider a simpli ed model in which the notation is less burdensome. Namely, consider the problem of studying the space of solutions to a \crossing symmetry" equation of the form where k are allowed to vary over the entire real line. We will assume that the functions G (z) and their derivatives can be well approximated by polynomials in , so we have X k 2kG k (z) = c(z) ; z=1=2 2N X p =0 =: P^i( ) ; where we have assumed that for a given range of values of i, each such polynomial has degree less than or equal to some xed even number 2N .39 C.1.1 The primal problem: ruling out solutions To constrain the space of solutions to such a problem, we consider acting with a linear functional on both sides of the equality and check for contradictions. The problem can be formalized as follows, If the minimum turns out to be negative then our toy problem has no solution. Taking [f (z)] := Pin=0 ai@zi f (z) z=1=2, we can reformulate the optimization problem as follows minimize [c(z)] such that minimize ai aici such that aiP^i( ) 0 8 8 : : (C.10) (C.11) (C.12) (C.13) 39For the sake of comparison, we note that in the actual crossing symmetry equations encountered in this work we have an additional z coordinate, as well as sums over spins and possibly avor symmetry channels. Also the values of k are bounded below in a given channel by unitarity bounds. However, these complications do not conceptually change this discussion. In terms of the vector ~ = (1; ; 2; : : : N )t, the second line of (C.13) requires the existence of a symmetric, positive semide nite matrix P^ such that P^( ) = ~ tP ~ with P This allows us to reformulate the polynomial inequalities as a semide nite program. We begin by introducing two sets of matrices in terms of which the problem is naturally reformulated. For N > 1, the matrix P is not completely xed by (C.15) because there are only 2N +1 components in P^( ) whereas P has (N +1)(N +2)=2 independent components. This redundancy in P can be parametrized by matrices Q satisfying HJEP03(216)8 ~ tQ ~ = 0 8 : Examples of such matrices Q are the 3 3 matrices with ( 1; 2; 1) on the cross-diagonal, or the 4 4 matrix with (1; 1; 1; 1) on the cross-diagonal. All other matrices Q take a similar form, and the rst set of matrices we must introduce is a complete basis for such ^ Q. We denote the elements of this basis as Qi. The second set of matrices are in one-to-one correspondence with the polynomials P^i( ). They take the form: By construction these matrices satisfy the condition P i := BB 0 0 pi0 12 pi1 0 BB 21 pi1 p 2 1 i 2 3 p 21 pi3 p 4 BB 0 0 : : :1 0 : : :C C 12 pi5 : : :CC : i6 : : :CC . .. . . A . P^i( ) = ~ tP i ~ : aiP^i( ) = ~ t aiP i + b^iQ^i ~ ; minimize ai;b^i aici ^ such that aiP i + b^iQi 0 ; (C.14) (C.15) (C.16) (C.17) (C.18) (C.19) (C.20) (C.21) Armed with these matrices we can write down the most general matrix that, upon contraction from both sides with ~ , gives the requisite polynomial: where the b^i are arbitrary real parameters. rephrased as The optimization (C.13) can now be which we recognize to be precisely a semide nite program of the form given in (C.7), with xi (ai; b^i) ; F i (P i; Q^i) ; F 0 = 0 : The constraints in (C.20) are invariant under an overall rescaling of the (ai; b^i), so the optimal value is either zero or negative in nity. To render the primal formulation bounded we can introduce an additional normalization constraint Tr(P ) = aiTr(P i) + b^iTr(Q^i) = 1 : (C.22) This condition is always enforceable because a nonzero, positive semide nite matrix has strictly positive trace. Although other normalization conditions are possible, we will see that (C.22) is particularly natural from the perspective of the dual problem. In practice, we can simply solve the additional constraint for, say, a1 to end up with a bounded variation The dual problem: constructing solutions Let us now address the dual problem to (C.20) with the additional constraint (C.22). After a little rewriting, the problem is as follows: maximize ;Y such that Y + I 0 ; Tr(P i Y ) = ci Tr(Q^i Y ) = 0 8 i; 8 ^i : This is a well-known form of a feasibility problem, which is the search for a matrix Y 0 subject to linear constraints. If the optimal value of comes out non-positive then such a matrix Y exists (i.e., there is a feasible solution), otherwise it does not. In standard applications the reason for introducing a variable multiplying the identity matrix I is to ensure that a strictly feasible solution will always exist, because for 0 the matrix Y + I 0. Its appearance in (C.23) is a consequence of the trace constraint (C.22) in the primal problem. Whereas the primal problem amounted to the search for functionals that certify the absence of solutions to crossing symmetry, dual problem is related to constructing solutions to crossing symmetry [13]. Let us observe how this works for these semide nite programs. We rst solve the constraints Tr(Q^i Y ) = 0. The most general solution is given by Y = y Y ; = 0; : : : ; 2N; with arbitrary coe cients y and with matrices Y de ned as B 0 1 0 0 0 Y0 = BB 0 0 0 0 1 C C CC ; B 0 0 1 0 0 B 1 0 0 0 Y1 = BB 0 0 0 0 1 C C CC ; B 0 0 0 1 0 B 0 1 0 0 Y2 = BB 1 0 0 0 1 C C CC ; BB 0 0 0 0 @ ... ... ... ... . . .A C C BB 0 0 0 0 @ ... ... ... ... . . .A C C BB 0 0 0 0 @ ... ... ... ... . . .A C C (C.23) (C.24) : (C.25) Now let us choose tuples ( 2k; k) so that y = k k k k k ~ tk and the additional constraints of the form Tr(P i Y ) = ci become 2 ~ We then have Y = X k 2kP^i( k) = ci : (C.26) This is precisely the crossing symmetry equation (C.10) after truncating to a nite number of derivatives. Finally, let us comment on the duality gap and the interpretation of solutions to this problem. The freedom to set to a large positive number ensures that the above formulation of the dual problem is strictly feasible. It is, however, not obviously bounded. From the formulation of the problem it is clear that this is related to the existence of solutions to crossing symmetry where c(z) = 0. More precisely, the problem is unbounded if there is a positive semide nite matrix Y that satis es Tr(P i Y ) = 0 and Tr(Q^i Y ) = 0 for all i and ^i. In the absence of such solutions the problem is bounded, Slater's condition is satis ed, and there is no duality gap, so for the optimal values we nd that This equation makes intuitive sense. Indeed, suppose the dual formulation does not nd = aici. a solution to crossing symmetry. This happens when = aici < 0 and therefore the primal formulation indeed provides a functional that proves that such a solution cannot exist. Similarly, suppose we do nd a matrix Y 0 satisfying all the above constraints. In that case = aici 0, so no functional can be found in the primal problem. Extremal functionals. In the applications of this framework to study interesting physical theories, there are often additional parameters in the problem such as assumed gaps in the spectrum for certain spins. In such cases we are usually interested in nding the boundary in the space of such parameters between regions where crossing symmetry can and cannot be satis ed. Precisely at the boundary = aici = 0. This turns out to imply that the corresponding solution to crossing symmetry is completely determined by the zeroes of the extremal functional [35, 41]. This is because the absence of a duality gap implies Tr(X Y ) = 0 which together with the above assumption on the form of Y leads to aiP i( k) = 0 : (C.27) the ~ tkX ~ P 2 k k C.2 Notes on implementation The solution to crossing symmetry encoded in Y therefore involves precisely those values of for which the extremal functional vanishes. This observation leads to the following algorithm for nding the solution to crossing symmetry: one rst lists the k for which k = 0, and then nding the 2k reduces to solving the linear problem y ( k) . Note that we require both the X and the Y matrix here. In this work we have utilized the dual formulation of the semide nite program associated to crossing symmetry. We rst solved all the linear constraints analogous to those appearing in (C.23), leading to a smaller set of independent parameters that we denote z ^ and corresponding matrices Z^. The nonzero ci lead to an inhomogeneous term that we may call Z^0. The complete semide nite program is then as above with xi ) (z ^; ) ; F i ) (Z ^; I) ; F 0 ) Z^ ; 0 (C.28) Parameter maxIteration epsilonStar lambdaStar omegaStar lowerBound upperBound betaStar betaBar gammaStar epsilonDash precision Value 1000 10 12 108 106 1030 1030 relevant for the SDPA-GMP solver. and a cost vector such that only is extremized. Since we were unable to rigorously show that the dual problem was bounded in all cases, we added an additional constraint In the primal problem this additional constraint transforms the trace equality (C.22) into the inequality Tr(P ) 1. With this condition the optimal value will be zero if a solution exists and no functional is found, or strictly negative if the opposite happens. We used SDPA and SDPA-GMP solvers [120, 121], which use an interior point method that simultaneously optimizes both the primal and dual problems, and that terminates when the duality gap is below a certain (small) threshold. This requires a strictly feasible solution to both the primal and the dual problem, and our formulation of the problem ensures that such strictly feasible solutions exist. Furthermore, we found that a normalization of the form given in (C.22) improves numerical stability compared to other normalizations such as, e.g., aici = 1. We ascribe this di erence to the fact that aici naturally tends to zero in physically interesting regions, and so setting it to one as a normalization leads to large numbers elsewhere.40 In order to achieve maximal numerical stability we `renormalized' many of the numbers fed into the problem. For example, the polynomials P i( ) can be rede ned by multiplying with an overall (positive) constant, by a ne rede nitions of , and by choosing a di erent basis for the space of derivatives. Altogether these reparametrizations give us the freedom to transform the problem according to P i( ) ! MjiP j (a + b) ; ci ! Mjicj : (C.29) We choose Mji, a, and b so as to minimize the potential for numerical inaccuracies. Numerical stability can be further improved by rescaling the normalization condition Tr(X) = 1 to Tr(X) = for a positive real . (In the dual problem becomes the cost vector, so this parameter is introduced through the optimization of instead of .) In order to avoid 40Our normalization is not suitable for obtaining bounds on OPE coe cients. In that case we need to normalize the functional as described in section 5. large numerical di erences between the primal and the dual formulation, we choose large so that X, which is a matrix of size O(103), can have O(1) entries on its diagonal. In previous implementations of the numerical bootstrap as a semide nite program [39], it was necessary to employ the arbitrary precision solver SDPA-GMP to avoid numerical instabilities. The setup described above, with Slater's condition satis ed and coe cients that are suitably renormalized, has allowed us to use the double precision SDPA program for low and intermediate values of . Since working at machine precision is signi cantly faster than working at arbitrary precision, we were able to explore a much greater range of the parameter space given our computational resources. We still found it necessary to switch to SDPA-GMP for higher values of , with the exact transition value somewhat dependent on the problem at hand. For example, we had to switch at = 16 for the bounds on theories with e6 avor symmetry shown in section 6, but were able to obtain reliable results with double precision numerics up to = 22 for some of the bounds on theories with su(2) avor symmetry. Typical settings for the parameters of both the SDPA and SDPA-GMP solvers can be found in table 8. D Polynomial approximations and conformal blocks takes the following form, The semide nite programming approach to the numerical bootstrap depends on our ability to approximate conformal blocks of xed spin ` and varying conformal dimension by polynomials in [39, 44]. This appendix includes a brief review of these approximations and some details relevant to the special cases of interest. The goal is to express the conformal blocks and their derivatives in a factorized form, with one factor being a function that can be well approximated by a polynomial in , and the other a non-polynomial term that is strictly positive and independent of the choice of derivative. We denote the by Pm(`;)n( ) and the non-polynomial term by ( ; `), so the approximation ( ; `)Pm(`;)n( ) : (D.1) The starting point for this approximation scheme is a recursion relation for derivatives of the hypergeometric functions appearing in conformal blocks, d 2 dz2 + 1 a z 1 b d dz 2 + abz 1) z 2F1 ( a; b; 2 ; z) = 0 : (D.2) This recursion relation follows immediately from the fact that the 2F1 hypergeometric function is a solution to Euler's di erential equation. Using this relation, any derivative of the above hypergeometric function at xed z can be expressed as the sum of the zeroth and rst order derivatives of the same hypergeometric function, each with some polynomial in as a prefactor. Thus the only non-polynomial feature of any derivative of the hypergeometric function can be expressed in terms of the value of the hypergeometric function itself and that of its rst derivative. To approximate conventional conformal blocks we follow exactly the same steps as in [39]. From (B.1) any derivative of a conformal block @zm@znG(`)(z; z) can be rewritten, by recursive use of (D.2) with a = b = 0, in terms of the hypergeometric functions and their rst derivatives. These functions encode all of the non-polynomial dependence on . We can then pull out factors out of the blocks such that the leftover expression can be well approximated by polynomials. To start we factor out the following term 1 z 2F1( ; ; 2 ; z) z 2F1( ; ; 2 ; z) z= 12 Here we have 2+` , ` 2 . This is positive for all for any conformal block appearing in a unitary theory. After factoring out this positive non-polynomial term, the remaining non-polynomial dependence is isolated in the following 1, and so it is positive ratio (and a similar one for ), K z 2F1( ; ; 2 ; z) z= 12 ' p 1 YM ( 2 j=0 ( rj ) sj ) NM ( ) DM ( ) The coe cient rj is the j-th zero of 2F1( ; ; 2 ; z) and sj the j-th zero of @@z z 2F1( ; ; 2 ; z).41 The rational function DNMM (( )) is an approximation of K obtained by restricting to the rst M zeroes of both the numerator and denominator. The approximation becomes arbitrarily good as M is increased, and converges very quickly, as described in [39]. The last step is to multiply by D( )D( ), which is strictly positive for the same range of and . In this way we have factored out all of the nonpolynomial dependence of @zm@znG(`)(z; z), which de nes ( ; `) in (D.1), and are left with a polynomial in Pm(`;)n( ), whose degree is controlled by the number of terms M kept in the approximation (D.4). Exactly this approximation is used for the blocks in the 2^ channel for the Er , ! + 4). For superconformal blocks in the 1^ channel given in (4.3) the procedure is analogous. This time we use (D.2), where now a = 1 and b = 1, to write all of the block derivatives in terms of the zeroth and second derivatives of the hypergeometric function. In this case we de ne 2 +`+2 and 2 ` . The rst step is again to factor out z= 12 : : (D.3) (D.4) z 2F1( ; ; 2 ; z) z 2F1( ; ; 2 ; z) ; (D.6) z= 12 which is positive for all possible values of and occurring in the relevant OPE ( ; 1). The remaining nonpolynomial dependence is then encoded by ratios of hypergeometric functions and their second derivatives. As it happens, an application of various identities 41In practice we compute the zeros of the latter by making use of the following identity, which relates it to another hypergeometric function dn h dzn z a+n 12F1 ( where in this case we want to use n = 1, and we have a = 0. a; b; 2 ; z)i = ( a)n z a 12F1 ( a + n; b; 2 ; z) ; (D.5) for hypergeometric functions (cf. [122]) allows us to express this nonpolynomial quantity in terms of the same function K , so we utilize the same approximation of (D.4) and nd 2F1 2F1 1; + 1; + 1; 2 ; 1 + 1; 2 ; 1 2 2 1 + 4( 4 + 8( 1)K 1)K DM ( ) + 4( ' 4DM ( ) + 8( 1)NM ( ) 1)NM ( ) : (D.7) Here we used (D.5) to relate the second derivative of the hypergeometric function to a di erent hypergeometric function. A similar ratio appears for the dependent hypergeometric functions, which we approximate in the same way. After approximating K by (D.4) we can again factor out another strictly positive denominator (4DM ( ) + 8( 1)NM ( ))(4DM ( ) + 8( 1)NM ( )) for the same range of ; . The approximation for the braided superconformal block goes in the same way. (We will now ignore the dependence since it is simply obtained by sion below.) We start by noting that braiding the block has the following e ect on the ! in the discushypergeometric functions [122] z z 1 2F1 1; + 1; 2 ; = (1 1 2F1 ( 1; 1; 2 ; z) : (D.8) The next step is now to write all derivatives in terms of the zeroth and second derivatives of the hypergeometric function by means of (D.2) with a = 1, b = 1. We can then again factor out any nonnegative and nonpolynomial terms, beginning with z 2F1( 1; 1) which is strictly positive for 1. The residual non-polynomial dependence is then 1; 2 ; 12 )( given by 2F1 2F1 1; 1; + 1; 2 ; 1 1; 2 ; 1 2 2 = 4 2F1 2F1 1; + 1; + 1; 2 ; 1 + 1; 2 ; 1 2 2 ' 4 DM ( ) + 4( 4DM ( ) + 8( 1)NM ( ) 1)NM ( ) ; where we have rewritten, through (D.5), the second derivative of the hypergeometric function as z 22F1 ( 1; + 1; 2 ; 1=2), and used several hypergeometric identities. For the relevant range of the denominator in the above equation is strictly positive, and it is the nal term to be factored out. The ratio rj =sj tends to one extremely fast, and we observed that truncating the product in (D.4) at M = 4 was already accurate enough for all 22. For 22 < we found that M = 5 was su cient. In a number of cases we repeated the numerical analysis with M = 6 and veri ed that there was no change in the results. E Exact OPE coe cients for the N = 2 chiral ring The OPE coe cients of Coulomb branch chiral ring operators in four-dimensional N = 2 SCFTs satisfy four-dimensional tt? equations [117, 118]. In this appendix we limit our attention to the case of theories with a conformal manifold that has one complex dimension, i.e., theories with just a single E2 multiplet. In such cases there is a close connection between the chiral ring OPE coe cients and the Zamolodchikov metric on the conformal manifold. (D.9) where g is the only nonvanishing component of the Zamolodchikov metric on the conformal manifold.42 On the right-hand side we recognize the expression for the scalar curvature of the Zamolodchikov metric. The bounds reported in section 7 for 2 therefore provide HJEP03(216)8 lower and upper bounds on this curvature. Let us consider a few examples, starting with the theory of n free vector multiplets. The superconformal primary of the avor singlet E2 multiplet in this theory is 'a'a(x), with '(x) the scalar operator in the vector multiplet. We can compute performing Wick contractions, whereupon we nd 2 E4 directly by E4 N = 4 super Yang-Mills: 2E4 = 2 + 1 c n free vector multiplets: n = 2 + 2 3c : In the last equality we have used the precise value of the central charge in this theory: c = n=6. In any N = 2 superconformal gauge theory with gauge group G, the tree-level value for this OPE coe cient takes the same form, tree level gauge theory: 2E4 = 2 + 4 dim(G) 2 + 2 3c : The inequality is a consequence of the fact that the central charge of a superconformal gauge theory is always greater than that of the vector multiplets alone. In N = 4 supersymmetric Yang-Mills theory, the central charge is c = 14 dim(G). In this special case, extended supersymmetry prevents the OPE coe cient in question from being renormalized. Consequently the exact value (for all values of the complex gauge coupling) is given by the tree-level result, After diagonalization of the elds, the OPE of the (unit normalized) chiral operators takes the form E2(x)E2(0) = E4 E4(0) + : : : ; and we are interested in the squared OPE coe cient 2 . Precisely this coe cient is part of a solvable subsector of the tt equations and it takes the form 2E4 = 2 + g = 2 2 the Kahler metric on the conformal manifold and the S4 partition function [123], In many N = 2 SCFTs, this OPE coe cient is made accessible by the relation between It is frequently the case that the partition function ZS4 can be computed exactly using supersymmetric localization [124]. As an example, consider N = 2 SCQCD with Nf = 4 avors (sometimes referred to in the text as the so(8) theory ). The Nekrasov instanton 42In the notations of [118], this is the metric written as gij. This di ers from the true Zamolodchikov metric Gij by a factor of 192. (E.1) (E.2) (E.3) (E.4) (E.5) (E.6) partition function that features in the localization result is related to four-point Virasoro conformal blocks [125]. These in turn are e ciently computed using the recursion relations developed in [126]. Altogether, one ultimately nds the following expression for the S4 partition function, log ZS4 (q) = log Z 1 1 da a2j16qj2a2 G(1 + 2ia)2 2 G(1 + ia)8 H(a; q)H(a; q) + f ( ) + f ( ) ; (E.7) where the functions f ( ) are Kahler transformations that drop out in the computation of the curvature, and G(z) is Barnes' G-function.43 The function H(a; q) has been de ned in [126] by means of a somewhat intricate recursion relation that we will not review here. It is a building block of the Virasoro four-point conformal block with c = 25, all four external dimensions equal to one, and internal dimension equal to 1 + a2. The rst few terms in its series expansion take the form H(a; q) = 1 + 12 a2 + 2 q 2 (4a2 + 9)2 4 (4a2 + 9)2 (4a2 + 25)2 One should note that the parameter q is qIR which is not the same parameter as the parameter qUV used in [124] and in [117, 118].44 The relation between the two is given in [125], and also in [126], HJEP03(216)8 qIR = exp(i IR) = exp K(1 qUV) K(qUV) Here K(m) is the complete elliptic integral of the rst kind.45 The explicit form of this transformation is in fact not particularly relevant for our purposes because the scalar curvature is a di eomorphism invariant. But it is IR that is valued in the fundamental domain for the action of SL(2; Z) on the upper half plane. Namely, under S- and T transformations we have T : S : IR ! 1= IR ; IR ! IR + 1 ; qUV ! 1 qUV ; qUV ! qUV 1 : The transformations of qUV describe the action of crossing symmetry on the Liouville The value of the OPE coe cient 2 ( ) can be computed numerically to arbitrary accuracy at any value of the coupling. The free- eld value is given by 2 ( = 1) = 10=3. E4 The OPE coe cient decreases monotonically as a function of the gauge coupling and becomes stationary at the self-dual points. To get reasonable accuracy we need to expand H(q) to order q8, resulting in the following stationary values: E4 2 ( = i) = 2:8983769 : : : E4 2 ( = ei =3) = 2:8940994 : : : (E.12) 43This function is implemented in Mathematica as BarnesG[z]. 44An early discussion of this point can be found in [127]. (E.10) (E.11) function of the exactly marginal complexi ed gauge coupling = 2 + 4g2i , and the fundamental domain for the action of SL(2; Z)-duality on the coupling plane is outlined in red. This OPE coe cient is plotted in gure 27. 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Christopher Beem, Madalena Lemos, Pedro Liendo. The \( \mathcal{N}=2 \) superconformal bootstrap, Journal of High Energy Physics, 2016, 183, DOI: 10.1007/JHEP03(2016)183