[论文解读] Non-lattice simulation of supersymmetric gauge theories as a probe to quantum black holes and strings

Jun Nishimura|arXiv (Cornell University)|Dec 1, 2009

Black Holes and Theoretical Physics被引用 9

一句话总结

本文提出了一种用于一维16个超荷超对称规范场论的非晶格模拟方法，采用矩阵模型正则化来研究强耦合的大N杨-米尔斯理论。该方法通过从规范理论侧重现黑洞热力学和史瓦西半径，证实了规范-重力对偶性，并通过大N约化将该方法扩展至在$R\times S^3$上模拟${\cal N}=4$超杨-米尔斯理论，无需微调，验证了非重整化定理，并为量子黑洞和弦提供了第一性原理探测。

ABSTRACT

In the past decade we have witnessed remarkable developments in the gauge-gravity duality, which suggested a new approach to superstring theory and quantum space-time. In this context it is important to study supersymmetric large-N gauge theories in the strongly coupled regime. I will summarize the results and insights obtained so far by non-lattice simulations. A simple example of the gauge-gravity duality is the one between 1d U(N) gauge theory with 16 supercharges and the so-called black 0-brane solution in type IIA supergravity. In order for this duality to be valid, one has to take the 't Hooft large-N limit and to take the strong coupling limit on the gauge theory side. The gauge theory can be regularized by fixing the gauge completely thanks to one dimension, and by introducing a Fourier mode cutoff. One can then use the standard RHMC algorithm to simulate the system. The energy calculated as a function of the temperature was compared with the results obtained from the gravity side based on the black hole thermodynamics. This confirmed the gauge-gravity duality with high accuracy and provided the microscopic origin of the black hole thermodynamics. From the calculation of the Wilson loop, one obtains the Schwarzschild radius of the dual geometry. One can actually use the present 1d model with supersymmetric mass deformation to study \\mathcal{N}=4 super Yang-Mills theory on R \ imes S^3 based on a novel large-N reduction, which generalizes the original idea of Eguchi and Kawai. It is remarkable that we can now simulate the 4d superconformal field theory, which appears in the most typical case of the gauge-gravity duality known as the AdS/CFT correspondence. In particular, no fine-tuning is required unlike previous proposals based on the lattice regularization.

研究动机与目标

• 开发一种用于模拟强耦合区域超对称规范场论的非晶格方法，避免晶格正则化中固有的超对称性破缺。
• 通过将非晶格模拟结果与超引力预测进行比较，测试规范-重力对偶性，特别是关于黑洞热力学和几何结构的预测。
• 利用大N约化技术将该方法扩展至高维超共形场论，实现在$R\times S^3$上对4d ${\cal N}=4$超杨-米尔斯理论的模拟，且无需微调参数。
• 探究黑洞熵的微观起源，并在AdS/CFT对应框架下测试非重整化定理。

提出的方法

• 通过完整规范固定和傅里叶模态截断对一维U(N)超对称杨-米尔斯理论（16个超荷）进行正则化，实现非晶格模拟。
• 在大N极限下，应用有理混合蒙特卡罗（RHMC）算法对矩阵模型进行随机模拟。
• 使用大N约化技术，推广埃古奇-卡瓦约约化，将$R\times S^3$上的4d ${\cal N}=4$ SYM映射为有限矩阵尺寸的一维矩阵模型。
• 在约化模型中计算标量初级算符和威尔逊环的关联函数，以提取与AdS中引力对应的可观测量。
• 将蒙特卡罗结果与超引力和弱耦合极限下的解析预测进行比较，以检验对偶性和非重整化定理的有效性。
• 使用无量纲时间$\mu t$和't Hooft耦合$\lambda_{\rm SYM}$分析标度行为，并研究向无限N和无限温度极限的收敛性。

实验结果

研究问题

• RQ1 一维超对称杨-米尔斯理论的非晶格模拟能否重现IIA型超引力中黑洞0-膜的热力学性质？
• RQ2 在一维矩阵模型中计算的威尔逊环是否能正确重现对应黑洞几何的史瓦西半径？
• RQ3 大N约化技术能否在不微调参数的情况下用于模拟$R\times S^3$上的4d ${\cal N}=4$超杨-米尔斯理论？
• RQ4 约化模型中标量初级算符的关联函数是否与AdS/CFT在强耦合下预测的非重整化定理一致？
• RQ5 约化模型中的矩形威尔逊环在大T极限下是否表现出符合共形对称性的线性增长？

主要发现

• 一维SYM模型中的能量-温度关系与对偶黑洞0-膜的黑洞热力学高度一致，证实了规范-重力对偶性。
• 威尔逊环计算结果与对偶几何的史瓦西半径一致，为规范理论可观测量与黑洞几何之间建立了直接联系。
• 对$\lambda_{\rm SYM}$的两个不同取值（0.24和6.4）进行的约化模型蒙特卡罗模拟，得到的标量初级算符两点函数几乎完全相同，表明该模型中非重整化定理成立。
• 约化模型中两点函数的结果与弱耦合极限下的解析结果吻合良好，并在$k,n,\nu \to \infty$极限下趋近于${\cal N}=4$ U(∞) SYM的预期结果。
• 约化模型中的矩形威尔逊环随时间$T$表现出线性增长，与共形预测$\langle W(T\times R)\rangle \propto \exp(\gamma T/R)$一致，尽管$\gamma$的确切值尚未计算。
• 结果表明，约化模型中剩余的一半超对称性在无限矩阵尺寸极限下得以恢复，意味着无需微调即可恢复完整的${\cal N}=4$超对称性。