Abstract
We use methods inspired from complex Tauberian theorems to make progress in understanding the asymptotic behavior of the magnitude of heavy-light-heavy three point coefficients rigorously. The conditions and the precise sense of averaging, which can lead to exponential suppression of such coefficients are investigated. We derive various bounds for the typical average value of the magnitude of heavy-light-heavy three point coefficients and verify them numerically.
Highlights
In [3], the modular covariance of torus one point function is used to estimate the asymptotic behavior of heavy-light-heavy three point coefficients, they found exponential suppression, which depends on dimension of heavy operator ∆, central charge of the CFT (c) and the dimension (∆χ) of the operator, χ such that it produces the light operator upon doing operator product expansion with itself and it has the least dimension among all such operators producing the light operator upon doing operator product expansion with itself
We have proved a rigorous lower bound on the asymptotic behavior of the magnitude of the heavy-light-heavy three point coefficients
One might wonder whether considering the negative of the light operator O would turn the lower bound into an upper bound
Summary
We consider a CFT on a torus, where the spatial cycle is of length 2π and the thermal cycle is of length β. The light part captures the leading behavior of torus expectation value in high temperature limit and yields the approximation of the weighted three point coefficient B This is done via a suitable choice of β as a function of ∆ where we take ∆ → ∞. Where we have introduced σ(∆ ) as a shorthand for the following quantity: σ(∆ ) = To relate this to the torus one point function, we have to add to the both sides of the inequality (2.2) the contribution coming from the states, not in the interval. The readers who want to circumnavigate the technical details for their first read can skip directly to the section 6
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