发表机构
Physique Théorique et Mathématique and International Solvay Institutes, Université Libre de Bruxelles (ULB)（理论物理与数学系及国际索尔维研究所，布鲁塞尔自由大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过Picard-Lefschetz理论分析类时刘维尔理论零模扇区的积分围道，解释了DOZZ公式中振荡因子的起源，并揭示了不同β值下鞍点结构与Stokes现象的联系。
AI 中文摘要
类时刘维尔理论的二维球面配分函数允许两种互补的半经典描述，它们之间的关系呈现出一个非微扰之谜。围绕圆二球面鞍点的直接路径积分展开重现了解析延拓的Dorn-Otto-Zamolodchikov-Zamolodchikov (DOZZ)公式的微扰小-$\eta$展开（其中$\eta$为刘维尔耦合常数），而后者包含一个额外的振荡因子，暗示存在第二个鞍点贡献。我们利用Picard-Lefschetz理论在刘维尔零模扇区中研究该因子的起源。以由零模积分的逆伽马表示所启发的Hankel型围道作为输入，我们确定了其依赖于$\eta$的Lefschetz-thimble分解。对于$0<\eta<1$，该分解精确重现了解析延拓的DOZZ结果中所编码的鞍点结构。我们将积分围道识别为零模鞍点描述之间差异的来源。当$\eta$趋近于1时，零模鞍点移动到场空间的边界，而对于$\eta>1$，同一围道由单个贡献的thimble表示。我们进一步将这种Stokes重组与类时刘维尔理论的最新概率构造联系起来。
英文摘要
The two-sphere partition function of timelike Liouville theory admits two complementary semi-classical descriptions whose relation presents a non-perturbative puzzle. A direct path-integral expansion around the round two-sphere saddle reproduces the perturbative small-$β$ expansion, with $β$ the Liouville coupling, of the analytically continued Dorn-Otto-Zamolodchikov-Zamolodchikov (DOZZ) formula, while the latter contains an additional oscillatory factor suggestive of a second saddle contribution. We investigate the origin of this factor in the Liouville zero-mode sector using Picard-Lefschetz theory. Taking as input the Hankel-type contour motivated by the inverse-Gamma representation of the zero-mode integral, we determine its Lefschetz-thimble decomposition depending on $β$. For $0<β<1$, it precisely reproduces the saddle structure encoded in the analytically continued DOZZ result. We identify the integration contour as the source of the difference between the zero-mode saddle descriptions. As $β$ approaches $1$, the zero-mode saddle moves to the boundary of field space, while for $β>1$ the same contour is represented by a single contributing thimble. We further relate this Stokes reorganisation to recent probabilistic constructions of timelike Liouville theory.
Comments35 pages, 7 figures, two saddles
