Thermal Bootstrap in Finite-Temperature CFT

• Thermal Bootstrap is a framework that reconstructs finite-temperature observables from zero-temperature CFT data using consistency conditions like KMS periodicity and analyticity.
• It leverages the Euclidean thermal manifold (S¹_β×ℝ^(d-1)) to analyze thermal correlators, one-point coefficients, and thermodynamic quantities such as free-energy density.
• This approach extends to modular bootstrap in 2D CFTs and the statistical bootstrap of hadronic matter, highlighting its broad applicability in theoretical physics.

Searching arXiv for key thermal bootstrap papers and formulations. Searching arXiv for "thermal bootstrap finite temperature CFT KMS". Thermal bootstrap denotes a family of bootstrap programs that constrain or reconstruct finite-temperature observables from general consistency conditions rather than from a direct microscopic solution. In modern conformal-field-theory usage, the central setting is a CFT on the Euclidean thermal manifold $S^1_\beta\times \mathbb R^{d-1}$, where thermal correlators obey the Kubo–Martin–Schwinger condition, admit local operator-product expansions, and contain new dynamical data in the form of thermal one-point coefficients $b_{\mathcal O}$. In that setting, the thermal bootstrap asks how zero-temperature CFT data $$(\Delta_{\mathcal O},J_{\mathcal O},f_{\mathcal O\mathcal O'\mathcal O''})$$, together with KMS periodicity, analyticity, and heavy-operator asymptotics, determine thermal one- and two-point functions and thermodynamic quantities such as the free-energy density (Iliesiu et al., 2018). In broader usage, the term also covers modular $S$-transformation bootstrap for torus observables in two-dimensional CFTs, bootstrap derivations of emergent thermality in heavy states, and the older statistical bootstrap description of hadronic matter and its limiting temperature (Brehm et al., 2019).

1. Finite-temperature CFT formulation

The modern thermal bootstrap is formulated on

$$S^1_\beta\times \mathbb R^{d-1}, \qquad \beta=\frac1T,$$

with Euclidean time $\tau\sim \tau+\beta$. Thermal expectation values are defined by

$$\rho = e^{-\beta H}, \qquad Z(\beta)=\operatorname{Tr} e^{-\beta H},$$

and [ \langle \mathcal O_1(x_1)\cdots \mathcal O_n(x_n)\rangle_\beta = Z^{-1}(\beta)\operatorname{Tr}!\