Bounds on Superconformal Theories with Global Symmetries
Recently, the conformal-bootstrap has been successfully used to obtain generic bounds on the spectrum and OPE coefficients of unitary conformal field theories. In practice, these bounds are obtained by assuming the existence of a scalar operator in the theory and analyzing the crossing-symmetry constraints of its 4-point function. In superconformal theories with a global symmetry there is always a scalar primary operator, which is the top of the current multiplet. In this paper we analyze the crossing-symmetry constraints of the 4-point function of this operator for theories with global symmetry. We analyze the current-current OPE and write the superconformal blocks, generalizing the work of Fortin, Intriligator and Stergiou to the non-Abelian case. Moreover we find new contributions to the OPE which can appear both in the Abelian and non-Abelian cases. We then use these results to obtain lower bounds on the coefficient of the current 2-point function.
Keywords:Supersymmetry, Conformal Bootstrap
Recently, there has been much interest in generating numerical constraints on conformal field theories using the conformal bootstrap Rattazzi:2008pe (); Rychkov:2009ij (); Caracciolo:2009bx (); Poland:2010wg (); Rattazzi:2010gj (); Rattazzi:2010yc (); Poland2011 (); Vichi:2011ux (); ElShowk:2012hu (); ElShowk:2012ht (); Liendo:2012hy (); Beem:2013qxa (); Kos:2013tga (); El-Showk:2013nia (); Alday:2013opa (); Gaiotto:2013nva (). In these works the existence of a scalar primary operator of dimension is assumed. Using the conformal bootstrap on the 4-point function of , it is then possible to generate numerically bounds on operator dimensions and OPE coefficients of operators in the OPE as a function of . The crucial ingredient which allows us to generate these bounds is the knowledge of all the scalar conformal blocks Dolan:2000ut (); Dolan:2003hv (), which encode the dependence of the -point function of on each conformal family in the OPE111More precisely, for doing numerics it is sufficient to have a systematic way to approximate the conformal blocks. For scalar 4-point functions this can be done efficiently in any dimension ElShowk:2012ht (); Kos:2013tga (); Hogervorst:2013kva ()..
It would interesting to apply these methods without introducing any assumption on the operator spectrum. For instance, one would like to analyze the four-point function of the stress-tensor, which exists for any CFT. More generally, assuming the CFT has some global symmetry one would like to understand the constraints of conformal invariance arising from application of the conformal bootstrap to the -point function of the global symmetry current. Unfortunately, to this date there are no closed form expressions for the conformal blocks of non-scalar operators (see however Costa:2011dw (); SimmonsDuffin:2012uy ()), so these interesting directions cannot be pursued in a straightforward way yet.
However, in supersymmetric theories the situation is better since in some cases, symmetry currents reside in multiplets whose superconformal-primary (sprimary) is a scalar field. For instance in supersymmetric Yang-Mills the energy-momentum tensor resides in a multiplet whose sprimary is a scalar in the representation of the -symmetry group. The bootstrap constraints for this case were recently analyzed in Beem:2013qxa ().
Similarly, in any four-dimensional superconformal theory, a global symmetry current resides in a real multiplet
which satisfies , and the omitted terms in the equation above are determined by this constraint. The sprimary is a dimension two real scalar field in the adjoint representation of the symmetry group.
In this paper we will use the conformal bootstrap to constrain four dimensional superconformal theories with an global symmetry222The generalization to other symmetry groups is straightforward.. In particular, we will place lower bounds on the current “central charge” defined as the coefficient of the current 2-point function333Similar bounds were generated in Poland:2010wg (); Poland2011 (), assuming the existence of a charged scalar primary. Here we assume supersymmetry instead.
The decomposition of the 4-point function of into conformal blocks is constrained by supersymmetry. In particular, the OPE coefficients in of different primary operators in a super-multiplet are not independent and the corresponding conformal blocks are re-packaged into the so-called superconformal blocks. These constraints were already analyzed in detail in Fortin:2011nq () for the case, and will generalize those results to the non-Abelian case. In addition, we find new operators which generally appear in the OPE which were not found in Fortin:2011nq ().
The form of the bounds we find is . Qualitatively, the existence of a lower bound means there is a minimal amount of “charged stuff” which must exist in any such theory. A free chiral superfield has, in our normalization, . We do not know of any theory with and it would be very interesting to understand whether those exist, or alternatively to prove that in general.
The paper is organized as follows. In section 2 we briefly review the conformal bootstrap and set up our conventions. In addition, we determine the sum-rules which result from applying crossing-symmetry to the 4-point function of a scalar primary in the adjoint representation of . In section 3 we discuss the constraints imposed by superconformal invariance on the OPE and superconformal blocks. In section 4 we present the lower bounds we obtained on and a short discussion.
2.1 Conformal Bootstrap
In this section we spell out our normalization conventions and briefly summarize the conformal bootstrap constraint for a general CFT. The reader is referred to Rattazzi:2008pe () for a more extensive treatment.
Consider a general CFT in four Euclidean dimensions, and in particular the subset of operators consisting of spin- primary operators , which are symmetric-traceless rank- tensors (i.e. in the representation of the Lorentz group ). The index labels the primary operators in the CFT, and we will denote the complex conjugate operator by a barred index .
We set the normalization of such operators by demanding that their -point function is of the form
where on the RHS the indices and should be symmetrized with the traces removed, and denotes the dimension of . The -point function of a spin- primary with two scalar primaries , of equal dimension is
where again the Lorentz indices on the RHS should be symmetrized with the traces removed, and are arbitrary labels.
In above equation the identity operator contains the information on the -point function and the sum over primaries encodes all the information on -point functions. The operator is entirely determined by conformal symmetry, and encodes the contributions of all the descendants of to the OPE.
In a unitary theory if the scalars are real but are complex then the OPE coefficients are generally complex and satisfy . If we choose a real basis of operators, then the OPE coefficients must be real . The -point function is non-zero only for of integer spin, and (odd) even spins correspond to the (anti-)symmetric combination of and (i.e. is (anti-)symmetric in for (odd) even spins).
The crossing-symmetry constraints for the 4-point function are obtained by using the OPE in the (“-channel”) and (“-channel”) channels and equating the results,
are the two conformal cross-ratios. In (8) the summation is over all primary operators in the OPE444If an operator is complex then its complex conjugate should also be included in the sum as an independent primary operator. .
2.2 Bootstrap for Scalars in the Adjoint of
In this section we discuss a specific case of the general bootstrap constraint (8) in which is a real scalar primary in the adjoint representation555The indices label the adjoint representation of and are (anti-)fundamental indices. We will sometimes find it more convenient to work in the fundamental basis with , where is a generator in the fundamental of . Our normalization convention is . The structure constants are and . of . We will later apply the results of this section to the case in which this scalar is the top of the current multiplet in theories. The crossing-symmetry relations in CFTs with global symmetries were considered in full generality in Rattazzi:2010yc (), and we apply these results to our case of interest.
The operators which appear in the OPE can be decomposed into any of the irreducible representations in the product of two adjoint representations of . Each such representation arises from either the symmetric or anti-symmetric product. Operators in the OPE which are in a (anti-)symmetric representation must be of (odd) even spin from Bose symmetry. The reader is referred to appendix A for details regarding the tensor product of two adjoint representations and our notations.
Let be an operator in representation which appears in the OPE, with labeling the elements of the representation. We denote the corresponding OPE coefficient (defined in (5)) as and split it to a universal group factor times some coefficient,
where is the relevant Clebsch-Gordan coefficient, and is the same for any operator in the representation , while the coefficient is the same for each element of the representation. The sum-rule in (8) becomes
Each term in the second sums in equation (14) has the same sign from the factor, as this only depends on whether is in the symmetric or anti-symetric product of two adjoints666The adjoint representation appears both in the symmetric and anti-symmetric product and we count those as distinct in the sum over representations in (14).. We use this property to write the above sum-rule as
where is the sum over conformal blocks in a given representation, and is just the identity matrix in the representation projected to adjoint representation indices up to a sign, which can be determined by reflection positivity. Explicit expressions for these identity matrices are given in (62).
After plugging (62) into the sum-rule (15), it can be decomposed into several equations by equating the coefficients of the independent delta-functions in the identity matrices. We do this in the next subsections paying attention to the special cases and .
The resulting sum-rules are conveniently expressed in terms of the functions
We have verified that all the sum-rules written below are obeyed by both the four point function of a free scalar field in the adjoint of , and that of the adjoint bilinear in the theory of a free fundamental scalar .
For the sum-rule is the usual one for the 4-point function of a real scalar operator:
where we separated the contribution of the identity operator for which and .
For we have corresponding to the representations , and the trivial representation. Setting all the terms which correspond to the other representations in (15) to zero, plugging in the expressions for the identity matrices (62) and equating independent coefficients, we can express the result as three independent sum-rules777Equivalent sum-rules were also worked out in Rattazzi:2010yc () for scalars in the fundamental of . Our result is consistent with Rattazzi:2010yc (), but we work in slightly different convention such that , which amounts to a rescaling of all the OPE coefficients in the trivial representation by a factor of .,
For the representation does not exist so we set it to zero in (15). The resulting sum-rules are given by
For all the 7 representations listed in appendix A can appear in the OPE, and we find
3 Conformal Bootstrap for Conserved Currents in SCFTs
Consider an superconformal field theory with global symmetry group . In this section we will analyze the bootstrap constraints for the -point function of , which is the top of the current multiplet defined in (1). In particular, we extend the results of Fortin:2011nq () for to the non-Abelian case, and also find additional possible operators in the OPE. We use the notations and conventions of Fortin:2011nq ().
3.1 Current-Current OPE in SCFTs
The general form of the 3-point function of sprimary operators was found in Osborn:1998qu (). For the 3-point function of two conserved currents with some other sprimary in some representation the result is
where the superspace coordinates are , and we define
The quantities , and are functions of the superspace coordinates given by
and labels the Lorentz representation of .
The function has to scale appropriately with respect to dilatations and transformations888The scaling is , where , and . The R-charge and dimension of are related to by and .. Moreover, because of current conservation, , the correlator (33) satisfies a differential equation. As shown in Osborn:1998qu (), this equation can be translated to the following differential equation for :
Note that is symmetric under . Therefore, since is either symmetric or anti-symmetric under , we need to find which is either symmetric or anti-symmetric under . Under we have , with . It is therefore useful to define
which are manifestly odd and even under , respectively.
The above constraints are sufficient to completely determine up to an overall numerical factor. In particular, Fortin:2011nq () found999We find a slightly different coefficient then Fortin:2011nq () for the second term in the square brackets of (41). two structures corresponding to spin- sprimary operators with zero R-charge, which take the form101010Round brackets around Lorentz indices denote symmetrization, which is defined by averaging over all permutations.
Therefore if is (anti-)symmetric in and , then in (33), the structure appears for (odd) even and for (even) odd .
The case is special since there is no structure for which is odd under (see appendix B). Therefore in this case only scalar sprimaries in representations which arise from the symmetric product of two adjoints can contribute to (33) with the structure
The scalar in the adjoint representation corresponds to111111There could be other conserved currents in the theory, but those would appear in the singlet representation. . In that case the 3-point function is completely determined (for the canonically normalized current) by the Ward identities to be Osborn:1998qu (),
where is the ‘t Hooft anomaly and is defined through the 2-point function of the canonically normalized current (2).
Let us discuss some properties of the operators listed in table 1. The structure in the second entry of table 1 (and its conjugate) actually arises from a larger family of structures , which satisfies all the constraints121212We use the notation: .. These structures correspond to operators with dimension which violate the unitarity bound for , . The structure however, corresponds to a chiral operator (), in which case the unitarity bound is modified to . The corresponding operator saturates this bound, so it is in fact a free chiral fermion.
When the zero R-charge operators saturate the unitarity bound , they decompose into two short representations as follows:
The shortening condition is . The resulting structure for the short representation appears as the third entry of table 1, so these two series of structures are actually related. A similar story holds for the representations and . This decomposition into short multiplets matches the one described, e.g. on Gadde:2010en (), which also specifies where the spin conformal primaries reside after the decomposition.
Short representations such as , can certainly appear, at least for free theories. They can be constructed in the following way, using the current as the basic building block
Symmetrization over Lorentz is to be understood. One can verify that these are superconformal primaries and satisfy the shortening condition , by using the superconformal algebra and the fact that