发表机构
Indian Institute of Technology Madras; Centre for Operator Algebras, Geometry, Matter and Spacetime; Asia Pacific Center for Theoretical Physics (APCTP)（印度马德拉斯理工学院; 算子代数、几何、物质与时空中心; 亚太理论物理中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种系统方法，通过Shapovalov形式直接计算共形块并固定Fuchsian ODE的附加参数，从而数值与精确地求解Virasoro最小模型中的编织F矩阵，并猜想其元素平方位于有理数的分圆扩张中。
AI 中文摘要
我们发展了一种系统的方法来计算有理共形场论（RCFT）中四点函数的编织矩阵。该方法针对Virasoro最小模型的情形加以实现。共形块级数的前几项通过Shapovalov形式直接计算得到。对于三阶及更高阶的Fuchsian（BPZ）常微分方程（ODE），其附加参数由这一直接计算所固定。由此，我们绕过了从零态条件推导的过程，而该推导对于更高阶的零态会变得繁琐。编织F矩阵是ODE在两个奇点处的Frobenius解之间的连接矩阵。F矩阵首先通过数值方法计算，结果与使用Coulomb气体形式体系的Dotsenko–Fateev结果一致。我们发现，在基变换之后，F矩阵变为酉矩阵，这确定了三点结构常数（的乘积）。我们猜想，酉F矩阵的元素的平方位于有理数的分圆扩张中。这使我们能够将F矩阵的数值估计转换为精确结果。该形式体系通过Virasoro最小模型中的众多例子加以说明。我们还讨论了如何将这些方法推广到涉及扩展Virasoro对称性的对称性的情形，以及由全纯模自举程序产生的可行例子。
英文摘要
We develop a systematic method for computing the braiding matrices of four-point functions in RCFTs. This is implemented for the case of Virasoro minimal models. The first few terms of the conformal block series are directly computed through the Shapovalov form. The Fuchsian (BPZ) ODEs for order three and higher have accessory parameters that are fixed by this direct computation. We thus bypass the derivation from the null-state condition which gets tedious for higher-level null states. The braiding F-matrices are connection matrices between the Frobenius solutions at two singular points of the ODE. The F-matrices are computed numerically at first, agreeing with the results of Dotsenko--Fateev, which uses the Coulomb-gas formalism. We find that after a change of basis, the F-matrix is rendered unitary, which determines (products of) the three-point structure constants. We conjecture that the squares of the entries of the unitary F-matrix lie in a cyclotomic extension of the rational numbers. This enables us to convert our numerical estimates for the F-matrix into exact ones. The formalism is illustrated through numerous examples in Virasoro minimal models. We also discuss how these methods can be extended to cases involving symmetries that extend the Virasoro symmetry as well as for tenable examples that arise from the holomorphic modular bootstrap program.
Comments40 pages
