Neural Networks, Dispersion Relations and the Thermal Bootstrap

Abstract: We review a framework for the conformal bootstrap that does not rely on positivity and treats the infinite tower of high-dimension OPE contributions to conformal correlators through dispersion relations and neural networks. We apply it to scalar thermal two-point functions on $S^1\times \mathbb R^{d-1}$. We discuss the stability properties of the relevant non-convex optimisation scheme and potential relations to recent discussions of smoothness properties in CFT correlators. We illustrate the numerical application of the method to Generalized Free Fields and 4d holographic CFTs. This is a proceedings contribution to the ``Athens Workshop in Theoretical Physics: 10th Anniversary", held at the National and Kapodistrian University of Athens on December 17-19 2025.

• The paper introduces a novel thermal bootstrap framework that transforms infinite OPE tails into dispersion relations enhanced by neural network tail functions.
• It leverages a non-convex optimization approach via specialized MBMLP architectures to accurately capture spin-dependent contributions while enforcing the thermal KMS condition.
• Empirical tests in generalized free fields and holographic CFTs demonstrate controlled error estimates and robust convergence, highlighting the method’s broad applicability.

Deep Thermal Bootstrap: Neural Networks, Dispersion Relations, and the Thermal KMS Condition

Theoretical Motivation and Framework

The conformal bootstrap has achieved notable success in nonperturbatively constraining CFTs via feasibility analyses of crossing equations, frequently reformulated as convex semi-definite programming problems exploiting OPE coefficient positivity. However, contexts such as finite-temperature CFTs, defect/boundary theories, and higher-point correlators lack positivity and necessitate a primal bootstrap approach for reconstructing full correlators from an infinite set of sum rules. Standard OPE truncations inevitably introduce uncontrolled systematic errors. The present framework circumvents these limitations by transforming the infinite tower of high-dimension OPE contributions into a combination of dispersion relations and dynamically modelled neural network tail functions (2605.13183).

For thermal correlators on $S^1 \times \mathbb{R}^{d-1}$, the KMS condition is recast as a crossing equation $g(z,\bar z) = g(1-z, 1-\bar z)$ for the scalar two-point function. The thermal OPE is reorganized via a split: low-spin ($J \leq J_*$) contributions are further decomposed into explicit (exposed) CFT data and smooth tails parameterizing the high-$\Delta$ sector, while the contributions above spin $J_*$ are packaged into an integrated discontinuity via a subtracted dispersion relation. This leads to an exact representation of the correlator with freely tunable cutoffs, affording controlled error estimates.

Numerical Approach: Neural Network Tail Functions and Non-Convex Optimization

The tail functions are efficiently represented by Multi-Branch Multi-Layer-Perceptron (MBMLP) architectures, with each branch specializing to a particular spin-dependent tail. The optimization problem is formulated by discretizing the crossing region and defining the loss functional as the mean absolute deviation or dot-product-based minimizers for the KMS constraint. This converts the bootstrap into a non-convex optimization problem over exposed CFT coefficients and neural network parameters.

Notably, the non-convexity preserves the multi-solution structure of the KMS equation, absent in linear regression approaches. The inclusion of the discontinuity term injects specific theory-dependent information and preserves degenerate solution sectors, essential for holographic CFTs exhibiting continuous families of bootstrap solutions.

Empirical Tests: Generalized Free Fields and Holographic CFTs

Generalized Free Fields (GFF) Benchmark:

In the GFF case, analytic two-point functions and OPE coefficients allow for stringent consistency checks. The approximate KMS condition, evaluated with analytic tails, achieves loss values that decrease rapidly as the spin cutoff $J_*$ increases. Bootstrapping the tails from the KMS constraint alone (with only the identity exposed) recovers higher-spin tails with high accuracy; leading tails exhibit broader variance that can be sharply reduced by anchoring tail values at intermediate radii ([anchor](https://www.emergentmind.com/topics/anchor) points), stabilizing the optimization.

Quantitative Results:

Exposing the combined coefficient $a_{1,0} + 3 a_{0,2}$ in $d=4$, $\Delta_\phi=1.68$ yields consistent convergence to analytic values:

• At $J_*=8$, $g(z,\bar z) = g(1-z, 1-\bar z)$0 reaches $g(z,\bar z) = g(1-z, 1-\bar z)$1, with the extracted coefficient $g(z,\bar z) = g(1-z, 1-\bar z)$2 closely matching the analytic result $g(z,\bar z) = g(1-z, 1-\bar z)$3.
• Stability and accuracy critically depend on the anchor; incorrect anchor values steer the solution away from the physical correlator, broadening the variance across initializations.

Holographic CFTs:

The framework is applied to holographic CFTs using theory input from the AdS/CFT correspondence for the energy-momentum sector and multi-trace operators. The discontinuity is sourced only by the energy-momentum sector and the identity. With Einstein gravity input ($g(z,\bar z) = g(1-z, 1-\bar z)$4, $g(z,\bar z) = g(1-z, 1-\bar z)$5), the optimization procedure utilizes a ReLU loss penalizing deviations from a GFF reference anchor at intermediate radius.

Numerical Extraction of Double-Twist Data:

The exposed combination $g(z,\bar z) = g(1-z, 1-\bar z)$6 is determined at the stability minimum of the ReLU tolerance parameter:

• The result, $g(z,\bar z) = g(1-z, 1-\bar z)$7, disagrees with bulk calculations ($g(z,\bar z) = g(1-z, 1-\bar z)$8) at about $g(z,\bar z) = g(1-z, 1-\bar z)$9, indicating systematic error due to truncation and reference anchor choice.
• The stability criterion (minimizing ensemble spread as function of penalty parameter) is robust, allowing clear identification of the physical solution.

Neural Spectral Bias, Anchors, and Multifaceted Bootstrap Methods

A complementary approach exploits neural spectral bias: gradient-based optimization on smooth MLPs preferentially selects low-frequency (smooth) solutions among the infinite family of crossing-symmetric correlators. Recent work (Ghosh et al., 20 Apr 2026, Ghosh et al., 20 Apr 2026) demonstrates that, with a single anchor at an intermediate point, neural networks reconstruct CFT correlators to percent-level accuracy across a wide range of theories and dimensions. The smoothness prior, quantified by spectral norm penalties, underlies this empirical success.

Combining the deep thermal bootstrap framework with spectral-bias-based methods enables hybrid strategies, leveraging dispersion relations, spin-dependent tail functions, and smoothness priors for efficient and accurate reconstruction of thermal correlators. This approach has immediate extensions to defect/boundary CFTs and higher-point bootstrap equations, where positivity is also absent.

Implications and Future Directions

The framework provides a pathway for analyzing thermal correlators in non-unitary regimes, defect and boundary CFTs, and more generally for bootstrap scenarios outside the convex positivity paradigm. Controlled treatment of infinite OPE tails, dynamic modeling via neural networks, and integration with dispersion relations yields strong numerical results and exposes the solution structure of the thermal bootstrap. The empirical utility of anchor-based stability and smoothness priors points to practical hybrid bootstrap pipelines for extracting thermal spectra in holographic and physical CFTs, as well as for exploring open problems in boundary/defect setups and higher-point crossing constraints.

Theoretical implications include a more granular understanding of the solution space of bootstrap equations and their sensitivity to theory-specific information, with spectral bias analysis motivating further investigation of the functional smoothness of conformal correlators. Practical implications pertain to robust determination of spin-resolved OPE data and thermal one-point functions in models relevant for quantum critical dynamics, condensed matter, and holographic duals.

Conclusion

The combination of dispersion relations, neural network tail functions, and anchor-based optimization provides a rigorous, systematically improvable framework for bootstrapping thermal correlators in CFTs. By eschewing positivity and uncontrolled OPE truncations, this approach captures infinite contributions and resolves degenerate solution sectors, with strong quantitative verification in GFF and holographic CFTs. The synergy with neural spectral bias and anchor strategies offers robust, accurate methods for analyzing thermal observables and paves the way for future developments in non-unitary, defect/boundary, and higher-point bootstrap contexts.