Bounds on CFTs with W3 algebras and AdS3 higher spin theories

Abstract

The scaling dimension of the first excited state in two-dimensional conformal field theories (CFTs) satisfies a universal upper bound. Using the modular bootstrap, we extend this result to CFTs with W3 algebras which are generically dual to higher spin theories in AdS3. Assuming unitarity and modular invariance, we show that the conformal weights h,h of the lightest charged state satisfy h < c/12 + O(1) andh <c/12 + O(1) in the limit where the central charges c,c are large. Furthermore, we show that in this limit any consistent CFT with W3 currents must contain at least one state whose W3 charge w obeys |w| > 4|h − c/24|/ √ 10πc + O(1). We discuss hints on the existence of stronger bounds and comment on the interpretation of our results in the dual higher spin theory.

Modular constraints on conformal field theories with currents

“…We find that the numerical upper bounds depend on the choice of W(g)-algebra when the central charge is small, namely c rank(g). For the case of W(A 2 )-algebra, this problem has been recently discussed in [23,24].…”

“…The two-dimensional CFTs with W(A 2 ) = W(2, 3) symmetry have been studied recently in [23,24]. To investigate the universal constraints on the spectrum of higher-spin "irrational" CFTs, i.e., c > 2, the authors of [23,24] apply the modular bootstrap method to the torus two-point function…”

“…To investigate the universal constraints on the spectrum of higher-spin "irrational" CFTs, i.e., c > 2, the authors of [23,24] apply the modular bootstrap method to the torus two-point function…”

Constraints on higher spin CFT2

“…It is in principle straightforward (although in practice tedious) to generalize this to higher N . A similar modular bootstrap strategy for W 3 CFTs was discussed in [14]. We will follow the modular bootstrap approach initiated in [9], where modular invariance of the torus partition was used to place an upper bound on the dimension of the lightest non-identity primary operator.…”

“…(This state may have largeh 1 , so in particular it may have scaling dimension ∆ > (c − N + 1)/12, the threshold for the lightest black hole.) Specializing to the case of W 3 , we obtain further constraints on the spectrum by applying modular bootstrap techniques [9][10][11][12][13][14], not to the partition function, but to the torus two-point function of W 0 , the zero mode of the spin-3 current. This relies on the results of [15,16], which computed the modular transformation properties of this and similar objects.…”