宇宙学、簇代数与 $\oldsymbol{u}$
Cosmology, Cluster Algebras, and $\boldsymbol{u}$
- Kavli Institute for Cosmological Physics, Department of Astronomy and Astrophysics, The University of Chicago（芝加哥大学天文学与天体物理学系卡弗里宇宙学物理研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文利用 $u$-变量建立德西特宇宙学运动学与簇代数的映射，发现链图对应 $A_{2n-2}$ 型、循环图对应 $B_{2n-1}$ 型，并借助簇相容性引导符号，揭示波函数与相关函数在链情形共享结构而循环情形不同。
AI中文摘要：
我们探索了德西特空间中共形耦合标量场的波函数和相关函数中存在的簇代数结构。为了将宇宙学运动学与簇变量联系起来，我们利用了在两边都出现的所谓 $u$-变量。由于 $u$-变量参数化了一个刚性空间，很自然地可以识别那些从宇宙学图出现的变量和那些从簇代数产生的变量。这提供了运动学变量和簇变量之间的映射。对于 $n$-位点链式费曼图，这给出了与 $A_{2n-2}$ 型簇代数的坐标不变关系，而对于 $n$-位点循环图动量积分，相关的簇代数是 $B_{2n-1}$，正如先前所发现的。由于运动学变量被映射到簇变量的非线性组合，簇相容性对可能的符号条目施加了强条件。我们利用这种相容性概念来引导符号。在链式情形中，宇宙学波函数和低点函数的乘积都满足簇相容性，而相关函数是特定的组合，因此波函数和相关函数共享相同的簇结构。然而，在循环情形中，波函数似乎被簇结构唯一地选择，而相关函数不显示与波函数相同的相容性。
英文摘要：
We explore the cluster algebra structures present in the wavefunction and correlators of conformally coupled scalar fields in de Sitter space. To connect cosmological kinematics to cluster variables, we utilize the so-called $u$-variables that appear on both sides. Since $u$-variables parameterize a rigid space, it is natural to identify the ones that appear from cosmological graphs and those that arise from cluster algebras. This provides a mapping between kinematic variables and cluster variables. For $n$-site chain Feynman graphs, this gives a coordinate-invariant relation to $A_{2n-2}$-type cluster algebras, while in the case of $n$-site cycle graph momentum integrands, the relevant cluster algebra is $B_{2n-1}$, as found previously. Since kinematic variables are mapped to nonlinear combinations of cluster variables, cluster compatibility imposes strong conditions on possible symbol entries. We use this notion of compatibility to bootstrap the symbol. In the chain case, both the cosmological wavefunction and products of lower-point functions satisfy cluster compatibility, with the correlator a particular combination, so that the wavefunction and correlator share the same cluster structure. In the cycle case, however, the wavefunction appears to be uniquely selected by the cluster structure, and the correlator does not display the same compatibility as the wavefunction.
