Bound on asymptotics of magnitude of three point coefficients in 2D CFT

Abstract

We use methods inspired from complex Tauberian theorems to make progress in understanding the asymptotic behavior of the magnitude of heavy-lightheavy 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. arXiv:1906.11223v2 [hep-th] 1 Jul 2019 -2 -

Universal dynamics of heavy operators in CFT2

“…Various authors have previously considered the asymptotic behaviour of three point coefficients in each of these three separate limits [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The asymptotic formulas which were obtained generally relied on detailed computations of the conformal blocks, and -while correct -required assumptions about the behaviour of the blocks in certain kinematic regimes or the simplification of large central charge.…”

“…We expect, however, that a much stronger version is true, where one needs to integrate only over a small window; results that establish this kind of behaviour go under the general name of Tauberian theorems (see e.g. [16,[33][34][35][36][37] for recent applications of Tauberian theorems in this context). In the present case we would require new results for several variables, adapted to the Virasoro crossing transforms.…”

“…Given this object, the modular S transformation acts on the spectral density as 16) and modular covariance of…”

“…Our derivation emphasizes that this object comes from the representation theory of the Virasoro algebra, describing the decomposition of the trivial representation after modular transformation. 16 In the remainder of the paper, we will show that another representation theoretic object similarly controls the OPE coefficients in a variety of limits. Now, our argument for the asymptotic formula (2.14) was very imprecise, and indeed the result is simply false if interpreted literally, so we briefly discuss the sense in which it holds.…”

“…This is the density in the P,P variables, so a Jacobian is required to convert to density in h,h. For an asymptotic formula in dimension ∆ = h +h only, insensitive to spin, one simply integrates (2.14) over the circleP 2 +P 2 = ∆ − c−112 , obtaining the Bessel function formulas of, for example,[35,59] 16. In fact, ρ0 has a purely representation theoretic characterization: it is the Plancherel measure on the space of representations of the Virasoro algebra[22].…”