AI 中文总结
本文将混合连续/离散潜变量构造的神经网络场论（NN-FT）应用于两个紧致玻色子，分别重现了Berezinskii-Kosterlitz-Thouless相变和玻色弦的相关物理性质，表明搭配不同离散拓扑数据的局域神经采样器可产生不同紧致理论。
AI 中文摘要
神经网络场论（NN-FT）通过网络架构及其潜变量上的概率密度定义场论。对于紧致理论，局域高斯部分只是整体的一部分：还必须对离散拓扑部分求和。我们综述了一种混合连续/离散潜变量构造，并将其应用于两个紧致玻色子。对于Berezinskii-Kosterlitz-Thouless相变，由随机傅里叶特征自旋波采样器补充显式库仑气体涡旋部分，可重现T_c以下的高斯临界线、T_c以上的涡旋增殖、关联长度的本质奇点以及Nelson-Kosterlitz跃变。对于玻色弦，振荡器模式辅以动量-缠绕标签可重现圆T对偶性、常环面背景下的Buscher变换、自对偶流代数增强以及一个T-fold模型。共同结论是，相同的局域神经采样器搭配不同的离散拓扑数据，可得到物理上不同的紧致理论。本报告基于arXiv:2604.02313。
英文摘要
Neural network field theory (NN-FT) defines a field theory by a network architecture together with a probability density on its latent variables. For compact theories the local Gaussian sector is only part of the story: one must also sum over discrete topological sectors. We review a mixed continuous/discrete latent-variable construction and apply it to two compact bosons. For the Berezinskii--Kosterlitz--Thouless transition, a random Fourier feature spin-wave sampler supplemented by an explicit Coulomb gas vortex sector reproduces the Gaussian critical line below $T_c$, vortex proliferation above $T_c$, the essential singularity of the correlation length, and the Nelson--Kosterlitz jump. For the bosonic string, oscillator modes augmented by momentum--winding labels reproduce circle T-duality, Buscher transformations on constant toroidal backgrounds, self-dual current algebra enhancement, and a toy T-fold. The common lesson is that the same local neural sampler, paired with different discrete topological data, yields physically distinct compact theories. These proceedings are based on arXiv:2604.02313.
Comments11 pages, for Proceedings of "DANGER: Data, Numbers, and Geometry'' workshop, Banff International Research Station, April 2026, based on arXiv:2604.02313
