提出新型神经网络架构，实现二维共形对称性与伏拉索罗代数。
Virasoro Symmetry in Neural Network Field Theories
- 设计旋转不变谱先验，强制局部共形对称性
- 数值验证中心荷为0.9958±0.0196，接近理论值1
- 首次实现超伏拉索罗代数与共形边界条件
神经网络场论（NN-FT）通常描述无局域应力-能量张量的广义自由场，阻碍了二维中伏拉索罗对称性的实现。本文提出「对数核」（Log-Kernel, LK）架构，通过特定旋转不变谱先验 $p(k) /propto |k|^{-2}$ 强制局部共形对称性。我们从神经网络系综统计中解析推导出伏拉索罗代数的出现。数值模拟验证该构造：实测中心荷 $c_{exp} = 0.9958 \± 0.0196$（理论值 $c=1$），并确认顶点算符的标度维度。此外，我们证明有限宽度修正产生 $1/N$ 量级的相互作用。最后，将框架扩展至费米子与边界条件，实现超伏拉索罗代数。通过测量超电流关联函数，以96%精度验证 $\ ext{N}=1$ 超伏拉索罗代数；在上半平面实现共形边界条件，边界费米子与玻色子传播子符合率达99%。
原文摘要 · Abstract (English)
Neural Network Field Theories (NN-FTs) typically describe Generalized Free Fields that lack a local stress-energy tensor in two dimensions, obstructing the realization of Virasoro symmetry. We present the ``Log-Kernel'' (LK) architecture, which enforces local conformal symmetry via a specific rotation-invariant spectral prior $p(k) \propto |k|^{-2}$. We analytically derive the emergence of the Virasoro algebra from the statistics of the neural ensemble. We validate this construction through numerical simulation, computing the central charge $c_{exp} = 0.9958 \pm 0.0196$ (theoretical $c=1$) and confirming the scaling dimensions of vertex operators. Furthermore, we demonstrate that finite-width corrections generate interactions scaling as $1/N$. Finally, we extend the framework to include fermions and boundary conditions, realizing the super-Virasoro algebra. We verify the $\mathcal{N}=1$ super-Virasoro algebra by measuring the supercurrent correlator to $96\%$ accuracy. We further demonstrate conformal boundary conditions on the upper half-plane, achieving 99\% agreement for boundary fermion and boson propagators.
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