论文标题
liouville共形场理论的随机量化
Stochastic quantization of Liouville conformal field theory
论文作者
论文摘要
我们研究了一个非线性随机热方程，该方程是由封闭表面上的时空白噪声强迫的，非线性$ e^{βU} $。该方程对应于Liouville量子重力（LQG）度量的随机量化。 （i）我们首先在$ l^2 $制度$ 0 <β<\ sqrt {2} $中首先重新审视liouville共形田间理论（LCFT）中LQG度量的构建。这在这种制度中统一了David-Kupiainen-Rhodes-Vargas（2016），David-Rhodes-Vargas（2016）和Guillarmou-rhodes-Vargas（2019）的方法，这些方法分别用属于属0、1和$> 1 $的封闭式案例。此外，我们的论点表明，该度量与大型平滑近似类别的近似过程无关。 （ii）我们证明了抛物线随机动力学的全局良好，以及在这种随机流下的度量的不变性。特别是，我们的结果改善了Garban（2020）在带有规范指标的球体和圆环的情况下获得的先前结果，并且对于具有较高属的封闭表面而言是新的。
We study a nonlinear stochastic heat equation forced by a space-time white noise on closed surfaces, with nonlinearity $e^{βu}$. This equation corresponds to the stochastic quantization of the Liouville quantum gravity (LQG) measure. (i) We first revisit the construction of the LQG measure in Liouville conformal field theory (LCFT) in the $L^2$ regime $0<β<\sqrt{2}$. This uniformizes in this regime the approaches of David-Kupiainen-Rhodes-Vargas (2016), David-Rhodes-Vargas (2016) and Guillarmou-Rhodes-Vargas (2019) which treated the case of a closed surface with genus 0, 1 and $> 1$ respectively. Moreover, our argument shows that this measure is independent of the approximation procedure for a large class of smooth approximations. (ii) We prove almost sure global well-posedness of the parabolic stochastic dynamics, and invariance of the measure under this stochastic flow. In particular, our results improve previous results obtained by Garban (2020) in the cases of the sphere and the torus with their canonical metric, and are new in the case of closed surfaces with higher genus.
