The superconformal bootstrap
We develop the conformal bootstrap program for six-dimensional conformal field theories with supersymmetry, focusing on the universal four-point function of stress tensor multiplets. We review the solution of the superconformal Ward identities and describe the superconformal block decomposition of this correlator. We apply numerical bootstrap techniques to derive bounds on OPE coefficients and scaling dimensions from the constraints of crossing symmetry and unitarity. We also derive analytic results for the large spin spectrum using the lightcone expansion of the crossing equation. Our principal result is strong evidence that the theory realizes the minimal allowed central charge for any interacting theory. This implies that the full stress tensor four-point function of the theory is the unique unitary solution to the crossing symmetry equation at . For this theory, we estimate the scaling dimensions of the lightest unprotected operators appearing in the stress tensor operator product expansion. We also find rigorous upper bounds for dimensions and OPE coefficients for a general interacting theory of central charge . For large , our bounds appear to be saturated by the holographic predictions obtained from eleven-dimensional supergravity.
Keywords:conformal field theory, supersymmetry, conformal bootstrap
1 Introduction and summary
In this work we introduce and develop the modern conformal bootstrap program for superconformal theories in six dimensions. These theories provide a powerful organizing principle for lower-dimensional supersymmetric dynamics. From their existence one can infer a vast landscape of supersymmetric field theories in various dimensions and rationalize many deep, nonperturbative dualities that act within this landscape (see, e.g., Gaiotto:2009we (); Gaiotto:2009hg (); Dimofte:2011ju (); Bah:2012dg (); Gadde:2013sca ()). Despite their increasingly central role, the theories have proved stubbornly resistant to study by traditional quantum field theory techniques. This situation, coupled with the high degree of symmetry and conjectured uniqueness of these theories, suggests that the theories may be a prime target for the conformal bootstrap approach.
It has been known since the work of Nahm Nahm:1977tg () that superconformal algebras can only be defined for spacetime dimension less than or equal to six. In six dimensions, the possible superconformal algebras are the algebras . The maximum six-dimensional superconformal algebra for which there can exist a stress tensor multiplet is the algebra Cordova_banff (). Thus the six-dimensional theories are singled out as the maximally supersymmetric local CFTs in the maximum number of dimensions.
While it is easy to identify a free field theory that realizes superconformal symmetry – namely the abelian tensor model – the existence of interacting theories was only inferred in the mid 1990s on the basis of string theory constructions Witten:1995zh (); Strominger:1995ac (). A decoupling limit of Type IIB string theory on ALE spaces predicts the existence of a list of interacting theories labelled by the simple and simply-laced Lie algebras Witten:1995zh (). The model can also be realized as the low-energy limit of the worldvolume theory of coincident M5 branes in M-theory Strominger:1995ac (). Two important properties of these theories can be deduced almost immediately from their string/M-theory constructions:
On , the theory of type has a moduli space of vacua given by the orbifold
where is the rank of the simply laced Lie algebra and its Weyl group. At a generic point in the moduli space, the long distance physics is described by a collection of decoupled free tensor multiplets.
Upon supersymmetric compactification on , the theory of type reduces at a low energies to a maximally supersymmetric five-dimensional Yang-Mills theory with gauge algebra , whose dimensionful gauge coupling is controlled by the radius, .
The string/M-theory constructions give physically compelling evidence for the existence of the interacting theories, but the evidence is indirect and requires an extensive conceptual superstructure. Moreover, at present these constructions do not provide tools for computing anything in the conformal phase of the theory beyond a very limited set of robust observables, such as anomalies.111 The ’t Hooft anomalies for the -symmetry, as well as the gravitational and mixed anomalies, are encoded in an eight-form anomaly polynomial. First obtained for the and theories by anomaly inflow arguments in M-theory Duff:1995wd (); Witten:1996hc (); Freed:1998tg (); Harvey:1998bx (); Yi:2001bz (), the anomaly polynomial can also be reproduced by purely field-theoretic reasoning Intriligator:2000eq (); Ohmori:2014kda (), relying only on properties (i) and (ii). A similar field-theoretic derivation has recently been performed for the -type Weyl anomaly Cordova:2015vwa (), i.e., the coefficient in front of the Euler density in the trace anomaly. Finally, the -type anomalies, i.e., the coefficients in front of the the three independent Weyl invariants in the trace anomaly, are proportional to each other and related by supersymmetry to the two-point function of the canonically normalized stress tensor, see (2) below.
At large , the theories can be described holographically in terms of eleven-dimensional supergravity on Maldacena:1997re ().222There is an analogous conjecture for the theories, relating them to supergravity on Witten:1998xy (); Aharony:1998rm (). The AdS/CFT correspondence then renders these large theories extremely tractable. However, the extension to finite is presently only possible at leading order. Higher order corrections require a method for computing quantum corrections in M-theory. An intrinsic field theoretic construction of the theories would therefore be indispensable. Turning this logic around, such an independent definition would offer a window into quantum M-theory.
However, most standard field theory methods are inadequate for describing the theories. The mere existence of unitary, interacting quantum field theories in appears surprising from effective field theory reasoning – conventional local Lagrangians are ruled out by power counting. The theories are isolated, intrinsically quantum mechanical conformal field theories, which cannot be reached as infrared fixed points of local RG flows starting from a gaussian fixed point. This is in sharp contrast to more familiar examples of isolated CFTs in lower dimensions, such as the critical Ising model in three dimensions. It is unclear whether an unconventional (non-local?) Lagrangian for the interacting theories could be written down (see, e.g., Lambert:2010wm (); Ho:2011ni (); Chu:2012um (); Bonetti:2012st (); Samtleben:2012fb (); Saemann:2013pca () for a partial list of attempts), but in any event it would be unlikely to lend itself to a semiclassical approximation.
There have been several attempts at field-theoretic definitions of the theories. The most concrete proposal Aharony:1997th (); Aharony:1997an () relates the Discrete Light Cone Quantization (DLCQ) of the theory (with units of lightcone momentum) to supersymmetric quantum mechanics on the moduli space of instantons – to recover the theory on one is instructed to send . This moduli space is singular due to small instanton singularities. A specific regularization procedure has been advocated in Aharony:1997an (), but its conceptual status seems somewhat unclear, and one may be concerned that important features of the theory might be hidden in the details of the UV regulator. The DLCQ proposal remains largely unexplored due to the inherent difficulty of performing calculations in a strongly coupled quantum mechanical model, and the need to take the large limit. To the best of our knowledge, the only explicit result obtained in this framework is the calculation of the half BPS spectrum of the theory, carried out in Aharony:1997an (). According to another little-explored proposal, the theory on (with a finite-size ) can be “deconstructed” in terms of four-dimensional quiver gauge theories ArkaniHamed:2001ie (). Finally there is the suggestion Douglas:2010iu (); Lambert:2010iw () that five-dimensional maximally supersymmetric Yang-Mills theory is a consistent quantum field theory at the nonperturbative level, without additional UV degrees of freedom, and that it gives a complete definition of the theory on . The conceptual similarities shared by these three proposals have been emphasized in Lambert:2012qy (). It would be of great interest to develop them further, ideally to the point where quantitative information for the non-protected operator spectrum could be derived and compared to the bootstrap results obtained here.
In the present work, we will eschew the problem of identifying “the fundamental degrees of freedom” of the theories. Indeed, we are not certain that this question makes sense. Instead, we will do our best to rely exclusively on symmetry. We regard the theories as abstract conformal field theories (CFTs), to be constrained – ideally, completely determined – by bootstrap methods. The conformal bootstrap program was formulated in the pioneering papers Ferrara:1973yt (); Ferrara:1973vz (); Polyakov:1974gs () and has undergone a modern renaissance starting with Rattazzi:2008pe (). The algebra of local operators, which is defined by the spectrum of local operators and their operator product expansion (OPE) coefficients, is taken as the primary object. The conformal bootstrap aims to fix these data by relying exclusively on symmetries and general consistency requirements, such as associativity and unitarity.333 In principle, the bootstrap for local operators could be enlarged by also including non-local operators (such as defects of various codimensions), or by considering non-local observables (such as partition functions in non-trivial geometries), or both. Apart from the calculation of new interesting observables, such an enlarged framework may yield additional constraints on the local operator algebra itself. This is familiar in two-dimensional CFT, where modular invariance imposes additional strong restrictions on the operator spectrum. Constraints arising from non-trivial geometries are however much less transparent in higher-dimensional CFTs, and their incorporation in the bootstrap program remains an open problem. Refs Liendo:2012hy (); Gaiotto:2013nva (); Gliozzi:2015qsa () contain some work on the bootstrap in the presence of defects. Intuitively, we expect this approach to be especially powerful for theories that are uniquely determined by their symmetries and perhaps a small amount of additional data, such as central charges. The theories, with their conjectured ADE classification and absence of exactly marginal deformations, are an attractive target.
We are making the fundamental assumption that the theories admit a local operator algebra satisfying the usual properties. This may warrant some discussion, in light of the fact that the theories require a slight generalization of the usual axioms of quantum field theory Witten:1996hc (); Aharony:1998qu (); Witten:1998wy (); Moore:2004jv (); Freed:2006yc (); Witten:2009at (); Henningson:2010rc (); Freed:2012bs (). In contrast to a standard QFT, each theory does not have a well-defined partition function on a manifold of non-trivial topology (with non-trivial three-cycles), but it yields instead a vector worth of partition functions. This subtlety does not affect the correlation functions of local operators on where there are no interesting three-cycles to be found. That the theories of type and have a conventional local operator algebra is manifest from AdS/CFT, at least for large . The extrapolation of this property to finite is a very plausible conjecture. Ultimately, in the absence of an alternative calculable definition of the theories, we are going to take the existence of a local operator algebra as axiomatic. Our work will give new compelling evidence that this is a consistent hypothesis.
In this paper, we focus on the crossing symmetry constraints that arise from the four-point function of stress tensor multiplets. This is a natural starting point for the bootstrap program, since the stress tensor is the one non-trivial operator that we know for certain must exist in a theory. By superconformal representation theory, the stress tensor belongs to a half BPS multiplet, whose superconformal primary (highest weight state) is a dimension four scalar operator in the two-index symmetric traceless representation of . It is equivalent but technically simpler to focus on the four-point function of , which contains the same information as the stress tensor four-point function. (Indeed, the two-, three- and four-point functions of all half BPS multiplets are known to admit a unique structure in superspace Eden:2001wg (); Arutyunov:2002ff (); Dolan:2004mu ().)
The two- and three-point functions of the stress tensor supermultiplet are uniquely determined in terms of a single parameter, the central charge . Normalizing to be one for the free tensor multiplet, the central charge of the theory of type is given by Beem:2014kka ()444This is the unique expression compatible with the structure of the eight-form anomaly polynomial and with the explicit knowledge of for the theories at large , which is available from a holographic calculation Henningson:1998gx (); Henningson:1998ey ().
where , and are the dimension, dual Coxeter number, and rank of , respectively. For example, this gives , which exhibits the famous growth of degrees of freedom. Equation (2) gives the value of the central charge for all the known theories, but we do not wish to make any such a priori assumptions about the theories we are studying. We will therefore treat the central charge as an arbitrary parameter, imposing only the unitarity requirement that it is real and positive.
Building on previous work Ferrara:2001uj (); Eden:2001wg (); Arutyunov:2002ff (); Dolan:2004mu (); Heslop:2004du (), we impose the constraints of superconformal invariance on the four-point function of and decompose it in a double OPE expansion into an infinite sum of superconformal blocks. Schematically we have
The sum is over all the superconformal multiplets allowed by selection rules to appear in the OPE. Each multiplet is labelled by the corresponding superconformal primary , with associated superconformal block , a known function of the two independent conformal cross ratios. Finally denotes the OPE coefficient. We further assume that no conserved currents of spin appear in this expansion. Very generally, the presence of higher-spin conserved currents in a CFT implies that the theory contains a free decoupled subsector Maldacena:2011jn (), while we wish to focus on interacting theories. There are three classes of supermultiplets that contribute in the double OPE expansion (3):
An infinite set of BPS multiplets, whose quantum numbers are known from shortening conditions and whose OPE coefficients can be determined in closed form using crossing symmetry, as functions of the central charge . For this particular four-point function, this follows from elementary algebraic manipulations, but there is a deeper structure underlying this analytic result. The operator algebra of any theory admits a closed subalgebra, isomorphic to a two-dimensional chiral algebra Beem:2013sza (); Beem:2014kka (). Certain protected contributions to four-point functions of BPS operators of the theory are entirely captured by this chiral algebra. In the case at hand, the operator corresponds to the holomorphic stress tensor of the chiral algebra, with central charge , while the operators map to products of derivatives of .
Another infinite tower of BPS multiplets . The operators have spin , while the single operator is a scalar.555Here we use an abbreviated notation for the supermultiplets, dropping their -charge quantum numbers. The translation to the precise notation introduced in section 2 is as follows: , , . All their quantum numbers (including the conformal dimension) are known from shortening conditions, but their contribution to the four-point function is not captured by the chiral algebra.
An infinite set of non-BPS operators , labelled by the conformal dimension and spin . Only even contribute because of Bose symmetry. Both the set of entering the sum and the OPE coefficients are a priori unknown. Unitarity gives the bound . It turns out that
so we can view the BPS contributions of type (ii) as a limiting case of the non-BPS contributions.
We then analyze the constraints of crossing symmetry and unitarity on the unknown CFT data . Some partial analytic results can be derived by taking the Lorentzian lightcone limit of the four-point function. As shown in Fitzpatrick:2012yx (); Komargodski:2012ek (), crossing symmetry relates the leading singularity in one channel with the large spin asymptotics in the crossed channel. By this route we demonstrate that the BPS operators are necessarily present – at least for large – and compute the asymptotic behavior of the OPE coefficients as . To proceed further we must resort to numerics. There is by now a standard suite of numerical techniques to derive rigorous inequalities in the space of CFT data, following the blueprint of Rattazzi:2008pe (). The numerical algorithm requires us to choose a finite-dimensional space of linear functionals that act on functions of the conformal cross ratios. We parametrize the space of functionals by an integer . Greater corresponds to a bigger space of functionals, and hence more stringent bounds.
Summary of results
The most fundamental bound is for the central charge itself. We derive a rigorous lower bound . This is the bound for the maximal value of allowed by our numerical resources. However, it turns out that the lower bound on has a very regular dependence on the cut-off , see Fig. 1. This leads to the compelling conjecture that it converges exactly to as . This is precisely the central charge for the theory, the smallest central charge of all the known interacting theories.666 Recall that for the free tensor theory, but we are excluding free theories in our ansatz by not allowing for higher spin conserved currents in the operator algebra.
This result rules out the existence of exotic theories with a central charge smaller than that of the theory. A much stronger conclusion follows if one accepts the standard bootstrap wisdom Poland:2010wg (); ElShowk:2012hu () that the crossing equation has a unique unitarity solution whenever a bound is saturated. We are then making the precise mathematical conjecture that for the CFT data contained in (3) are completely determined by the bootstrap equation. We test this conjecture by extrapolating various observables to using different schemes, always finding a consistent picture. As an example, we determine numerically the dimension of the leading-twist scalar non-BPS operator, , see Fig. 10. (The range of values reflects our estimate of the uncertainty in the extrapolation.) One is tempted to further speculate that all other crossing equations will also have unique solutions, i.e., that the theory can be completely bootstrapped.
An important check of our claim that for we are bootstrapping the theory follows from examining the dependence of the OPE coefficient . We find that it is zero for . In fact, it is precisely the vanishing of this OPE coefficient that is responsible for the existence of a lower bound on . For , the squared OPE coefficient becomes negative, violating unitarity. Now the operator is a chiral ring operator, and the chiral spectrum of the theory of type has been computed Bhattacharyya:2007sa () assuming (1) and a standard folk theorem relating chiral operators to holomorphic functions on the moduli space. This analysis reveals that is absent in the theory, in nice agreement with the vanishing of for .
We also derive bounds on operator dimensions and OPE coefficients in the entire range . For , our bounds must be compatible with the existence of a known unitary solution of the crossing equation, given by the holographic four-point correlator of evaluated from supergravity. For strictly infinite central charge, this is simply the “generalized free field theory” answer, for which the four-point function factorizes into products of two-point functions. The supergravity answer gives a non-trivial correction to the disconnected result. We find compelling evidence that the holographic answer – including the correction – saturates the best possible numerical bounds of this type. The same phenomenon has been observed for SCFTs in four dimensions Beem:2013qxa ().
In summary, we find strong evidence that both for small and for large central charge the bootstrap bounds are saturated by actual SCFTs – the theory for and the theory for . It is natural to conjecture that all theories saturate the bounds at the appropriate value (2) of the central charge. The bounds depend smoothly on , and when they are saturated one expects to find a unique unitary solution of this particular crossing equation. Presumably, only the discrete values (2) of the central charge will turn out to be compatible with the remaining, infinite set of bootstrap equations.
The bootstrap results derived from the four-point function are completely universal. The only input is the existence of itself, which is tantamount to the existence of a stress tensor. An obvious direction for future work is to make additional spectral assumptions, leveraging what is conjecturally known about the theories. A relevant additional piece of data is the half BPS spectrum, which is easily deduced from (1). In the theory of type , the half BPS ring is generated by operators. These generators are in one-to-one correspondence with the Casimir invariants of and have conformal dimension , where is the order of the Casimir invariant. The operator corresponds to the quadratic Casimir, it is always the lowest-dimensional generator and is in fact the unique generator for the theory.
The natural next step in the bootstrap program is then to consider four-point correlators of higher-dimensional half BPS operators – both individual correlators and systems of multiple correlators.777The study of multiple correlators has proved extremely fruitful in the bootstrap of CFTs. For example, they have led the world’s most precise determination of critical exponents for the critical Ising model, with rigorous error bars Kos:2014bka (); Simmons-Duffin:2015qma (). The protected chiral algebra associated to the theory of type has been identified with the algebra Beem:2014kka () and will be an essential tool in the analysis of these more general correlators. The chiral algebra controls an infinite amount of CFT data, which would be very difficult to obtain otherwise. For the four-point function only the universal subalgebra generated by the holomorphic stress tensor is needed, but more complicated correlators make use of the very non-trivial structure constants of the algebra. We have seen that there is a sense in which the theory is uniquely cornered by the vanishing of the OPE coefficient , which reflects the chiral ring relation that sets to zero the quarter BPS operator . The higher-rank theories admit analogous chiral ring relations, which imply certain relations amongst the OPE coefficients appearing in a suitable system of multiple correlators. At least in principle, this gives a strategy to bootstrap the general theory of type .
The remainder of this paper is organized as follows. In section 2 we provide some useful background on the six-dimensional theories and discuss how to formulate the corresponding bootstrap program in full generality. Sections 3, 4, and 5 contain the nuts and bolts of the bootstrap setup considered in this paper: they contain, respectively, the detailed structure of the correlator, its superconformal block decomposition, and a review of the numerical approach to the bootstrap. The results of our numerical analysis are then presented in section 6, and supplementary material can be found in the appendices. Casual readers may limit themselves to section 2 and the discussion surrounding Figs. 1, 2, and 10 in section 6.
2 The bootstrap program for theories
A great virtue of the bootstrap approach to conformal field theory is its generality. Indeed, this will be the reason that we can make progress in studying the conformal phase of SCFTs despite the absence of a conventional definition. Thus in broad terms this work will mirror many recent bootstrap studies Rychkov:2009ij (); Vichi:2009zz (); Caracciolo:2009bx (); Poland:2010wg (); Rattazzi:2010gj (); Rattazzi:2010yc (); Vichi:2011ux (); Poland:2011ey (); ElShowk:2012ht (); Liendo:2012hy (); ElShowk:2012hu (); Beem:2013qxa (); Gliozzi:2013ysa (); Kos:2013tga (); El-Showk:2013nia (); Alday:2013opa (); Gaiotto:2013nva (); Berkooz:2014yda (); El-Showk:2014dwa (); Gliozzi:2014jsa (); Nakayama:2014lva (); Nakayama:2014yia (); Alday:2014qfa (); Chester:2014fya (); Chester:2014mea (); Kos:2014bka (); Caracciolo:2014cxa (); Paulos:2014vya (); Bae:2014hia (); Simmons-Duffin:2015qma (); Gliozzi:2015qsa (); Bobev:2015vsa (); Bobev:2015jxa (); Kos:2015mba (). We will not review the basic philosophy in any detail here. Instead, the purpose of this section is to describe in fairly general terms how supersymmetry affects the bootstrap problem, and also to review some aspects of the known theories that are relevant for this chapter of the bootstrap program. In subsequent sections we will provide a more detailed account of the specific crossing symmetry problem we are studying, culminating in a bootstrap equation that can be fruitfully analyzed.
2.1 Local operators
The basic objects in the bootstrap approach to CFT are the local operators, which are organized into representations of the conformal algebra. The local operators in a unitary SCFT must further organize into unitary representations of the superconformal algebra. A unitary representation of is a highest weight representation and is completely determined by the transformations of its highest weight state (the superconformal primary state) under a maximal abelian subalgebra. For generators of the maximal abelian subalgebra we take the generators of rotations in three orthogonal planes in , generators and of a Cartan subalgebra of , and the dilatation generator . We define the quantum numbers of a state with respect to these generators as follows,888See Appendix A for our naming conventions and more details about these representations.
There are five families of unitary representations, each admitting various special cases. These families are characterized by linear relations obeyed by the quantum numbers of the superconformal primary state Dobrev:2002dt (); Minwalla:1997ka ():
Representations of type are called long or generic representations, and their scaling dimension can be any real number consistent with the inequality in (6). The other families of representations are short representations.999In the literature a distinction is sometimes drawn between short and semi-short representations. We make no such distinction here. These representations have additional null states appearing in the Verma module built on the superconformal primary by the action of raising operators in . These representations are also sometimes called protected representations or – in an abuse of terminology – BPS representations.
Now let us review what is understood about the spectrum of local operators in the known theories.
BPS operators and chiral rings
Perhaps the most familiar short representations are those of type . The highest weight states for these representations are scalars that are half BPS (annihilated by two full spinorial supercharges) if , and they are one quarter BPS (annihilated by one full spinorial supercharge) otherwise. As an example, the multiplet is just the abelian tensor multiplet, whose primary is a free scalar field (with scaling dimension ) transforming in the of . A more important example in the present paper is the multiplet. The superconformal primary in this multiplet is a scalar operator of dimension four that transforms in the of . This multiplet also contains the -symmetry currents, supercurrents, and the stress tensor for the theory, so such a multiplet should always be present in the spectrum of a local theory.
The BPS operators form two commutative rings – the half and quarter BPS chiral rings. The OPE of BPS operators is non-singular, and multiplication in the chiral rings can be defined by taking the short distance limit of the OPE of these BPS operators.101010Alternatively, one may define these rings cohomologically by passing to the cohomology of the relevant supercharges. In the known theories of type , these rings can be identified as the coordinate rings of certain complex subspaces of the moduli space of vacua – they take relatively simple forms Bhattacharyya:2007sa ():
where is the rank of and is the Weyl group acting in the natural way.
Knowing these rings for a given theory determines the full spectrum of -type multiplets in said theory.111111It isn’t quite the case that the ring elements are in one-to-one correspondence with the multiplets. This is because half BPS operators have descendants that are quarter BPS. Note, however, that the ring structure for these operators does not determine numerically the value of any OPE coefficients since the normalizations of the BPS operators corresponding to particular holomorphic functions on the moduli space are unknown. To put it another way, if we demand that all BPS operators be canonically normalized, then the structure constants of the chiral ring are no longer known.
Chiral algebra operators
There is a larger class of protected representations that participate in a more elaborate algebraic structure than the chiral rings – namely the protected chiral algebra introduced in Beem:2014kka () (extending the analogous story for SCFTs in four dimensions Beem:2013sza ()). Each of the following representations contains operators that act as two-dimensional meromorphic operators in the protected chiral algebra:121212Here we are reverting to the conventions for quantum numbers used in Appendix A. The are linearly related to the above in (70).
We observe that all half BPS operators and certain quarter BPS operators make an appearance in both the chiral rings and the chiral algebra, but the chiral algebra also knows about an infinite number of -type multiplets that are probably less familiar.
In Beem:2014kka () the chiral algebras of the known theories were identified as the affine -algebras of type , where is the same simply laced Lie algebra that labels the theory, so the spectrum of the above multiplets is known.131313This is the same chiral algebra that appears in the AGT correspondence in connection with the theory of type Alday:2009aq (); Wyllard:2009hg (). The precise connection between these two appearances of the same chiral algebra remains somewhat perplexing. We will have more to say about the information encoded in the chiral algebra below.
General short representations and the superconformal index
The full spectrum of short representations is encoded in the superconformal index up to cancellations between representations that can recombine (group theoretically) to form a long representation Bhattacharya:2008zy (). The allowed recombinations are reviewed in Appendix A. This means that the full index unambiguously encodes the number of representations of the following types:
It also provides lower bounds for the number of operators transforming in any short representation that appears in a recombination rule.
A proposal has been made for the full superconformal index of the theories in Kim:2012qf (); Kim:2013nva (); Lockhart:2012vp (). To our knowledge, this proposal has not yet been systematically developed to the point where it will produce the unambiguous spectral data mentioned above. For our purposes, we will not need the full superconformal index, but for future generalizations of our bootstrap approach it would be very helpful to develop the technology to such a point.141414Refs. Kim:2012ava (); Bullimore:2014upa () contain proposals for the index in an unrefined limit, where the enhanced supersymmetry leads to a simpler expression.
The situation for generic representations is much worse than that for short representations. Namely, in the known theories, the spectrum of long multiplets is almost completely mysterious. Outside of the the holographic regime, we are not aware of a single result concerning the spectrum of such operators. This is precisely the kind of information that one hopes will be attainable using bootstrap methods.
In the large limit of the theories, the full spectrum of local operators is known from AdS/CFT. Local operators are in one-to-one correspondence with single- and multi-graviton states of the bulk supergravity theory. Single-graviton states are the Kaluza-Klein modes of eleven-dimensional supergravity on Gunaydin:1984wc () and correspond to “single-trace” operators of the boundary theory. They can be organized into an infinite tower of half BPS representations of the superconformal algebra. Similarly, multi-graviton states in the bulk are dual to “multi-trace” operators of the boundary theory. They can be organized into a list of (generically long) multiplets of the superconformal algebra. At strictly infinite , the bulk supergravity is free so the energy of a multi-graviton state is the sum of the energies of its single-graviton constituents. This translates into an analogous statement for the conformal dimension of the dual multi-trace operator. The first finite correction to the conformal dimensions of these operators can be computed from tree-level gravitational interactions in the bulk.
Of particular interest to us will be the double-trace operators that are constructed from the superconformal primaries of the stress tensor multiplet:
Similar results can be obtained for more general double trace operators at large – see Heslop:2004du (). However, it should be noted that because the holographic dual is realized in M-theory, there is at present no method – even in principle – to generate the higher order corrections. This stands in contrast to the case of theories with string theory duals, where there is at least a framework for describing higher order corrections at large central charge.
2.2 OPE coefficients
The OPE coefficients of the known theories generally appear even more difficult to access than the spectrum of operators. Indeed, until recently the only three-point functions that were known for the finite rank theories were those that were fixed directly by conformal symmetry, i.e., those encoding the OPE of a conserved current with a charged operator.
Some information is available in the form of selection rules that dictate which superconformal multiplets can appear in the OPE of members of two other multiplets. These selection rules are nontrivial to derive, and provide useful simplifications when studying, e.g., the conformal block decomposition of four-point functions.
We are not aware of a complete catalogue of selection rules for the superconformal multiplets. However, an algebraic algorithm has been developed in Heslop:2004du () based on writing three-point functions in analytic superspace, and this should be sufficient to determine the selection rules in all cases. For some special cases the selection rules have been determined explicitly, cf. Ferrara:2001uj (); Eden:2001wg (); Arutyunov:2002ff (); Heslop:2004du (). We will use a particular case of this below in our discussion of half BPS operators.
OPE coefficients from the chiral algebra
Going beyond selection rules, the numerical determination of some OPE coefficients (aside from those mentioned above) has recently become possible as a consequence of the identification of the protected chiral algebras of the theories Beem:2014kka (). In particular, up to choice of normalization, the three-point couplings between three chiral-algebra-type operators in a theory will be equal to the three-point coupling of the corresponding meromorphic operators in the associated chiral algebra. The structure constants of the algebras are completely fixed in terms of the Virasoro central charge, which is related by general arguments to the -type Weyl anomaly coefficient of the corresponding theory, cf. (2).
Aside from these, the three-point functions of other protected operators are generally unknown – it would be very interesting if it were possible to determine, e.g., the three-point functions of quarter BPS operators not described by the chiral algebra, perhaps using some argument related to supersymmetric localization.
Undetermined short operators and generic representations
The situation for generic representations is again much worse, and aside from the OPEs encoding the charges of operators under global symmetries, we are not aware of any results for any OPE coefficients involving long multiplets outside of the holographic limit. At large many OPE coefficients can be computed at leading order in the expansion – see Heslop:2004du () for example. We will be particularly interested later in this paper in the three-point functions coupling two stress tensor multiplets and certain generalized double traces that turn out to realize the and multiplets mentioned above. In particular, we have
Let us introduce the slightly awkward convention that and . Then the values of the three-point couplings (squared) for these double traces take the following form:
In particular, for the first few low values of these are given by
2.3 Four-point functions of half BPS operators
In this paper we will be focusing on the four-point function of stress tensor multiplets. However, many nice features of the bootstrap problem for stress tensor multiplets occur more generally in studying the four-point function of arbitrary half BPS operators, and ultimately the approach developed in this paper should be extendable to this more general class of correlators without great conceptual difficulty. Therefore let us first give a schematic description of the crossing symmetry problem in this more general context, before specializing to the specific case of interest.
It is a convenient fact that the full superspace structure of a three-point function involving two half BPS multiplets and any third multiplet is completely determined by the three-point function of superconformal primary operators Eden:2001wg (). Since the superconformal primaries of half BPS representations are spacetime scalars, it follows that each such superspace three-point function is determined by a single numerical coefficient.
This simplicity of three-point functions (or equivalently of the OPE between half BPS operators) in superspace has pleasant consequences for the conformal block expansion of four-point functions and the associated crossing symmetry equation. In particular, it means that the superconformal block associated to the exchange of all operators in a superconformal multiplet is fixed up to a single overall coefficient, even though the exchanged operators could live in several conformal multiplets and transform in different representations of .
Thus, the conformal block decomposition will take the form of a sum over superconformal multiplets of a single real coefficient times a superconformal block. Schematically this takes the following form
Here runs over the irreducible representations appearing in both the and selection rules, runs over the different representations appearing in the supermultiplet , and we have introduced projection tensors . The precise form of these projectors is not particularly important – in the technical analysis of Section 4 we will introduce some additional structure in the form of complex -symmetry polarization vectors to simplify manipulations of these superconformal blocks.
The superconformal blocks for each are functions of conformal cross ratios, and their form is fixed in terms of the representations of the four external operators and . These functions also have fixed relative normalizations. This is a manifestation of the simplification mentioned above, and it means that there is only a single free numerical parameter that determines the contribution of a full supermultiplet to the four-point function.
The representations that can appear in the sum on the right hand side of (15) are constrained by the superconformal selection rules mentioned previously. For the particular case of the OPE of two half BPS operators, these selection rules have been studied starting with the work of Eden:2001wg (); Ferrara:2001uj (); Arutyunov:2002ff (), with the complete answer being given in Heslop:2004du (). The results are as follows:
An interesting point that will become relevant in a moment is that every short representation appearing on the right hand side of (2.3) is of one of two types:
Representations that appear in the decomposition a long multiplet in (2.3) at the appropriate unitarity bound.
Representations from the list (2.1) that include chiral algebra operators.
Notice in particular that for long multiplets that decompose at the unitarity bound (cf. Appendix A), only one of their irreducible components is allowed by the selection rules. This implies that the superconformal blocks for those short multiplets of type (A) above will simply be obtained as the limit of a long superconformal block when its scaling dimension is set to the unitarity bound.151515A consequence of this general structure is that the number of independent, unknown functions of conformal cross ratios appearing in the superconformal block expansion of a four-point function of half BPS operators is just equal to the number of different -symmetry representations in which the long multiplets in (2.3) transform.
Fixed and unfixed BPS contributions
The two types of short operators mentioned above will participate in the crossing symmetry problem very differently. Operators of type (B) will have their three-point functions with the external half BPS operators determined by the chiral algebra, so subject to identification of the chiral algebra their contribution to the four-point function will be completely fixed. For the known theories the chiral algebra has been identified, so for a general four-point function of half BPS operators we can put in some fairly intricate data about the theory we wish to study by fixing the chiral algebra part of the correlator.
Operators of type (A) on the other hand are not described by their chiral algebra and we cannot a priori fix their contribution to the four-point function. Indeed, we can see that as the dimension of the -symmetry representation of the external half BPS operators is increased, an ever larger number of short operators that are not connected to the chiral algebra will appear in the OPE. This situation stands in contrast to analogous bootstrap problems in four dimensions Beem:2013qxa (); Beem:2014zpa (), where the entirety of the short operator spectrum contributing to the desired four-point functions is constrained by the chiral algebra.
From the point of view of the conformal block decomposition, since these unfixed short multiplets occur in the decomposition of long multiplets at threshold, their conformal blocks will be limits of long conformal blocks and so they do not introduce any truly new ingredients into the crossing symmetry problem. However, since these are protected operators there is a chance that we may know something about their spectrum. We will explain below that precisely such a situation can occur for the four-point function of stress tensors multiplets.
2.4 Specialization to stress tensor multiplets
Let us now restrict our attention to the special case of interest, which is the four-point function of stress tensor multiplets, i.e., multiplets. This leads to some simplifications in the structure outlined above. We will see these simplifications in much greater detail in the coming sections.
The sums in (2.3) truncate fairly early when . Additionally some representations are ruled out by the requirement that the OPE be symmetric under the exchange of the two identical operators. This leaves the following selection rules:
The representations that are struck out contain higher spin conserved currents, and so will be absent in interacting theories.
Referring back to (2.1), we see that almost all of the short representations allowed by these selection rules are included in the chiral algebra. Their OPE coefficients will consequently be determined by the corresponding chiral algebra correlator. An important feature of this special case is that the chiral algebra correlator that is related to this particular four-point function is the four-point function of holomorphic stress tensors. This is important because the four-point function of holomorphic stress tensors is determined uniquely up to a single constant – the Virasoro central charge. Thus the chiral algebra contribution to this four-point function can be completely characterized in terms of the central charge, with no additional dependence on the theory being studied. This is in contrast to the case of general four-point functions, where the chiral algebra correlator is some four-point function in the algebra that may in principle look rather different for different choices of .
Thus the contributions of chiral algebra operators to the four-point function will be fixed in terms of . What about the unfixed BPS operators? For this example there are not that many options – namely, there is the quarter BPS multiplet and the -series multiplets . The superconformal blocks for these multiplets are the limits of the blocks for the long multiplets at the unitarity bound. Thus the only unknown superconformal blocks appearing in the expansion of this four point function are those contributed by long multiplets , possibly with . Later this will allow us to write the crossing symmetry equation for this four point function in terms of a single unknown function of conformal cross ratios.
Finally, let us make an auspicious observation about the quarter BPS multiplet that is allowed in this correlation function. Since the chiral rings of the ADE theories are thought to be known, we may test for the presence of this multiplet in these theories. In general, there is a single such operator in the known theories. For example, in the theories the quarter BPS ring is given by
Note that this is the same as the ring of -invariant functions of two commuting, traceless matrices and (which is the same as the part of the chiral ring of super Yang-Mills in four dimensions that is generated by two of the three chiral superfields, say and ). In these terms, a single quarter BPS operator is then generally present and can be written as
However, for the special case of , this operator is identically zero. What this tells us is that in precisely the theory, there will be no conformal block coming from a multiplet in the four-point function of stress tensor multiplets. This observation will have major consequences in our interpretation of the bootstrap results later in this paper.
3 The four-point function of stress tensor multiplets
We are now in a position to describe the detailed structure of the four-point function of stress tensor multiplets. Maximal superconformal symmetry guarantees that (i) the stress tensor belongs to a multiplet whose superconformal primary is a Lorentz scalar operator; (ii) the four-point function of this supermultiplet admits a unique structure in superspace. Consequently we lose no information by restricting our attention to the four-point function of the scalar superconformal primary, which is a dramatically simpler object.161616In fact, the technology to bootstrap four-point functions involving external tensorial operators, while conceptually straightforward, has not yet been fully developed. Rapid progress is being made in the area – see Costa:2011mg (); Dolan:2011dv (); Costa:2011dw (); SimmonsDuffin:2012uy (); Siegel:2012di (); Osborn:2012vt (); Hogervorst:2013sma (); Fitzpatrick:2013sya (); Hogervorst:2013kva (); Fitzpatrick:2014oza (); Khandker:2014mpa (); Elkhidir:2014woa (); Costa:2014rya (); Dymarsky:2013wla (); Echeverri:2015rwa (). This is a huge simplification in bootstrap studies, and has already been exploited for maximally superconformal field theories in four Beem:2013qxa () and three Chester:2014fya (); Chester:2014mea () dimensions. The results described in this section rely heavily on the previous works Arutyunov:2002ff (); Dolan:2004mu ().
The superconformal primary operator in the multiplet is a half BPS scalar operators of dimension four that transforms in the of the . We denote these scalar operators as , where are fundamental indices, and the brackets denote symmetrization and tracelessness. A convenient way to deal with the indices is to contract them with complex polarization vectors and define
The polarization vectors can be taken to be commutative due to symmetrization of the two indices, and tracelessness is encoded by the null condition:
With these conventions, the two-point function of is given by
Homogeneity of correlators with respect to simultaneous rescalings of the allows us to solve the null constraint as follows Dolan:2004mu (),
with an arbitrary three-vector. The two-point function is now given by
The normalization in (22) has led to unit normalization in these variables.
3.1 Structure of the four-point function
Conformal Ward identities and -symmetry conservation dictate that the four-point function can be written as Dolan:2004mu ()
where and are related to the canonical conformally invariant cross-ratios,
and and obey a similar relation with respect to “cross-ratios” of the polarization vectors,
Although not manifest in this notation, the dependence of the full correlator on the is polynomial by construction.
The constraints of superconformal invariance were investigated thoroughly in Dolan:2004mu (). Ultimately, the consequence of said constraints is that the four-point function must take the form
Here and are second order differential operators defined according to
The entire four-point function is determined in terms of a two-variable function and a single-variable function . The superconformal Ward identities impose no further constraints on these functions.
As described in Beem:2014kka (), there exists a specific -symmetry twist such that the correlation functions of devolve into those of a two-dimensional chiral algebra. For the two-point function (24) this twist is implemented by taking , leading to
where denotes the twisted operator in the chiral algebra. We see that behaves as a meromorphic operator of dimension two – it is in fact the Virasoro stress tensor of the chiral algebra Beem:2014kka (). For the four-point function we set and obtain
The dependence on the two-variable function completely drops out and the chiral correlator is determined by the derivative of the single-variable function introduced above.
3.2 Constraints from crossing symmetry
The correlation function (25) must be invariant under permutations of the four operators. Interchanging the first and the second operators implies the constraint
whereas invariance under interchanging the first and the third operators requires
Additional permutations do not give rise to any additional constraints.
Using equations (28) and (29) we can express these constraints in terms of and . Much as in Beem:2013qxa (); Beem:2014zpa (), we find that from each of the above equations a constraint can be extracted that applies purely to the single variable function ,
Defining , these constraints take the form
This is precisely the crossing symmetry constraint for the four-point function of a chiral operator of dimension two. Of course none of this is a coincidence – both the structure of this equation and the existence of a decoupled crossing relation for are a direct consequence of the chiral algebra described in Beem:2014kka (). We will solve (36) in the next subsection.
The remaining constraints from crossing symmetry amount to the following two relations for the two-variable function:
When reformulated in terms of the conformal block expansion the first equation is easily solved, while the latter is the non-trivial crossing symmetry equation that we will be the subject of the numerical analysis.
3.2.1 Solving for the meromorphic function
We will see in the next section that consistency with the six-dimensional OPE requires that is meromorphic in and admits a regular Taylor series expansion around with integer powers. This leads to the ansatz
The crossing symmetry constraints (36) imply that and . The full form of is then fixed in terms of according to
which implies that
where is an integration constant. From (29) we see that this constant does not affect and therefore can be set to any convenient value.
This leaves us with the determination of the parameters and . The former is determined from the normalization of the operator . We fixed this normalization in (24), which led to a normalization of the twisted operator as shown in (31). Compatibility of (32) with this equation implies that as , and therefore
The superconformal block decomposition described in the next section can be used to show that the parameter is a certain multiple of the squared OPE coefficient of the stress tensor. It would be relatively straightforward to work out the precise proportionality constant in this manner, but the chiral algebra provides an even more efficient way to find the same result. Indeed, the twisted correlator in (32) is proportional to the four-point function of Virasoro stress tensors in the chiral algebra Beem:2014kka (). The two-dimensional self-OPE of stress tensors takes the familiar form
with a two-dimensional central charge whose precise meaning will be discussed shortly. We chose to normalize the four-point function such that the unit operator appears with coefficient one, so in the chiral algebra . Comparing the above OPE with (39) with we find a match of the leading term, and at the first nontrivial subleading order we find
Supersymmetry dictates that is related to the coefficient in the two-point function of the canonical six-dimensional stress tensor, which takes the form Osborn:1993cr ()
The precise proportionality constant can be determined from the free tensor multiplet, for which Beem:2014kka () and Bastianelli:1999ab (). Therefore . For the theories of type this central charge was determined in Beem:2014kka () to be