Lessons from crossing symmetry at large N

JHE crossing symmetry at large Luis F. Alday 0 1 2 3 Agnese Bissi 0 1 2 3 Tomasz Lukowski 0 1 2 3 0 Woodstock Road , Oxford, OX2 6GG, U.K 1 Andrew Wiles Building , Radcliffe Observatory Quarter 2 Mathematical Institute, University of Oxford 3 Open Access , c The Authors We consider the four-point correlator of the stress tensor multiplet in N = 4 SYM. We construct all solutions consistent with crossing symmetry in the limit of large central charge c ∼ N 2 and large g2N . While we find an infinite tower of solutions, we argue most of them are suppressed by an extra scale Δgap and are consistent with the upper bounds for the scaling dimension of unprotected operators observed in the numerical superconformal bootstrap at large central charge. These solutions organize as a double expansion in 1/c and 1/Δgap. 1/N Expansion; Supersymmetric gauge theory; Conformal and W Symmetry 1 Introduction 2 Analytic solutions at large N 3 Interpretation 4 Conclusions A Gshort(u, v, c) D¯ -functions The superconformal bootstrap equation Solutions at large N Solutions in Mellin space Absence of other solutions Large n behavior Interpretation of our solutions Connection to causality and UV completion Conformal field theories (CFT) are one of the pillars of theoretical physics. Important motivations to study them are their role in phase transitions and their relation to renormalization group flows. Over the last two decades, it also became evident that they describe quantum gravity in AdS space, through the AdS/CFT correspondence. The main ingrecorrelators of these operators primary operators and the operator product expansion (OPE) coefficients cijk for any three primaries. In a unitary CFT this data satisfies certain constraints. In particular the OPE coefficients are real numbers and for the case to be studied in this paper for a primary operator of spin ℓ. Once the CFT data is given, the OPE allows to write, in principle, any higher point correlator. The idea of the conformal bootstrap program is to use crossing symmetry of correlation functions, together with unitarity and the structure of the OPE, in order to constrain the CFT data. In the simplest setting, which is also the relevant for this paper, we consider the correlator of four identical operators of scaling where we have introduced the cross-ratios u = v = The conformal blocks Gi(u, v) are completely fixed by conformal symmetry and depend only on the dimension and spin of the intermediate state. They encode the contribution of a given primary together with its tower of descendants. In the above expansion we have also singled out the contribution from the identity operator, always present in the OPE of two identical operators. We could have instead chosen to expand along the (13)(24) channel, and the result should have been the same. Indeed, crossing symmetry of the four-point function implies which results in the following non-trivial equation involving the CFT data X ci2 vΔGi(u, v) − uΔGi(v, u) = uΔ Note that the r.h.s. arose from the presence of the identity operator. Equation (1.7) is called the conformal bootstrap equation. So far the discussion has been pretty general. However, specific conformal field theories often possess extra symmetries which impose extra constraints. An important example is that of supersymmetric conformal field theories (SCFT). The subject of this paper will four dimensional conformal field theory, and is particularly interesting since it describes quantum gravity on AdS space. whose superconformal primary is a scalar operator O of protected dimension two and which transforms in the 20′ representation of the SU(4) R-symmetry group.1 In [1], the consequences of crossing-symmetry of the correlator hOOOOi were analyzed and were written in the form of a (super)conformal bootstrap equation: X aΔ,ℓ (GΔ,ℓ(u, v) − GΔ,ℓ(v, u)) = Fshort(u, v, c). 1In order to simplify the notation we will obviate the representation index. By using the OPE we can decompose the correlator (1.3) as a sum over intermediate states g(u, v) = 1 + X ci2Gi(u, v). Although the derivation is conceptually similar to the previous case, there are important differences. First, among the states in the OPE of O × O there is a rich spectrum of protected operators, belonging to short or semi-short multiplets, which do not acquire anomalous dimension and whose OPE coefficient is fixed due to superconformal Ward identities [2]. The r.h.s. of (1.8) resums the contribution from all such operators, instead of just the identity. The structure of Fshort(u, v, c) is very rich, but it is important to note that it is only a function of the central charge, and not of the coupling constant of the theory, and is explicitly known. Second, supersymmetry relates operators in different conformal towers. Therefore, the sum runs only over unprotected superconformal primaries in long non-negative as a consequence of unitarity. In spite of fitting in one line both (1.7) and (1.8) are form (...truncated)