Abstract
We consider the four-point correlator of the stress tensor multiplet in $$ \mathcal{N}=4 $$ SYM. We construct all solutions consistent with crossing symmetry in the limit of large central charge c ∼ N 2 and large g 2 N . 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. Our solutions are valid to leading order in 1/c and to all orders in 1/Δgap and reproduce, in particular, instanton corrections previously found. Furthermore, we find a connection between such upper bounds and positivity constraints arising from causality in flat space. Finally, we show that certain relations derived from causality constraints for scattering in AdS follow from crossing symmetry.
Highlights
The relevant for this paper, we consider the correlator of four identical operators of scaling dimension ∆
In N = 4 super Yang-Mills the energy-momentum tensor lies in a half-BPS multiplet, 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
We show that certain relations derived from causality constraints for scattering in AdS found in [12] follow from crossing symmetry
Summary
As already mentioned in the introduction, the conformal bootstrap equations for N = 4. In order to write down the explicit expressions for the superconformal blocks it is convenient to introduce variables z, z, with u = |z|2, v = |1 − z|2. The conformal bootstrap equation (2.1) is equivalent to the requirement. Let us emphasize that A(u, v) generally depends on the coupling constant and in order to compute it one usually has to resort to explicit computations. The superconformal bootstrap equation differs from the standard one in two aspects:. It involves superconformal blocks, instead of conformal blocks. For the present case they are proportional to the usual conformal blocks upon a shift ∆ → ∆ + 4. Fshort(u, v, c) has a much richer structure than its analogue in conformal field theories, which usually contains only the identity operator
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