The analytic bootstrap for large N Chern-Simons vector models

Abstract

Three-dimensional Chern-Simons vector models display an approximate higher spin symmetry in the large N limit. Their single-trace operators consist of a tower of weakly broken currents, as well as a scalar σ of approximate twist 1 or 2. We study the consequences of crossing symmetry for the four-point correlator of σ in a 1/N expansion, using analytic bootstrap techniques. To order 1/N we show that crossing symmetry fixes the contribution from the tower of currents, providing an alternative derivation of well-known results by Maldacena and Zhiboedov. When σ has twist 1 its OPE receives a contribution from the exchange of σ itself with an arbitrary coefficient, due to the existence of a marginal sextic coupling. We develop the machinery to determine the corrections to the OPE data of double-trace operators due to this, and to similar exchanges. This in turns allows us to fix completely the correlator up to three known truncated solutions to crossing. We then proceed to study the problem to order 1/N 2 . We find that crossing implies the appearance of odd-twist doubletrace operators, and calculate their OPE coefficients in a large spin expansion. Also, surprisingly, crossing at order 1/N 2 , implies non-trivial O(1/N ) anomalous dimensions for even-twist double-trace operators, even though such contributions do not appear in the four-point function at order 1/N (in the case where there is no scalar exchange). We argue that this phenomenon arises due to operator mixing. Finally, we analyse the bosonic vector model with a sextic coupling without gauge interactions, and determine the order 1/N 2 corrections to the dimensions of twist-2 double-trace operators.

Black holes and conformal Regge bootstrap

“…The Regge limit implies that the calculation is valid for n, l ≫ 1. These anomalous dimensions have been subsequently verified in [26,[62][63][64][65][66][67][68][69][70].…”

Closed-form expression for cross-channel conformal blocks near the lightcone

“…When the external scalars have identical scaling dimensions, we also provided the expression (27) for the Lorentzian inversion of a general cross-channel conformal block, which will be particularly useful for the investigations in d = 1, 2, 4 dimensions, as general compact expressions were not available in the past. They include the analytic conformal bootstrap of the Wilson-Fisher fixed points [37,42], conformal field theories in d = 3 dimensions [40,43] and d > 4 dimensions [15].…”

From bulk loops to boundary large-N expansion

“…Accordingly one should replace 1/N 2 with 1/N compared to the rest of the paper 13. This may be a tricky step considering that holographically reconstructed higher-spin theories are nonlocal in a conventional sense[32,63], hence, in principle, infinite-derivative terms can generate additional singularities.…”