QFT作为一组常微分方程：高维情况
QFT as a set of ODEs: higher dimensions
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中文总结 AI 辅助
将AdS₂中QFT数据在相关变形下的流动方程推广到AdS₃和AdS₄，展示了边界算子的合并湮灭和能级排斥机制，并提出用交叉方程替代OPE系数的ODE以及Padé近似改善收敛性，为强耦合QFT研究奠定基础。
中文摘要 AI 辅助
反德西特空间（AdS）中量子场论（QFT）的局域算符关联函数完全由QFT数据决定：边界算子的标度维度$\Delta_i$和OPE系数$C_{ijk}$，以及描述体场如何分解为边界算子的体-边界（BOE）系数$b^{\hat\Phi}_i$。本文将最初针对AdS$_2$推导的、控制体相关变形下QFT数据变化的常微分方程（ODE）推广到AdS$_3$和AdS$_4$情形。我们证明这些流动方程自然地捕捉了边界算子在达到边缘性时的合并湮灭机制，以及不同$\Delta_i$相互接近时的能级排斥。此外，我们讨论了该框架的实际实现：提出用交叉方程替代OPE系数的ODE以提高效率，并观察到Padé近似显著改善了边界算子求和的收敛性，至少在自由理论中如此。总之，这些进展为未来将流动方程应用于AdS中强耦合QFT及其平空间极限的研究奠定了基础。
英文摘要
Correlation functions of local operators in Quantum Field Theory (QFT) in Anti-de Sitter space (AdS) are completely fixed by the QFT data: the set of scaling dimensions $Δ_i$ and OPE coefficients $C_{ijk}$ of the boundary operators, and the bulk-boundary (BOE) coefficients $b^{\hatΦ}_i$ encoding how bulk fields decompose into boundary operators. In this work, we generalize the ordinary differential equations (ODEs) that govern the variation of the QFT data under a bulk relevant deformation, originally derived for AdS$_2$ \cite{Loparco:2026fki}, to the cases of AdS$_3$ and AdS$_4$. We demonstrate that these flow equations natively capture the mechanism of merger-annihilation when a boundary operator hits marginality, as well as level repulsion when different $Δ_i$'s approach each other. Furthermore, we address the practical implementation of the framework: we propose substituting the ODE for the OPE coefficients with the crossing equation for greater efficiency, and we observe that Padé approximants dramatically improve the convergence of the sums over boundary operators, at least in free theories. Altogether, these advances lay the groundwork for the future application of the flow equations to the study of strongly coupled QFTs in AdS and their flat space limits.
