将拓扑结构引入神经网络场论，揭示相变与对偶性现象
Topological Effects in Neural Network Field Theory
- 用离散参数标记拓扑量子数，构建拓扑化神经网络场论
- 复现贝雷津-科斯特里茨-索利相变及高温涡旋增殖现象
- 验证玻色弦T对偶性，包括自对偶半径下的代数增强
神经网络场论将场论表述为由网络架构和参数分布定义的场的统计系综。本文通过引入标记拓扑量子数的离散参数，将该构造扩展至拓扑情形。我们恢复了贝雷津-科斯特里茨-索利相变，包括自旋波临界线以及高温下涡旋的大量涌现。同时验证了玻色弦的T对偶性：在$S^1$上动量与绕数的交换不变性；在恒定环面背景上，σ模型耦合按Buscher规则变换；在自对偶半径处电流代数增强；以及非几何T折曲变换函数。
原文摘要 · Abstract (English)
Neural network field theory formulates field theory as a statistical ensemble of fields defined by a network architecture and a density on its parameters. We extend the construction to topological settings via the inclusion of discrete parameters that label the topological quantum number. We recover the Berezinskii--Kosterlitz--Thouless transition, including the spin-wave critical line and the proliferation of vortices at high temperatures. We also verify the T-duality of the bosonic string, showing invariance under the exchange of momentum and winding on $S^1$, the transformation of the sigma model couplings according to the Buscher rules on constant toroidal backgrounds, the enhancement of the current algebra at self-dual radius, and non-geometric T-fold transition functions.
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