Abstract
We introduce analytic functionals which act on the crossing equation for CFTs in arbitrary spacetime dimension. The functionals fully probe the constraints of crossing symmetry on the first sheet, and are in particular sensitive to the OPE, (double) lightcone and Regge limits. Compatibility with the crossing equation imposes constraints on the functional kernels which we study in detail. We then introduce two simple classes of functionals. The first class has a simple action on generalized free fields and their deformations and can be used to bootstrap AdS contact interactions in general dimension. The second class is obtained by tensoring holomorphic and antiholomorphic copies of d = 1 functionals which have been considered recently. They are dual to simple solutions to crossing in d = 2 which include the energy correlator of the Ising model. We show how these functionals lead to optimal bounds on the OPE density of d = 2 CFTs and argue that they provide an equivalent rewriting of the d = 2 crossing equation which is better suited for numeric computations than current approaches.
Highlights
In the last decade, a ruthless siege of the crossing equation has yielded a number of detailed insights into the structure of conformal field theories (CFTs) in general spacetime dimension d
We introduce analytic functionals which act on the crossing equation for CFTs in arbitrary spacetime dimension
The second class is obtained by tensoring holomorphic and antiholomorphic copies of d = 1 functionals which have been considered recently. They are dual to simple solutions to crossing in d = 2 which include the energy correlator of the Ising model. We show how these functionals lead to optimal bounds on the OPE density of d = 2 CFTs and argue that they provide an equivalent rewriting of the d = 2 crossing equation which is better suited for numeric computations than current approaches
Summary
A ruthless siege of the crossing equation has yielded a number of detailed insights into the structure of conformal field theories (CFTs) in general spacetime dimension d. It turns out that we can choose functionals to be dual to a particular sparse solution to crossing in d = 1, namely the fundamental field correlator of a generalized free field (GFF) In this case, duality means essentially that the functionals bootstrap the generalized free solution, as well as arbitrary small deformations away from it, see equations (2.11) and (2.12) below. An important result is that these functionals do not form a complete set, in the sense that they do not fully capture the constraints of crossing symmetry They do constrain possible solutions to crossing enormously, and in particular are sufficient to bootstrap general contact interactions in AdS for any dimension.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
