标题： 在$4d$ Einstein-Gauss-Bonnet (EGB) 和 Lovelock 理论中的三波和四波相互作用 显示英文标题

标题： Three-Wave and Four-Wave Interactions in the $4d$ Einstein Gauss-Bonnet (EGB) and Lovelock Theories

摘要： 我们推导了经典顶点满足的共形约束条件，这些顶点属于（爱因斯坦）高斯-博内理论的经典形式，在平坦空间中，一般维度下以及在 4 维高斯-博内 (EGB) 理论的 $d=4$ 情况下。 在 $4d$ 的 EGB 理论中，它们通过欧拉-庞加莱密度积分的奇异极限获得。 我们的分析利用了该理论与共形异常作用之间的关系，这使得我们可以揭示高斯-博内顶点在三次和四次水平上的一些有趣特征。 如果我们引入度规的共形分解，则得到的理论可以以两种不同的版本来表述，这两种版本依赖于正则化方式：局部版本是四次的标量场理论，非局部版本具有二次标量场。 非局部版本通过在执行奇异 $d\to 4$ 极限之前对高斯-博内密度进行有限重新定义，并加入一个 $(d-4)R^2$ 校正来获得。 在理论的局部版本中，我们展示了度规和标量场的独立动力学如何通过经典的迹恒等式交织在一起。 可以通过直接使用理论的 4 维 EGB 非局部版本，以一种非平凡的方式组织三重引力波相互作用。 这是可能的，因为这种表述的一致性——仅限于 3 点函数——直接继承自共形异常 (Riegert) 作用。 顶点所满足的约束条件是经典的、分层的沃德恒等式。 在四次水平上，类似的关联关系是通过对共形异常作用的 4T 相关函数反演项的分析借用而得出的，这是基于微扰展开定义的。 对于 $d\neq 4$，这些约束条件同样适用于洛夫洛克作用。 它们可以扩展到这类理论中的更高阶拓扑不变量。 显示英文摘要

摘要： We derive the conformal constraints satisfied by classical vertices of a (Einstein) Gauss-Bonnet theory around flat space, in general dimensions and at $d=4$ (4d EGB). In $4d$ EGB they are obtained by a singular limit of the integral of the Euler-Poincar\`e density. Our analysis exploits the relation between this theory and the conformal anomaly action, which allows to uncover some interesting features of the GB vertex at cubic and quartic level. If we introduce a conformal decomposition of the metric, the resulting theory can be formulated in two different versions, which are regularization dependent, a local one which is quartic in the dilaton field, and a nonlocal one, with a quadratic dilaton. The nonlocal version is derived by a finite redefinition of the GB density with the inclusion of a $(d-4)R^2$ correction, before performing the singular $d\to 4$ limit. In the local version of the theory, we show how the independent dynamics of the metric and of the dilaton are interwinded by a classical trace identity. Three-gravitational wave interactions can be organised in a nontrivial way by using directly the nonlocal 4d EGB version of the theory. This is possible thanks to the consistency of such formulation - only up to 3-point functions - directly inherited from the conformal anomaly (Riegert) action. The constraints satisfied by the vertices are classical, hierarchical Ward identities. At quartic level, similar relations are derived, borrowing from the analysis of the counterterms of the 4T correlators of the conformal anomaly action, as defined by a perturbative expansion. For $d\neq 4$ these constraints hold also for Lovelock actions. They can be extended to higher order topological invariants in such class of theories.