发表机构
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过复鞍点路径积分验证实射影平面上刘维尔理论的精确单点函数，解决了无实经典解的问题，并展示了复鞍点在类时刘维尔理论和二维德西特引力中的普遍性。
AI 中文摘要
实射影平面上刘维尔理论的精确单点函数早在很久以前就从自举方法推导出来，但从未通过半经典路径积分计算进行验证。这一检验并不像人们可能预期的那样直接。在实射影平面上，每个常曲率度量都是正曲率的，而经典刘维尔理论产生常负曲率的度量，因此不存在实的经典解。相反，正如我们所示，半经典路径积分接收到来自无穷多个具有负定度量的复鞍点的贡献。一旦选择积分围道使得路径积分收敛，这些鞍点就能重现精确的单点函数。复鞍点以同样的方式出现在类时刘维尔理论和二维德西特引力中，而这里处理的例子提供了一个难得的机会，可以对照已知的精确答案来检验其应用。
英文摘要
The exact one-point function of Liouville theory on the real projective plane was derived long ago from the bootstrap, yet it has never been verified against a semiclassical path-integral computation. The check is less straightforward than one might expect. On the real projective plane every constant-curvature metric is positively curved, whereas classical Liouville theory produces metrics of constant negative curvature, so no real classical solution exists. Instead, as we show, the semiclassical path integral receives contributions from infinitely many complex saddles with a negative-definite metric. Once the integration contour is chosen so that the path integral converges, these saddles reproduce the exact one-point function. Complex saddles arise in the same way in timelike Liouville theory and in two-dimensional de~Sitter gravity, and the example treated here offers a rare opportunity to test their use against a known exact answer.
Comments34 pages
