Scalar curvature, spectral zeta functions and local geometric invariants for noncommutative spaces

非交换空间的标量曲率、谱 zeta 函数和局部几何不变量

• 批准号: RGPIN-2014-04087
• 负责人: Khalkhali, Masoud
• 金额: $ 1.68万
• 依托单位: University of Western Ontario
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=610826
• 关键词: Scalar; curvature; spectral; zeta; functions

Geometry is about measurement of shapes, and spaces in general. Through classical differential geometry we have learned how to measure distances and volumes, as well as the curvature of a given space in all dimensions. How to define and compute the curvature of a noncommutative space? A noncommutative space is a much more complicated modern analogue of a classical space. Chaotic and fuzzy character of these new types of spaces, specially lack of classical points, renders almost all of the classical methods useless. Ideas from spectral geometry and quantum mechanics, namely information about a space encoded in the spectrum of its natural geometric operators like Dirac and Laplacian gives a clue as to how to proceed in the NC case. The celebrated Weyl's law on the asymptotic distribution of eigenvalues of the Hodge-de Rham Laplacian of a closed Riemannian manifold in terms of its volume is the first result of this kind. In general the short time asymptotic expansion of the trace of the heat kernel gives an infinite sequence of spectral invariants. Borrowing words of Marc Kac: one can hear the dimension, volume and scalar curvature. The essence of this situation is axiomatized by Alain Connes in noncommutative geometry (NCG) under the concept of spectral triple. With enough regularity condition spectral triples can be thought of as noncommutative spin Riemannian manifolds. Before 2010, no real computation of curvature in a curved NC space was known or seemed feasible. For the first time after more than 30 years into NCG, in the past 4 years we (and independently and simulataneously Connes and Moscovici) were able to obtain a formula for the curvature of a curved NC 2-d torus. We (Fathizadeh-Khalkhali) have also obtained a formula for the scalar curvature of a NC curved 4-d torus. These are formidable formulas which in no way can be obtained by deforming the classical curvature formulas. This is achieved by evaluating the value of the (analytic continuation of the) spectral zeta functional \zeta_a(s) := Trace(a \Delta-s) at s = 0. A new purely noncommutative feature here is the appearance of the modular automorphism group from the theory of type III von Neumann factors and quantum statistical mechanics in the final formula for curvature. A byproduct is a Gauss-Bonnet theorem for NC 2-d torus. Other tools like Connes' trace theorem, and a noncommutative Wodzicki residue has been also obtained by us. A totally fresh and unchartered territory is now opened with so many open and interesting fundamental problems waiting to be studied. I am planning to build upon the breakthroughs I had in the last 4 years and continue my research in understanding the curved geometry of noncommutative spaces. This includes: Constructing new noncommutative Riemannian manifolds, finding the NC Gauss-Bonnet density in dimension 4, computing the scalar curvature and Gauss-Bonnet theorem of noncommutative toroidal orbifolds, extending my Riemann-Roch theorem to all holomorphic line bundles on noncommutative 2-torus, verifying Chamseddine-Connes conjectures for spectral action for Robertson-Walker metrics, proving the conformal invariance of the eta invariant for noncommutative 3-torus, conceptual understanding of our curvature formula (it is extremely important to understand at a more conceptual level the amazing cancellations that occur in our calculations with noncommutative pseudodifferential symbols, and why at the end thousands of terms cancel), establishing higher order corrections to our Weyl's law for NC curved tori (analogue of Hormander's celebrated theorem), finding new examples of noncommutative Einstein manifolds, study of quantum Yang-Mills theory on noncommutative 4-torus.

几何是关于形状和空间的测量。通过经典的微分几何形状，我们学会了如何测量距离和体积，以及在所有维度中给定空间的曲率。如何定义和计算非共同空间的曲率？非交流空间是古典空间的现代类似物。这些新类型的空间的混乱和模糊特征，特别缺乏经典的观点，几乎所有经典方法都毫无用处。来自光谱几何和量子力学的想法，即有关在其自然几何算子（如Dirac和Laplacian）光谱中编码的空间的信息，这给了有关如何在NC情况下进行的线索。众所周知的关于hodge-de rham laplacian特征值的渐近分布的魏尔定律，就其数量而言，封闭的riemannian歧管是这种著名的律法。通常，热核痕迹的短时间渐近膨胀给出了光谱不变的无限序列。借用MARC KAC的词：可以听到维度，音量和标量曲率。 这种情况的本质是由Alain Connes在光谱三重概念下的非共同几何形状（NCG）中的公理。有足够的规律性条件光谱三元组可以被认为是非交通性的旋转riemannian歧管。在2010年之前，在弯曲的NC空间中没有真正的曲率计算，或者似乎是可行的。 进入NCG 30多年后，在过去的4年中，第一次（以及独立和模拟的Connes and Moscovici）能够获得弯曲NC曲率的公式 二维圆环。我们（Fathizadeh-khalkhali）也获得了NC弯曲的4-D圆环标量曲率的公式。这些是强大的公式，无法通过变形经典的曲率公式来获得。这是通过评估（分析延续）的价值来实现的 光谱zeta函数\ zeta_a（s）：= s = 0的跟踪（a \ delta-s）。 最终曲率公式中的III型von Neumann因子和量子统计力学理论的模块化自动形态群。副产品是用于NC 2-D圆环的高斯河网定理。我们还获得了其他工具，例如Connes的Trace Theorem和非交通性Wodzicki残留物。现在，一个完全新鲜且未经宪章的领土开放，有许多开放且有趣的基本问题等待着研究。 我打算基于过去四年的突破，并继续研究，以了解非交通空间的弯曲几何形状。这包括： 构建新的非交通性riemannian流形，在维度4中找到NC高斯 - 骨密度，计算非交通性螺旋孔的标量曲率和高斯 - 骨网定理，从罗伯逊步行者指标的光谱动作，证明了ETA不变的3道毛的不变性的形式不变性，对我们的曲率公式的概念性理解（在更概念性的层面上了解令人惊奇的取消是在我们的计算中以非交换性的命令和较高的命令纠正的范围，并且在较高的范围内发生了较高的命令，以及何时逐渐纠正了几千句话。 NC Curved Tori（霍曼德著名定理的类似物）， 找到非交通性爱因斯坦歧管的新例子，对非交通性4道的量子阳米尔斯理论的研究。