Search

We initiate the analytical functional bootstrap study of conformal field theories with large N limits. In this first paper we particularly focus on the 1D O(N) vector bootstrap. We obtain a remarkably simple bootstrap equation from the O(N) vector crossing equations in the large N limit. The numerical conformal bootstrap bound is saturated by the generalized free field theories, while its extremal functional actions do not converge to any non-vanishing limit. We study the analytical extremal functionals of this crossing equation, for which the total positivity of the SL(2, ℝ) conformal block plays a critical role. We prove the SL(2, ℝ) conformal block is totally positive in the limits with large ∆ or small 1 − z and show that the total positivity is violated below a critical value ∆ TP ∗ ≈ 0.32315626. The SL(2, ℝ) conformal block forms a surprisingly sophisticated mathematical structure, which for instance can violate total positivity at the order 10−5654 for a normal value ∆ = 0.1627! We construct a series of analytical functionals {αM} which satisfy the bootstrap positive conditions up to a range ∆ ⩽ ΛM. The functionals {αM} have a trivial large M limit. However, due to total positivity, they can approach the large M limit in a way consistent with the bootstrap positive conditions for arbitrarily high ΛM. Moreover, in the region ∆ ⩽ ΛM, the analytical functional actions are consistent with the numerical bootstrap results, therefore it clarifies the positive structure in the crossing equation analytically. Our result provides a concrete example to illustrate how the analytical properties of the conformal block lead to nontrivial bootstrap bounds. We expect this work paves the way for large N analytical functional bootstrap in higher dimensions. [ABSTRACT FROM AUTHOR]

Crossing symmetry (CS) is the main tool in the bootstrap program applied to CFT. This consists in an equality which imposes restrictions on the CFT data of a model, i.e., the OPE coefficients and the conformal dimensions. Reflection positivity (RP) has also played a role in this program, since this condition is what leads to the unitary bound and reality of the OPE coefficients. In this paper, we show that RP can still reveal more information, explaining how RP itself can capture an important part of the restrictions imposed by the full CS equality. In order to do that, we use a connection used by us in a previous work between RP and positive definiteness of a function of a single variable. This allows us to write constraints on the OPE coefficients in a concise way. These constraints are encoded in the conditions that certain functions of the cross-ratio will be positive defined and in particular completely monotonic. We will consider how the bounding of scalar conformal dimensions and OPE coefficients arise in this RP based approach. We will illustrate the conceptual and practical value of this view trough examples of general CFT models in d-dimensions. [ABSTRACT FROM AUTHOR]