用因式分解方法研究超弦理论中动力学紧化的简单模型中的复Action问题
A Study of the Complex Action Problem in a Simple Model for Dynamical Compactification in Superstring Theory Using the Factorization Method
摘要：
IIB矩阵模型提出了一种通过十维旋转对称性$\textrm{SO}(10)$的自发破缺来动态产生四维时空的机制。使用高斯展开法（GEM）的计算支持这一猜想。我们研究了一个简单的$\textrm{SO}(4)$不变矩阵模型，使用蒙特卡罗模拟，我们确认其旋转对称性打破，显示低维配置占主导地位的路径积分。该模型具有很强的复杂作用量问题，通过对时空转动惯量张量的适当归一化本征值$\tildeλ_n$的态密度$ρ_n(x)$采用因式分解方法，使计算成为可能。我们研究了$ρ_n(x)$的因子化项的标度性质，发现它们与简单的标度参数一致。这些结果可用于有限尺寸标度外推和研究被相位大起伏遮蔽的位形空间区域。$\tildeλ_n$的计算值与GEM的计算值相当吻合，并提出了一种比较相应ansatze自由能的数值方法。
英文摘要：
The IIB matrix model proposes a mechanism for dynamically generating four dimensional space--time in string theory by spontaneous breaking of the ten dimensional rotational symmetry $\textrm{SO}(10)$. Calculations using the Gaussian expansion method (GEM) lend support to this conjecture. We study a simple $\textrm{SO}(4)$ invariant matrix model using Monte Carlo simulations and we confirm that its rotational symmetry breaks down, showing that lower dimensional configurations dominate the path integral. The model has a strong complex action problem and the calculations were made possible by the use of the factorization method on the density of states $ρ_n(x)$ of properly normalized eigenvalues $\tildeλ_n$ of the space--time moment of inertia tensor. We study scaling properties of the factorized terms of $ρ_n(x)$ and we find them in agreement with simple scaling arguments. These can be used in the finite size scaling extrapolation and in the study of the region of configuration space obscured by the large fluctuations of the phase. The computed values of $\tildeλ_n$ are in reasonable agreement with GEM calculations and a numerical method for comparing the free energy of the corresponding ansatze is proposed and tested.
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