神经谱偏置与共形关联函数II：模块化与环面引导
Neural Spectral Bias and Conformal Correlators II: Modular and Annulus Bootstrap
AI总结：
该研究基于模块化不变性等开发神经网络引导框架重建二维CFT配分函数，对环面和环面配分函数分别处理，以锚定引导形式提出问题，用轻量级神经网络解决，利用谱偏置统一相关约束，能高精度从稀疏数据重建配分函数。
AI中文摘要：
我们基于模块化不变性和卡迪条件开发了一个神经网络引导框架，用于重建二维共形场论（CFT）的配分函数，这些条件被重铸为四点关联函数的交叉方程。对于环面配分函数，我们在对称轨道描述中使用扭场表示将模块化S不变性映射到四点交叉，并关注一条线上四个插入的对角运动学。对于环面配分函数，我们将开放/封闭通道对偶性表述为界面CFT中缺陷变化算符的混合四点函数的交叉对称性。在这两种情况下，重建问题都以锚定引导形式提出，其中交叉约束由最小谱输入（一个间隙）和锚定数据补充。我们通过使用轻量级前馈神经网络对关联函数及其相应的配分函数进行参数化来解决这个欠定问题。这种方法的一个关键要素是神经网络在懒惰训练 regime 中的谱偏置，它选择特定的交叉对称配置。这种重新表述将二维中的标准模块化和环面约束与用于CFT关联函数的锚定神经方法统一起来，提供了一种从稀疏数据中以显著精度重建完整配分函数的新方法。
英文摘要：
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.
