⚛️ high-energy theory

Neural Spectral Bias and Conformal Correlators II: Modular and Annulus Bootstrap

This paper introduces a neural network bootstrap framework that leverages the spectral bias of lightweight feed-forward networks to reconstruct two-dimensional CFT partition functions from sparse spectral data by recasting modular invariance and open/closed duality as crossing symmetry constraints for four-point correlators.

Original authors: Kausik Ghosh, Sidhaarth Kumar, Vasilis Niarchos, Andreas Stergiou

Published 2026-07-29

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CC BY 4.0

Original authors: Kausik Ghosh, Sidhaarth Kumar, Vasilis Niarchos, Andreas Stergiou

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible Lego set. Physicists call the rules that govern how these Lego bricks snap together "Conformal Field Theories" (CFTs). These rules describe how particles and forces behave when you zoom in so close that size and shape stop mattering, and only the relationships between things remain. For decades, scientists have tried to figure out which combinations of Lego bricks are actually allowed to exist in a stable universe. They use a method called the "bootstrap," which is like trying to solve a massive jigsaw puzzle where you don't have the picture on the box. You only have a few pieces and the rule that the edges must fit perfectly. If you try to force a piece where it doesn't belong, the whole picture falls apart. The goal is to find the specific patterns that hold together without breaking, revealing the hidden blueprints of reality.

In this new study, a team of researchers decided to try a different way to solve this puzzle. Instead of manually checking every possible Lego combination, they taught a computer brain—a neural network—to do the heavy lifting. They didn't just ask the computer to guess; they gave it the "rules of the game" (mathematical symmetries) and a tiny bit of real-world data, then watched to see what it built. The surprising discovery? The computer, on its own, seemed to "know" which patterns were real and which were fake, even though it was never explicitly told the answer. It's as if you handed a robot a few scattered puzzle pieces and the instruction "make a picture," and it magically assembled the exact image of a cat, simply because it had a natural bias toward smooth, sensible shapes.

The paper, titled "Neural Spectral Bias and Conformal Correlators II," explores this idea by applying it to two specific types of puzzles: the "torus" (a shape like a donut) and the "annulus" (a shape like a washer or a ring). In the world of physics, these shapes represent different ways of looking at the universe's energy and particles. The torus represents a closed loop, like a universe that wraps around itself, while the annulus represents a space with boundaries, like a ring floating in empty space.

Traditionally, checking if a theory works for these shapes involves complex math called "modular invariance" and "Cardy consistency." Think of modular invariance as a rule that says, "If you stretch or twist your donut-shaped universe, the physics inside must look exactly the same." The Cardy condition is similar but for the ring shape, ensuring that the view from the inside matches the view from the outside. The authors realized that these rules could be rewritten as a different kind of puzzle: a "crossing equation." Imagine four people standing in a square. The rule says that if you swap two of them, the story they tell must still make sense. By turning the donut and ring problems into these four-person stories, the researchers could use their neural network trick.

They set up a digital experiment where the neural network had to learn the "story" (the mathematical function) that satisfied these swap rules. They gave the network very little information: just the rule that the story must be consistent, a tiny hint about the "gap" (the smallest amount of energy allowed), and one single data point in the middle of the story to anchor it. They didn't tell the network the final answer.

The results were remarkably accurate. When the network tried to solve the puzzle for famous, well-known physics models—like the 2D Ising model (which describes how magnets work) or the Lee-Yang model (a more exotic, non-standard version)—it reconstructed the correct mathematical story with incredible precision. In many cases, the computer's guess was off by less than one percent. Even more interestingly, the network worked just as well for "non-unitary" models, which are theories that don't follow the usual rules of energy conservation in the standard way. This suggests the method is very flexible.

The researchers also tested the method on "non-compact" theories, which are like universes that stretch out infinitely rather than wrapping around. These are much harder to solve because the numbers can get wildly large or small. For a specific type of infinite universe called Liouville theory, the network still managed to find the right pattern, though it required some clever adjustments to the math to handle the extreme numbers.

The key takeaway isn't just that the computer got the right answers, but how it got them. The authors suggest that neural networks have a built-in "spectral bias." This is a fancy way of saying that when these computer brains learn, they naturally prefer smooth, simple, and physical solutions over messy, chaotic ones. Even when the math allows for thousands of different answers, the network consistently "chooses" the one that looks like a real universe.

This doesn't mean the problem is completely solved or that we can now predict every particle in the universe. The paper explicitly notes that the method relies on simulations and that the network's success is an observation of its behavior, not a rigorous mathematical proof. However, it suggests a powerful new direction: instead of searching through a vast, dark forest of possible theories, we can use these neural networks as a flashlight that naturally illuminates the paths that lead to physical reality. It's a new way to navigate the space of consistent theories, using the computer's own "intuition" to find the hidden blueprints of the cosmos.

Technical Summary: Neural Spectral Bias and Conformal Correlators II – Modular and Annulus Bootstrap

Problem Statement
The paper addresses the non-perturbative exploration of the space of consistent two-dimensional Conformal Field Theories (CFTs). While traditional conformal bootstrap methods search for valid spectra and Operator Product Expansion (OPE) coefficients by solving crossing equations as convex optimization problems, this approach is limited by the requirement of unitarity and positivity. The authors investigate whether a different philosophy—searching directly in the space of crossing-symmetric functions rather than CFT data—can be extended to modular invariance and boundary consistency conditions. Specifically, the work targets the reconstruction of torus partition functions (constrained by modular invariance) and annulus partition functions (constrained by Cardy consistency/open-closed duality) using sparse physical input.

Methodology
The core methodology reformulates partition function constraints as crossing equations for four-point correlators, allowing the application of a neural network (NN) framework previously developed for standard correlators [9, 10].

Geometric Reformulation:
- Torus: The modular $S$ -invariance of the torus partition function $Z(\tau, \bar{\tau})$ is mapped to the crossing symmetry of a four-point function of identical $\mathbb{Z}_2$ twist fields on a sphere. The complex modulus $\tau$ is related to the cross-ratio $z$ via the modular lambda function. The authors focus on the diagonal kinematics ( $z = \bar{z}$ , corresponding to purely imaginary $\tau$ ), reducing the problem to a single-variable crossing equation:
  $G(z) = \left(\frac{z}{1-z}\right)^{c/4} G(1-z)$
- Annulus: The Cardy condition (open/closed channel duality) is reformulated as the crossing symmetry of a mixed four-point function of defect-changing operators (DCOs) in an interface CFT. This relates the open-channel partition function to the closed-channel one via a crossing equation involving the cross-ratio $z$ .
Neural Network Ansatz:
- The reconstruction problem is under-determined, as many crossing-symmetric functions exist. To select physical solutions, the authors employ the "spectral bias" of neural networks in the "lazy training" regime.
- The target function is factorized into a known prefactor (removing universal kinematic singularities and descendant state contributions) and a smooth residual function parametrized by a lightweight feed-forward neural network (MLP with two hidden layers of width 64 and GELU activation).
- Anchored Bootstrap: The training is guided by a loss function comprising a crossing symmetry term and an "anchor" term. The anchor fixes the value of the correlator at a single interior point $z_0$ (typically $z_0=0.3$ ), supplemented by minimal spectral input (the gap $\Delta_{\text{gap}}$ of the lightest non-vacuum primary).
- Training: The networks are trained using Adam optimization with a StepLR schedule. The loss function minimizes the deviation from the crossing equation and the anchor constraint.

Key Contributions

Unification of Constraints: The paper demonstrates that modular invariance and Cardy consistency can be unified under the language of four-point crossing symmetry, enabling the use of neural spectral bias to solve these problems.
Extension to Partition Functions: It extends the neural bootstrap framework from local correlators to global partition functions (torus and annulus), showing that the method works for both compact and non-compact CFTs.
Handling Non-Unitarity and Non-Compactness: The framework successfully reconstructs partition functions for non-unitary theories (e.g., Lee-Yang model) and non-compact theories (e.g., free non-compact boson and Liouville theory), where traditional positivity-based bootstrap methods fail.
Annulus Reconstruction: It provides a novel simultaneous reconstruction of open and closed channel correlators for mixed systems, handling cases where the two channels have different spectra and vastly different magnitudes.

Results
The authors present extensive numerical evidence across various CFTs:

Compact CFTs: The method accurately reconstructs torus partition functions for the 2D Ising model, Lee-Yang model, generic ADE-series minimal models (including tricritical Ising and 3-state Potts), and Wess-Zumino-Witten (WZW) models ( $\widehat{su}(2)_k$ , $\widehat{su}(3)_1$ ). The reconstruction error is typically sub-percent, with the ensemble mean of predictions closely matching exact analytic results.
Annulus Partition Functions: The approach successfully reconstructs annulus partition functions for various Cardy boundary conditions in minimal models and WZW theories. The results show high accuracy for both open and closed channels, even when the channels are distinct.
Non-Compact CFTs: The method recovers the partition functions for the free non-compact boson and Liouville theory. For Liouville theory with $c=31.5$ , the authors demonstrate that with appropriate re-parameterization to handle small correlator values and channel disparities, the NN can achieve reconstruction accuracy within a few parts in $10^4$ .
Spectral Bias Observation: In specific cases (e.g., $\widehat{su}(2)_2$ ), the training landscape exhibits bimodality. The authors observe that runs terminating early (due to stagnation) tend to converge to the physical solution, while runs running the full epoch budget may converge to spurious solutions, suggesting a role for early stopping in selecting physical correlators.

Significance and Claims
The paper claims that neural networks, through their inherent spectral bias, provide a powerful and general operational tool for navigating the space of modular-invariant and crossing-symmetric functions.

Generality: Unlike traditional bootstrap methods, this approach does not rely on unitarity or positivity, making it applicable to a broader class of theories, including non-unitary and non-compact CFTs.
Efficiency: The method can reconstruct full partition functions from extremely sparse data (a single anchor point and the spectral gap).
New Paradigm: The work suggests a shift in the bootstrap philosophy: rather than searching directly over spectra and OPE coefficients, one can search directly in the space of functions satisfying consistency conditions.
Limitations: The authors modestly note that while the method shows remarkable accuracy, it does not currently provide rigorous error bounds. They also acknowledge that for non-compact theories with large central charges, the dynamic range of the target functions can pose numerical challenges requiring specific loss function modifications or rescaling.

The paper concludes that this neural approach offers a promising new direction for the modular bootstrap and the non-perturbative reconstruction of quantum field theory observables, with potential future applications to higher-genus partition functions and higher-point correlators.

Technical Summary: Neural Spectral Bias and Conformal Correlators II – Modular and Annulus Bootstrap

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