A conformal block Farey tail

TL;DR

The paper presents a modular-averaged approach to the conformal bootstrap in two dimensions, recasting crossing symmetry as PSL(2,Z) invariance on the upper half-plane and constructing crossing-symmetric four-point functions by averaging a light-block seed over the modular group. In minimal models this Farey-tail (Poincaré series) construction can exactly reproduce correlators and fix OPE data in several Ising-type theories, while in some cases the vacuum seed alone is insufficient, highlighting the need for seeds that include additional light exchanges. A group-theoretic perspective clarifies the construction as a projection onto PSL(2,Z)-invariant subspaces, with the representation structure determining when a unique correlator emerges. In the semiclassical regime, the Farey tail has a natural AdS3 gravity interpretation as a sum over bulk saddles corresponding to rational tangles, and explicit gravity analyses (including the h=c/32 case) connect the conformal blocks to bulk worldline actions and DOZZ-type coefficients, tying modular averaging to bulk topology and holographic data.

Abstract

We investigate the constraints of crossing symmetry on CFT correlation functions. Four point conformal blocks are naturally viewed as functions on the upper-half plane, on which crossing symmetry acts by PSL(2,Z) modular transformations. This allows us to construct a unique, crossing symmetric function out of a given conformal block by averaging over PSL(2,Z). In some two dimensional CFTs the correlation functions are precisely equal to the modular average of the contributions of a finite number of light states. For example, in the two dimensional Ising and tri-critical Ising model CFTs, the correlation functions of identical operators are equal to the PSL(2,Z) average of the Virasoro vacuum block; this determines the 3 point function coefficients uniquely in terms of the central charge. The sum over PSL(2,Z) in CFT2 has a natural AdS3 interpretation as a sum over semi-classical saddle points, which describe particles propagating along rational tangles in the bulk. We demonstrate this explicitly for the correlation function of certain heavy operators, where we compute holographically the semi-classical conformal block with a heavy internal operator.

Figures (6)

• Figure 1: Fundamental domain for $\Gamma(2)$ in the upper half $\tau$-plane, bounded by the blue dashed curves. $T^2$ identifies the left and right vertical lines, and $ST^2S$ identifies the semicircles. The three cusps at $\tau=i\infty$, $0$ and $\pm1$ correspond to $x=0,1,\infty$. The black dashed lines show how this domain breaks up further into six fundamental domains for the modular group $\Gamma$ (four of which are split in two across the blue lines). These six domains correspond to the images in the cross-ratio $x$-plane under the anharmonic group, shown in the right figure, where the marked points are at $x=0,1$.
• Figure 2: The blue dots show $c(k_{max})$, plotted agains $k_{max}$. The orange line on the RHS is approximately $0.372$ -- the exact OPE coefficient. The last point on the RHS is approximately $0.381$ which is about $2\%$ from $0.372$.
• Figure 3: The rational tangles $t_\infty$, $t_0$ and $t_{-29/74}$. The last diagram should be compared with the continued fraction $-1/(3-1/(2-1/(-4-1/3)))=-\frac{29}{74}$.
• Figure 4: The spherical braid group generators and relations: $B_4(S^2)$ is generated by $\sigma_1,\sigma_2,\sigma_3$, with the relations $\sigma_1\sigma_3=\sigma_3\sigma_1$, $\sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2$, $\sigma_2\sigma_3\sigma_2=\sigma_3\sigma_2\sigma_3$, and $\sigma_1\sigma_2\sigma_3^2\sigma_2\sigma_1=1$. The outer in inner circles represent cross-sections through a two-sphere so, for example, the last braid is trivial because the strand can be unwrapped around the front and back of the internal $S^2$.
• Figure 5: Examples of 2-tangles which are not rational, so may not be untangled only by moving the boundary points. No such worldline topologies appear as saddle points to the gravitational path integral.