In this note we compute the 1-loop partition function of spin-$s$ fields on Euclidean de Sitter space $S^{2n+1}$ using the quasinormal mode method. Instead of computing the quasinormal mode frequencies from scratch, we use the analytic continuation prescription $L_{\text{AdS}}\to iL_{\text{dS}}$, appearing in the dS/CFT correspondence, and Wick rotate the normal mode frequencies of fields on thermal $\text{AdS}_{2n+1}$ into the quasinormal mode frequencies of fields on de Sitter space. We compare the quasinormal mode and heat kernel methods of calculating 1-loop determinants, finding exact agreement, and furthermore explicitly relate these methods via a sum over the conformal dimension. We discuss how the Wick rotation of normal modes on thermal $\text{AdS}_{2n+1}$ can be generalized to calculating 1-loop partition functions on the thermal spherical quotients $S^{2n+1}/\mathbb{Z}_{p}$. We further show that the quasinormal mode frequencies encode the group theoretic structure of the spherical spacetimes in question, analogous to the recent analysis made for thermal AdS in (1910.07607) and (1910.11913).

中文翻译：

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在de Sitter量子引力中通过拟正规模式方法实现更高的自旋划分功能

在本说明中，我们使用准法则模式方法计算了欧几里得德西特空间$ S ^ {2n + 1} $上spin- $ s $字段的1-loop分区函数。代替从零开始计算准标准频率，我们使用解析连续处方$ L _ {\ text {AdS}} \至iL _ {\ text {dS}} $，出现在dS / CFT对应关系中，而Wick旋转法线$ \ text {AdS} _ {2n + 1} $上的热场的模态频率变为de Sitter空间上的准标准模态频率。我们比较了拟正态模态和热核方法计算一环行列式的方法，找到了精确的一致性，并通过保形维上的总和明确地关联了这些方法。我们讨论如何将热态\\ text {AdS} _ {2n + 1} $上的正常模式的Wick旋转推广到计算热球商$ S ^ {2n + 1} / \ mathbb上的1环分配函数{Z} _ {p} $。我们进一步表明，准法向模式频率编码了所讨论的球形时空的群理论结构，类似于最近对（1910.07607）和（1910.11913）中的热AdS进行的分析。