Deep Finite Temperature Bootstrap

TL;DR

This work develops a novel primal bootstrap framework for conformal field theories at finite temperature by combining spin-based dispersion relations with neural-network modeled tails that encode the contributions of infinitely many high-dimension operators. It circumvents positivity-based feasibility and hard truncations by treating tails and discontinuities as dynamical data to be learned, enabling a non-convex optimization over exposed CFT data and tail functions. The method is validated with Generalized Free Fields, where analytic checks confirm convergence and controlled truncation errors, and is extended to holographic CFTs to extract preliminary double-twist thermal data, including applications to Einstein-gravity and higher-derivative gravity scenarios. The approach yields a flexible, scalable route to thermal CFT bootstrap, with potential to illuminate N=4 SYM in the supergravity regime and other contexts where positivity constraints are absent or inapplicable, thus broadening the scope of bootstrap techniques in finite-temperature and holographic settings.

Abstract

We introduce a novel method to bootstrap crossing equations in Conformal Field Theory and apply it to finite temperature theories on $S^1\times \mathbb{R}^{d-1}$. The proposed approach does not rely on positivity constraints and does not employ uncontrolled truncation schemes. Instead, we capture the contribution of an infinite number of operators in conformal block expansions using suitable functions, which are bootstrapped (numerically) together with a finite number of exposed CFT data. Our approach at finite temperature employs three key ingredients: $(i)$ the Kubo-Martin-Schwinger (KMS) condition, $(ii)$ thermal dispersion relations and $(iii)$ Neural Networks that model spin-dependent tail functions within the conformal block expansions. We test the efficiency of the new method in the case of Generalized Free Fields and use it to perform a preliminary bootstrap analysis of double-twist thermal data in holographic CFTs.

Figures (12)

• Figure 1: The blue points represent a grid of 243 points on the complex $z$-plane used for optimization during NN training. The grid avoids the real axis and covers points in the region $|z|<0.95$, $|1-z|<0.95$. The orange points represent the validation grid.
• Figure 2: Plots depicting the results obtained with ${\cal L}_{\overline{\rm abs}}+{\cal L}_{\rm BC}[A]$ in the $d=4$ GFF theory with $\Delta_\phi=1.68$ and $J_*=6$ without any exposed operators, and asymptotic boundary conditions for the tail functions at $r=0.9999$. The results are based on the 10 lowest-loss configurations within a pool of 1K independent training runs for 50K epochs. The mean loss of these configurations is $1.93 \times 10^{-3} \pm 1.06\times 10^{-4}$, whereas the absolute loss of the analytic GFF configuration is $1.29 \times 10^{-3}$. The first 4 plots depict (in blue) the mean and 1$\sigma$ deviation of the predicted tail functions. The dashed red curves represent the exact, analytic result of the GFF theory. The middle plot on the second line depicts the combined contribution of the two leading tail functions, $A_0$ and $A_2$, to the conformal block expansion of the thermal correlator for $w=1$ and its comparison against the analytic expression (in red). The final plot at the bottom right is a heatmap of the relative difference \ref{['tailsAbsab']} between the predicted and analytic values in the training region.
• Figure 3: Plots of the predicted tail functions obtained with ${\cal L}_{{\rm dot}(0)}+{\cal L}_{\rm BC}$ for $J_*=6$ in the $d=4$ GFF theory for $\Delta_\phi=1.68$. The obtained results are based on the 10 lowest-loss configurations for 1K independent runs of 50K epochs. The mean loss, $3.54\times 10^{-9} \pm 6.4 \times 10^{-10}$, should be compared to the loss of the analytic GFF solution $3.35 \times 10^{-9}$. The middle plot in the second line represents the combined contribution to the 2-point function of the $A_0$, $A_2$ tails at $w=1$. The heatmap depicts the relative difference \ref{['tailsAbsab']} inside the training region.
• Figure 4: Results obtained with the loss ${\cal L}_{{\rm dot}(0)}+{\cal L}_{\rm BC}$, performing the same runs as in Fig. \ref{['fig:dot_no_exposed_no_input']}, but with the additional input of the analytic value $a_{1,0}+3a_{0,2}\simeq 15.06$.
• Figure 5: Results obtained with the dot-loss ${\cal L}_{{\rm dot}(0)}$, performing the same runs as in Fig. \ref{['fig:dot_no_exposed_no_input']} after replacing the condition on the asymptotic values of the tail functions at $r=0.9999$ with a condition that sets the values of the tail functions at $r_i=0.7$ to the analytic values of the GFF solution. Notice that the maximum of the color bar scale in the heatmap is now set to 0.05, compared to the higher value of 0.20 in the previous Figs. \ref{['fig:dot_no_exposed_no_input']}, \ref{['fig:dot_pinn_no_exposed_with_input']}.