The analytic functional bootstrap. Part I: 1D CFTs and 2D S-matrices

Abstract

We study a general class of functionals providing an analytic handle on the conformal bootstrap equations in one dimension. We explicitly identify the extremal functionals, corresponding to theories saturating conformal bootstrap bounds, in two regimes. The first corresponds to functionals that annihilate the generalized free fermion spectrum. In this case, we analytically find both OPE and gap maximization functionals proving the extremality of the generalized free fermion solution to crossing. Secondly, we consider a scaling limit where all conformal dimensions become large, equivalent to the large AdS radius limit of gapped theories in AdS 2 . In this regime we demonstrate analytically that optimal bounds on OPE coefficients lead to extremal solutions to crossing arising from integrable field theories placed in large AdS 2 . In the process, we uncover a close connection between asymptotic extremal functionals and S-matrices of integrable field theories in flat space and explain how 2D S-matrix bootstrap results can be derived from the 1D conformal bootstrap equations. These points illustrate that our formalism is capable of capturing non-trivial solutions of CFT crossing.

Analytic functional bootstrap for CFTs in d > 1

“…Below we will present complete bases of such functionals, where completeness means that they fully capture the constraints of crossing symmetry on the line. Some of these bases have already appeared in print, while the existence of others was only indicated [22][23][24]29]. In any case, all of them are reviewed and/or constructed (accordingly) in appendix A, to which we refer the reader for further details.…”

“…We also assume that these kernels decay sufficiently fast at infinity. Then by deforming the contours of integration (as in the 1d case [24], see our appendix A) this allows us to derive an alternative representation of the functional action:…”

“…These are determined as in the higher dimensional case by demanding that the action of the functional commutes with infinite sums of crossing vectors. 12 They demand [24]: 13 f…”

“…It is interesting to see how these product solutions actually solve the functional equations. We write the crossing equation as 24) and attempt to recover the correct OPE coefficients (5.23) using the functional equations.…”

“…The second class is obtained by tensoring holomorphic and antiholomorphic copies of 1d functionals, some of which have appeared in the literature [22][23][24]29] (including the ones mentioned above), others which we construct in detail. These product functionals exist in 2 Up to some important caveats as we shall see, namely that this functional set is not complete.…”