We extend the study of banana diagrams in coordinate representation to the case of curved space-times. If the space is harmonic, the Green functions continue to depend on a single variable – the geodesic distance. But now this dependence can be somewhat non-trivial. We demonstrate that, like in the flat case, the coordinate differential equations for powers of Green functions can still be expressed as determinants of certain operators. Therefore, not-surprisingly, the coordinate equations remain straightforward – while their reformulation in terms of momentum integrals and Picard-Fuchs equations can seem problematic. However we show that the Feynman parameter representation can also be generalized, at least for banana diagrams in simple harmonic spaces, so that the Picard-Fuchs equations retain their Euclidean form with just a minor modification. A separate story is the transfer to the case when the Green function essentially depends on several rather than a single argument. In this case, we provide just one example, that the equations are still there, but conceptual issues in the more general case will be discussed elsewhere.

中文翻译：

---

香蕉图作为测地线距离的函数


我们将坐标表示中的香蕉图的研究扩展到弯曲时空的情况。如果空间是谐波，则 Green 函数继续依赖于单个变量 - 测地线距离。但现在，这种依赖性在某种程度上可能并非微不足道。我们证明，与在平坦情况下一样，Green 函数幂的坐标微分方程仍然可以表示为某些运算符的行列式。因此，毫不奇怪，坐标方程仍然很简单——而它们根据动量积分和 Picard-Fuchs 方程的重新表述似乎是有问题的。然而，我们表明，费曼参数表示也可以推广，至少对于简谐空间中的香蕉图，因此 Picard-Fuchs 方程只需稍作修改即可保持其欧几里得形式。一个单独的故事是 Green 函数基本上依赖于多个参数而不是单个参数的情况。在这种情况下，我们只提供一个例子，方程仍然存在，但更一般情况下的概念问题将在其他地方讨论。