非單位共形自舉的數值最佳化方法

Abstract

在本篇文章中，我們提出一種數值方法用於共形自舉。首先，我們利用交叉對稱方程式計算給定的共形譜中的算符乘積展開係數，這些係數在不同交叉比的取樣下所形成的統計分布提供了一個衡量交叉對稱滿足程度的指標。基於此統計量，我們建構一個獎勵函數並通過遺傳演算法在給定的範圍裡搜尋能夠最大化該獎勵的共形譜。重要的是，此方法並不依賴單位性的假設。我們在中央荷小於 1的共形模型 (最小模型) 上驗證了此方法的有效性，所有找到的最佳解皆能對應到已知的最小模型。本方法本質上是透過線性變數的統計行為，來求解具非線性限制的方程組，其概念可推廣至更廣泛的自舉問題。

In this thesis, we introduce a numerical approach to bootstrapping the spectrum of general conformal field theories (CFT). Starting with a given spectrum we use crossing equations to compute the OPE coefficients. The statistical distribution of the latter, from sampling over cross-ratios, provides a robust measure of how close crossing symmetry is being solved. We define a reward function based on the statistics and utilize genetic algorithm to search for spectrum that optimize the reward. Importantly, the setup does not require unitarity. We demonstrate the utility of this approach on Virasoro blocks with c < 1, where all optimized points can be identified with known minimal models. This method of solving constrained equations of non-linear variables utilizing the statistics of linear variables can be applied to broader bootstrap problems.