Solving CFTs with Weakly Broken Higher Spin Symmetry

The method of large spin perturbation theory allows to analyse conformal field theories (CFT) by turning the crossing equations into an algebraic problem. We apply this method to a generic CFT with weakly broken higher spin (HS) symmetry, to the first non-trivial order in the breaking parameter. We show that the spectrum of broken currents, for any value of the spin, follows from crossing symmetry. After discussing a generic model of a single scalar field, we focus on vector models with $O(N)$ global symmetry. We rediscover the spectrum of several models, including the $O(N)$ Wilson-Fisher model around four dimensions, the large $O(N)$ model in $2<d<4$ and cubic models around six dimensions, not necessarily unitary. We also discuss models where the fundamental field is not part of the spectrum. Examples of this are weakly coupled gauge theories and our method gives an on-shell gauge invariant way to study them. At first order in the coupling constant we show that again the spectrum follows from crossing symmetry, to all values of the spin. Our method provides an alternative to usual perturbation theory without any reference to a Lagrangian.Comment: 30 pages,v2:typos correcte

Higher-spin correlators

We analyze the properly normalized three-point correlator of two protected scalar operators and one higher spin twist-two operator in N=4 super Yang-Mills, in the limit of large spin j. The relevant structure constant can be extracted from the OPE of the four-point correlator of protected scalar operators. We show that crossing symmetry of the four point correlator plus a judicious guess for the perturbative structure of the three-point correlator, allow to make a prediction for the structure constant at all loops in perturbation theory, up to terms that remain finite as the spin becomes large. Furthermore, the expression for the structure constant allows to propose an expression for the all loops four-point correlator G(u,v), in the limit u,v -> 0. Our predictions are in perfect agreement with the large j expansion of results available in the literature.Comment: 14 pages,v2:minor changes, refs adde

The superconformal bootstrap for structure constants

We report on non-perturbative bounds for structure constants on N=4 SYM. Such bounds are obtained by applying the conformal bootstrap recently extended to superconformal theories. We compare our results with interpolating functions suitably restricted by the S-duality of the theory. Within numerical errors, these interpolations support the conjecture that the bounds found in this paper are saturated at duality invariant values of the coupling. This extends recent conjectures for the anomalous dimension of leading twist operators.Comment: 14 pages, 4 figure

Conformal Bootstrap With Slightly Broken Higher Spin Symmetry

We consider conformal field theories with slightly broken higher spin symmetry in arbitrary spacetime dimensions. We analyze the crossing equation in the double light-cone limit and solve for the anomalous dimensions of higher spin currents $\gamma_s$ with large spin $s$. The result depends on the symmetries and the spectrum of the unperturbed conformal field theory. We reproduce all known results and make further predictions. In particular we make a prediction for the anomalous dimensions of higher spin currents in the 3d Ising model.Comment: 41 pages, 2 figures, %\draftmod

Generalized bootstrap equations for N=4 SCFT

We study the consistency of four-point functions of half-BPS chiral primary operators of weight p in four-dimensional N=4 superconformal field theories. The resulting conformal bootstrap equations impose non-trivial bounds for the scaling dimension of unprotected local operators transforming in various representations of the R-symmetry group. These bounds generalize recent bounds for operators in the singlet representation, arising from consistency of the four-point function of the stress-energy tensor multiplet.Comment: 18 pages, 3 figure

Loop Corrections to Supergravity on AdS5×S5AdS_5 \times S^5

We consider the four-point correlator of the stress-energy tensor multiplet in ${\cal N}=4$ SYM. In the planar limit and at large 't Hooft coupling such correlator is given by the corresponding holographic correlation function in IIB supergravity on $AdS_5 \times S^5$. We consider subleading corrections in the number of colours, i.e. order $1/N^4$, at large 't Hooft coupling. This corresponds to loop corrections to the supergravity result. Consistency conditions, most notably crossing symmetry, constrain the form of such corrections and lead to a complete determination of the spectrum of leading twist intermediate operators.Comment: 6 pages, minor changes, added reference

Modular interpolating functions for N=4 SYM

We construct interpolating functions fully compatible with S-duality. We then consider the problem of resumming perturbative expansions for anomalous dimensions of low twist non-protected operators in N=4 super Yang-Mills theory. When the rank of the gauge group is small, the interpolations suggest that anomalous dimensions of leading twist operators take their maximum value at the point $\tau =\exp(i\pi/3)$. For fixed spin and large enough rank, there is a level-crossing region, where the anomalous dimension of the leading twist operator reaches its maximum and then bounces back.Comment: 13 pages, 5 figures; v2: comments adde

Gravitational S-matrix from CFT dispersion relations

We analyse the double-discontinuities of the four-point correlator of the stress-tensor multiplet in N=4 SYM at large t' Hooft coupling and at order $1/N^4$, as a way to access one-loop effects in the dual supergravity theory. From these singularities we extract CFT-data by using two inversion procedures: one based on a recently proposed Froissart-Gribov inversion integral, and the other based on large spin perturbation theory. Both procedures lead to the same results and are shown to be equivalent more generally. Our computation parallels the standard S-matrix reconstruction via dispersion relations. In a suitable limit, the result of the conformal field theory calculation is compared with the one-loop graviton scattering amplitude in ten-dimensional IIB supergravity in flat space, finding perfect agreement.Comment: 39 pages, pretty figure