Universal asymptotics of three-point coefficients from elliptic representation of Virasoro blocks

Abstract

In (1 þ 1)-d CFTs, the 4-point function on the plane can be mapped to the pillow geometry and thereby crossing symmetry gets translated into a modular property. This feature arises from the Virasoro blocks in the elliptic representation. We use these modular features to derive a universal asymptotic formula for OPE coefficients in which one of the operators is averaged over heavy primaries. As an application, we demonstrate that the coarse-grained heavy channel then reproduces features of the holographic 2 → 2 S-matrix which has black holes as their intermediate states.

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.…”

“…The first factor exactly cancels a similar factor in the conformal blocks (F ≈ (16q) hs [66]), ensuring that the block expansion has the correct domain of convergence. A formula of this form for the asymptotics of the averaged heavy-light-light structure constants was first obtained in [9]. In that paper, the authors used the asymptotics of the Virasoro four-point blocks in the heavy limit h s → ∞ [66], subsequently taking a z → 1 limit to reproduce the OPE singularity from the T-channel identity operator.…”

Universality in asymptotic bounds and its saturation in 2D CFT

“…Next we will show that the heavy part is suppressed compared to the light part. We can proceed as we did for the 4 point correlator and bound the absolute value: 12) . (4.14)…”

“…In spite of absence of an example, significant progress has been made in understanding the heavy excited states in generic CFTs, with/without imposing the twist gap condition. A partial list of examples are Cardy formula [2][3][4][5][6][7][8], understanding physics related to ETH/black hole thermality [9][10][11][12][13][14][15][16][17][18][19][20][21][22], chaos at large central charge CFTs with sparse low lying spectrum [23], identifying the irrational behavior in large m minimal models [24].…”

Virasoro blocks and quasimodular forms

“…In the context of torus 1-and 2-point functions, this property has been used to find asymptotic formulae for OPE coefficients, pioneered by [10], and later adapted to various other cases [11][12][13][14]. Additionally, since crossing symmetry of the full 4-point sphere correlator can be expressed as a modular property, asymptotic constraints can also be obtained from bootstrapping the high "temperature" result [15].…”