Universal Asymptotics for High Energy CFT Data

TL;DR

This work develops a universal high-energy framework for higher-dimensional CFT data using a thermal effective action obtained by circle compactification. It derives leading and subleading density-of-states formulas across dimensions, including exact results in d=2 and d=3, and matches holographic and free-theory benchmarks. A novel genus-2 (higher-dimensional) partition function is introduced, with a hot-spot mechanism that isolates singular contributions and yields universal heavy OPE coefficients via genus-2 conformal blocks and a detailed inversion program. The approach reveals deep connections between thermodynamics, conformal blocks, and OPE statistics, with implications for bootstrap, holography, and bulk locality in thermal backgrounds. The results provide concrete, testable predictions for OPE data and thermal one-point functions in a broad class of CFTs.}

Abstract

Equilibrium finite temperature observables of a CFT can be described by a local effective action for background fields -- a "thermal effective action." This effective action determines the asymptotic density of states of a CFT as a detailed function of dimension and spin. We discuss subleading perturbative and nonperturbative corrections to the density, comparing with free and holographic examples. We furthermore show how to use the thermal effective action on more complicated geometries at special locations called "hot spots." The hot spot idea makes a prediction for a CFT partition function on a higher-dimensional version of a genus-2 Riemann surface, in a particular high temperature limit. By decomposing the partition function into a novel higher-dimensional version of genus-2 conformal blocks (which we compute at large scaling dimension), we extract the asymptotic density of heavy-heavy-heavy OPE coefficients in a higher-dimensional CFT. We also compute asymptotics of thermal 1-point functions using the same techniques.

Figures (13)

• Figure 1: The "genus-2" geometry and its "hot spots." The top is a ball $B^d$ with two balls removed. It is topologically equivalent to a three-punctured $S^d$. The bottom is the same. The top and bottom are glued together with three cylinders. In the limit that the cylinders get short, there are shrinking circles indicated in red that run down one cylinder and up another. The neighborhoods of each of these circles are "hot spots," where the thermal effective action receives a large contribution.
• Figure 2: The space $Y=B_3\backslash (B_1\cup B_2)$. The boundary of $Y$ has three $S^{d-1}$ components given by $\partial B_1,\partial B_2,\partial B_3$ with radii $1,1,2$, respectively. Note that in $d\geq 3$, $Y$ has an $\mathrm{SO}(d-1)$ rotational symmetry around the horizontal axis, and here we are depicting only a 2-dimensional slice.
• Figure 3: The manifold $M_2$ is obtained by taking two copies of $Y$ and gluing corresponding boundary components with cylinders $C_1,C_2,C_3$ of inverse temperatures $\beta_1,\beta_2,\beta_3$ and angular twists $h_1,h_2,h_3$, colored red, blue, and green, respectively. In the figure, we slightly bent the edges of the cylinders to help visualize them, but in the actual geometry the cylinders have constant radii. The figure also naively suggests that the lengths of the cylinders must be equal, but in reality they need not be related to each other.
• Figure 4: The sphere $S^d$ is Weyl-equivalent to a "capped cylinder" $\mathcal{C}_{r,\beta}$ with radius $r$ and length $r\beta$. Each end cap is a ball (the interior of an $S^{d-1}$) of radius $r$. The "closed junctions" where the cylinder meets the end caps are highlighted in red.
• Figure 5: An "open junction," where a cylinder meets the complement of a ball in a flat plane. The junction is highlighted in blue.