AI 中文总结
研究AdS$_{d + 1}$中服从诺伊曼边界条件的标量的单圈配分函数，通过解析延拓计算，对比狄利克雷边界条件情况，研究相关函数长程行为来探索不同维度标量场论在零温和有限温度下的相。
AI 中文摘要
我们对AdS$_{d + 1}$中服从诺伊曼边界条件的标量的单圈配分函数进行分析，并探索几个维度的标量场论在零温和有限温度下的相。配分函数计算涉及从对应狄利克雷边界条件的$\Delta_+$到对应诺伊曼边界条件的$\Delta_-$的解析延拓。我们表明这可通过拉普拉斯算子本征值（$\lambda$）积分轮廓的变形来实现，与狄利克雷边界条件在[arXiv:2201.09043]中的实积分相对。我们进一步将这些相与服从狄利克雷边界条件的标量情况出现的相进行对比，并通过研究相关函数的长程行为来证实这些相的出现。
英文摘要
We present an analysis of one-loop partition functions for scalars in AdS$_{d+1}$ obeying the Neumann boundary condition and explore phases of scalar field theories in several dimensions at zero and finite temperature. The partition function computation involves an analytic continuation from $Δ_+$ corresponding to the Dirichlet boundary condition to $Δ_-$ corresponding to Neumann boundary condition. We show that this can be implemented by deformations of the contour for integral over eigenvalue ($λ$) of the Laplace operator as compared to the integral over ${\mathbb R}$ in [arXiv:2201.09043] for the Dirichlet boundary condition. We further contrast these phases with those appearing for the case of scalars obeying the Dirichlet boundary condition and corroborate the occurrence of these phases by studying the long range behaviour of correlators.
Comments35 + 9 pages ; references added, factors of $L^2$ corrected in intermediate steps, final results remain unchanged and minor clarifications added
