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arXiv:2607.24913

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

Kausik Ghosh, Sidhaarth Kumar, Vasilis Niarchos B, Andreas Stergiou

Published Jul 27, 2026• 0 citations
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Abstract

We develop a neural network bootstrap framework for reconstructing partition functions of two-dimensional conformal field theories (CFTs) based on modular invariance and the Cardy condition, which are recast as crossing equations for four-point correlators. For torus partition functions, we use the twist-field representation in the symmetric-orbifold description to map modular S-invariance to four-point crossing and focus on the diagonal kinematics of four insertions on a line. For annulus partition functions, we formulate open/closed channel duality as crossing symmetry for mixed four-point functions of defect-changing operators in interface CFT. In both cases, the reconstruction problem is formulated in the anchored-bootstrap form, where the crossing constraints are supplemented by minimal spectral input (a gap) and anchor data. We solve this under-determined problem by using lightweight feed-forward neural networks to parametrise the correlators and their corresponding partition functions. A key ingredient of this approach is the spectral bias of the neural networks in the lazy training regime, which selects specific crossing-symmetric configurations. This reformulation unifies standard modular and annulus constraints in two dimensions with the anchored neural approach for CFT correlators, providing a new way to reconstruct full partition functions from sparse data with remarkable accuracy.

DOI
10.48550/arxiv.2607.24913

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We develop a neural network bootstrap framework for reconstructing partition functions of two-dimensional conformal field theories (CFTs) based on modular invariance and the Cardy condition, which are recast as crossing equations for four-point correlators. For torus partition functions, we use the twist-field representation in the symmetric-orbifold description to map modular S-invariance to four-point crossing and focus on the diagonal kinematics of four insertions on a line. For annulus partition functions, we formulate open/closed channel duality as crossing symmetry for mixed four-point functions of defect-changing operators in interface CFT. In both cases, the reconstruction problem is formulated in the anchored-bootstrap form, where the crossing constraints are supplemented by minimal spectral input (a gap) and anchor data. We solve this under-determined problem by using lightweight feed-forward neural networks to parametrise the correlators and their corresponding partition functions. A key ingredient of this approach is the spectral bias of the neural networks in the lazy training regime, which selects specific crossing-symmetric configurations. This reformulation unifies standard modular and annulus constraints in two dimensions with the anchored neural approach for CFT correlators, providing a new way to reconstruct full partition functions from sparse data with remarkable accuracy.
We demonstrate that simple feed-forward neural networks (NNs) can accurately compute correlation functions of conformal field theories (CFTs) on a line. Strikingly, by optimising a NN solely on crossing symmetry and providing only the scaling dimension of the leading non-trivial operator and the correlator's value at a single "anchor point", we can reconstruct target physical correlators to within a few percent. We establish the robustness of this minimal-data approach across a broad class of theories and dimensions, including generalised free fields, contact and one-loop Witten diagrams in AdS2_22​, unitary and non-unitary 2d minimal models, the 3d Ising model, and half-BPS correlators in 4d N=4\mathcal{N}=4N=4 super-Yang-Mills theory, together with several thermal two-point functions, notably including those of the 3d Ising model. We argue that this remarkable alignment between NNs and CFTs stems from the spectral bias of gradient-based training, which heavily favours smooth functions. To ground this connection, we analyse the smoothness of conformal correlators using fractional Sobolev semi-norms, Chebyshev spectral decompositions, and a measure based on curvature. Finally, we establish the broader reconstructive power of this technique by extending it beyond the diagonal kinematics of the line.
Two-dimensional Conformal Field Theories (CFTs) allow us to describe quantum field theories at critical points and in general systems without a characteristic scale. It is possible to implement such theories on topologically non-trivial surfaces. If we formulate a two-dimensional CFT on a toroidal surface, a new symmetry emerges known as modular invariance. This symmetry corresponds to the set of transformations that act without changing the toroidal surface, consequently, the partition function of the theory must remain invariant under these transformations. This invariance is used as a constraint by the modular bootstrap method to obtain information about the spectrum without using the Lagrangian and operator content of the theory. By imposing that the theory is consistent with modular symmetry, it is possible to determine the limits of the spectrum beyond which modular invariance can no longer be satisfied, thus constraining the space of admissible two-dimensional conformal field theories. To study CFTs that present discrete global symmetries, it is essential to introduce Topological Defect Lines (TDLs). These are one-dimensional, topologically invariant operators that implement the action of the symmetry group on the Hilbert space. The presence of TDLs modifies the topology of the system and introduces new partition functions associated with different symmetry sectors. Under modular transformations, the TDLs mix among themselves, and consequently, the corresponding partition functions also mix, thus promoting the modular constraint to a vector-valued condition. This formalism makes it possible to implement discrete symmetries also considering the possible presence of anomalies. The objective of this thesis is therefore to understand how discrete symmetries and anomalies affect the spectrum of a CFT. Based on the modular bootstrap framework developed by Lin and Shao for the Z2 symmetry, the analysis was first analytically extended to the Abelian case Z3. This was useful in extending the discussion to a more complex case, namely the non-Abelian symmetry S3. This extension constitutes the original theoretical contribution of the thesis, providing the explicit formulation of modular bootstrap equations for CFTs with non-Abelian discrete symmetry. The analytical modular constraints were reformulated numerically, bridging the modular bootstrap framework with semidefinite programming in SDPB. This analysis paves the way for a numerical study based on convex optimization techniques to explore the constraints arising from modular bootstrap equations.
We propose that simple neural networks (NNs) trained on crossing symmetry can reconstruct conformal correlators restricted to a line to remarkable accuracy. The input is minimal: an external scaling dimension, a spectral gap, and the value of the correlator at a single point. We present evidence across a wide range of conformal theories and dimensions, for both four-point and thermal two-point functions. We attribute these observations to the spectral bias of gradient-based NN training, which appears to align with an intrinsic smoothness property of conformal field theory. This suggests a novel variational principle for conformal correlators and opens a path towards a powerful new computational framework for non-perturbative quantum field theory.
We propose that simple neural networks (NNs) trained on crossing symmetry can reconstruct conformal correlators restricted to a line to remarkable accuracy. The input is minimal: an external scaling dimension, a spectral gap, and the value of the correlator at a single point. We present evidence across a wide range of conformal theories and dimensions, for both four-point and thermal two-point functions. We attribute these observations to the spectral bias of gradient-based NN training, which appears to align with an intrinsic smoothness property of conformal field theory. This suggests a novel variational principle for conformal correlators and opens a path towards a powerful new computational framework for non-perturbative quantum field theory.
A bstract We study two-dimensional conformal field theories (CFTs) with boundaries via the conformal bootstrap. We derive a positive semi-definite program from crossing symmetry of three observables: the annulus partition function, the two-point function of identical operators in the presence of a boundary, and the four-point function of the same operators on the infinite plane. The mixed-correlator system allows the numerical bootstrap to access new data, like the bulk-to-boundary Operator Product Expansion coefficients, and to strengthen the bounds on observables already contained in the partition function on the annulus, such as the boundary entropy. We test the method on the free boson CFT; then, as a first application, we produce new non-perturbative bounds on the entropy and the gaps in boundary CFTs with central charge c = 3/2, with special emphasis on the 𝔰𝔲(2) 2 WZW model.
We develop a functional-analytic framework for modular constraints in two-dimensional conformal field theory based on the reflection symmetry [Formula: see text]. This symmetry isolates a natural reflection-odd subspace of [Formula: see text] in which all deviations from modular self-duality live. For any modular-invariant partition function, the vacuum-heavy decomposition determines a uniquely defined reflection-odd modular defect [Formula: see text] whose completed Mellin transform satisfies an odd functional equation. Pairing D with compactly supported reflection-odd test functions yields a structured hierarchy of linear functionals. Under mild assumptions on the light spectrum, these pairings obey positivity conditions that lead to smeared Cardy inequalities comparing the heavy spectrum to the Cardy density through positive Laplace kernels. We also show that the vanishing of all such pairings is equivalent to averaged Cardy saturation in [Formula: see text], giving a complete reflection-odd characterization of modular self-duality. This provides a simple and flexible functional basis for modular bootstrap analyses, replacing ad hoc constructions with a unified reflection-odd hierarchy suited for both analytic arguments and numerical implementation.