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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=587596
• 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.

几何是关于一般形状和空间的测量。通过经典的微分几何，我们学会了如何测量距离和体积，以及给定空间在所有维度上的曲率。如何定义和计算非交换空间的曲率？非交换空间是经典空间的更复杂的现代类似物。这些新型空间的混乱和模糊特性，特别是缺乏经典点，使得几乎所有经典方法都失去了作用。来自谱几何和量子力学的想法，即在其自然几何算子（如狄拉克和拉普拉斯）的谱中编码的空间信息，为如何在 NC 情况下进行提供了线索。 著名的关于黎曼闭流形的 Hodge-de Rham 拉普拉斯算子的特征值渐近分布的外尔定律（就其体积而言）是此类结果的第一个结果。一般来说，热核迹线的短时渐近展开给出了谱不变量的无限序列。借用 Marc Kac 的话：我们可以听到尺寸、体积和标量曲率。 这种情况的本质是由 Alain Connes 在非交换几何 (NCG) 中根据谱三元组的概念公理化的。在足够的规律性条件下，谱三元组可以被认为是非交换自旋黎曼流形。 2010 年之前，弯曲 NC 空间中的曲率计算尚不为人所知或似乎可行。 进入 NCG 30 多年后，在过去的 4 年里，我们（以及 Connes 和 Moscovici 同时独立地）首次能够获得弯曲 NC 曲率的公式 二维环面。 我们（Fathizadeh-Khalkhali）还获得了 NC 弯曲 4 维环面的标量曲率公式。这些都是令人敬畏的公式，不可能通过变形经典曲率公式来获得。 这是通过评估（的分析延续）的值来实现的 谱 zeta 泛函 \zeta_a(s) := Trace(a \Delta-s) at s = 0。这里一个新的纯非交换特征是外观 最终曲率公式中的模自同构群由 III 型冯诺依曼因子理论和量子统计力学得出。副产品是 NC 二维环面的 Gauss-Bonnet 定理。我们还获得了其他工具，如康尼斯迹定理和非交换沃齐基留数。 现在，一个全新的、未知的领域已被打开，有许多开放且有趣的基本问题等待研究。 我计划在过去 4 年取得的突破的基础上继续研究非交换空间的弯曲几何。这包括： 构造新的非交换黎曼流形，求 4 维中的 NC 高斯-邦尼密度，计算非交换环形轨道的标量曲率和高斯-邦尼定理，将黎曼-罗赫定理扩展到非交换 2-环面上的所有全纯线束，验证 Chamseddine -罗伯逊-沃克光谱作用的康尼斯猜想度量，证明非交换 3-环面的 eta 不变量的共形不变性，对我们的曲率公式的概念性理解（在更概念的层面上理解我们使用非交换伪微分符号进行的计算中发生的惊人抵消，以及为什么在最后数千项取消），对 NC 弯曲环面的韦尔定律（类似于霍曼德著名定理）建立高阶修正， 寻找非交换爱因斯坦流形的新例子，研究非交换 4-环面的量子杨-米尔斯理论。