The Conformal Bootstrap at Finite Temperature

TL;DR

This work develops a finite-temperature conformal bootstrap framework by treating the KMS condition as a crossing constraint for thermal two-point functions on $S^1_\beta\times\mathbb{R}^{d-1}$. Central to the approach is a Lorentzian inversion formula that extracts thermal one-point data $b_{\mathcal{O}}$ and encodes the spectrum via $a(\Delta, J)$, enabling analytic control over large-spin sectors. The authors validate the method with Mean Field Theory, analyze large-$N$ CFTs (notably the $O(N)$ vector model) and holographic contexts, and develop a systematic large-spin perturbation theory for thermal data, including universal contributions from the unit operator and stress tensor to double-twist families. They illustrate the program with a detailed Ising-model case study and discuss future directions, including spinning operators, other compactifications, and transport phenomena, highlighting the potential of the thermal bootstrap to constrain finite-temperature CFT data in broad settings.

Abstract

We initiate an approach to constraining conformal field theory (CFT) data at finite temperature using methods inspired by the conformal bootstrap for vacuum correlation functions. We focus on thermal one- and two-point functions of local operators on the plane. The KMS condition for thermal two-point functions is cast as a crossing equation. By studying the analyticity properties of thermal two-point functions, we derive a "thermal inversion formula" whose output is the set of thermal one-point functions for all operators appearing in a given OPE. This involves identifying a kinematic regime which is the analog of the Regge regime for four-point functions. We demonstrate the effectiveness of the inversion formula by recovering the spectrum and thermal one-point functions in mean field theory, and computing thermal one-point functions for all higher-spin currents in the critical $O(N)$ model at leading order in $1/N$. Furthermore, we develop a systematic perturbation theory for thermal data in the large spin, low-twist spectrum of any CFT. We explain how the inversion formula and KMS condition may be combined to algorithmically constrain CFTs at finite temperature. Throughout, we draw analogies to the bootstrap for vacuum four-point functions. Finally, we discuss future directions for the thermal conformal bootstrap program, emphasizing applications to various types of CFTs, including those with holographic duals.

Figures (13)

• Figure 1: The OPE on $S^1_\beta \times \mathbb{R}^{d-1}$ is valid if the two operators lie inside a sphere. The largest possible sphere has diameter $\beta$, wrapping entirely around the $S^1$ such that it is tangent to itself. Here, we illustrate such a sphere (blue) in $d=2$.
• Figure 2: Lightlike trajectories moving in the $x_L$ direction and around the thermal circle. One trajectory is $z=0$ and the other is $\overline{z}=0$. Poles in the Lorentzian inversion formula come from the neighborhood of these trajectories.
• Figure 3: Contour manipulations for the inversion formula in the $w$ plane. In (a) we show the original contour which lies along the circle $\vert w\vert=1$. For the $w^J$ terms in \ref{['eq:euclideaninversion']}, we deform the contour as in (b), and for the $w^{-J}$ terms in \ref{['eq:euclideaninversion']}, we deform the contour as in (c).
• Figure 4: For fixed $r\in(0,1)$, the $s$-channel OPE (expansion around $z=\overline{z} = 0$) implies that the thermal two-point function $g(z,\overline{z})$ is analytic in an annulus in the $w$ plane between radii $r$ and $1/r$ (shaded blue). The $t$-channel OPE (expansion around $z=\overline{z} = 1$), together with symmetry under $w\leftrightarrow -w$, implies analyticity in the red-shaded regions, except for cuts running along $(-\infty,-1/r), (-r,0), (0,r), (1/r,\infty)$ (indicated with zig-zags). In this section, we argue for analyticity everywhere in the upper and lower half planes.
• Figure 5: An illustration of the relation between $s$- and $t$-channels in the $\langle \phi\phi\rangle_\beta$ correlator. The two channels are related by moving the external operators around the thermal circle (gray). A single term in the $t$-channel OPE $\mathcal{O}\in \phi\times \phi$ inverts to the sum over the $[\phi\phi]_n$ families in the $s$-channel. Alternatively, the sum over the $[\phi\phi]_n$ families in the $s$-channel reproduces the $\mathcal{O}$ term in the $t$-channel.