用机器学习优化方法在无已知例子的范围内发现新二维共形场论。
Descending into the Modular Bootstrap
- 将模不变性转为损失函数，用新优化器搜索可能的算子谱。
- 在中心荷1到8/7间构造出候选理论，支持存在连续解空间。
- 发现c=1附近能隙约束比已有结果更严格，适合理论物理研究者。
本文通过类机器学习的优化方法，高效搜索二维共形场论（2d CFT）的数值解，以探索其理论景观。2d CFT的环面划分函数由初级算子谱和中心荷 $c>1$ 的Virasoro代数决定。我们将模不变性要求转化为损失函数，并通过最小化该函数识别可能的初级谱。提出两项关键技术：一是估算截断至低维算子时的不确定性；二是采用基于奇异值的新优化器Sven，相比梯度下降更擅长处理损失函数的层级结构。我们数值构建了中心荷在1到$\frac{8}{7}$之间的候选截断CFT划分函数，此区间此前无已知例子，且证据表明这些候选可能来自连续的模自洽解空间。此外，还提供了更强的谱隙约束证据，其在 $c = 1$ 附近超越现有界限 $Δ_{\rm gap} \le \frac{c}{6} + \frac{1}{3}$。
原文摘要 · Abstract (English)
In this paper, we attempt to explore the landscape of two-dimensional conformal field theories (2d CFTs) by efficiently searching for numerical solutions to the modular bootstrap equation using machine-learning-style optimization. The torus partition function of a 2d CFT is fixed by the spectrum of its primary operators and its chiral algebra, which we take to be the Virasoro algebra with $c>1$. We translate the requirement that this partition function is modular invariant into a loss function, which we then minimize to identify possible primary spectra. Our approach involves two technical innovations that facilitate finding reliable candidate CFTs. The first is a strategy to estimate the uncertainty associated with truncating the spectrum to the lowest dimension operators. The second is the use of a new singular-value-based optimizer (Sven) that is more effective than gradient descent at navigating the hierarchical structure of the loss landscape. We numerically construct candidate truncated CFT partition functions with central charges between 1 and $\frac{8}{7}$, a range devoid of known examples, and argue that these candidates likely come from a continuous space of modular bootstrap solutions. We also provide evidence for a more stringent constraint on the spectral gap near $c = 1$ than the existing bound of $Δ_{\rm gap} \le \frac{c}{6} + \frac{1}{3}$.
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