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

几何学是关于形状和空间的测量。通过经典的微分几何，我们已经学会了如何测量距离和体积，以及在所有维度上给定空间的曲率。如何定义和计算非对易空间的曲率？非对易空间是古典空间的一种更为复杂的现代模拟。这些新类型的空间的混沌和模糊的性质，特别是缺乏经典点，使得几乎所有的经典方法无用。来自谱几何和量子力学的思想，即关于编码在其自然几何算子（如狄拉克和拉普拉斯算子）谱中的空间的信息，为如何在NC情况下进行提供了线索。著名的外尔定律关于闭黎曼流形的Hodge-de Rham Laplacian的特征值关于其体积的渐近分布是这类的第一个结果。一般来说，热核迹的短时渐近展开给出了谱不变量的无穷序列。借用马克·卡茨的话：人们可以听到维度、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）中的谱三元组概念下公理化的。在足够的正则性条件下，谱三元组可以被认为是非对易的自旋黎曼流形。在2010年之前，没有已知的或看起来可行的弯曲NC空间中的曲率的真实的计算。第一次超过30年后进入NCG，在过去的4年里，我们（和独立和similataniously康纳斯和莫斯科维奇）能够获得一个公式的曲率弯曲NC 2-D环面。我们（Fathizadeh-Khalkhali）还得到了一个公式的标量曲率的NC曲线的4-D环面。这些公式都是很难计算的，它们绝不能通过对经典曲率公式进行变形而得到。这是通过计算在s = 0时的谱ζ泛函\zeta_a（s）：= Trace（a \Delta-s）的值来实现的。一个新的纯粹的非交换功能，在这里是外观的模自同构群从理论的第三型冯诺依曼因素和量子统计力学在最终公式的曲率。一个副产品是NC 2-D环面的高斯-博内定理。其他工具，如Connes的迹定理，和一个非交换Wodzicki剩余也被我们获得。一个全新的、未经特许的领域现在被打开了，有如此多的开放和有趣的基本问题等待研究。我计划在过去4年中取得的突破的基础上继续我的研究，以理解非对易空间的弯曲几何。这包括：构造新的非对易Riemannian流形，求4维NC Gauss-Bonnet密度，计算非对易Toroidal orbifold的纯量曲率和Gauss-Bonnet定理，将我的Riemann-Roch定理扩展到非对易2-torus上的所有全纯线丛，验证Robertson-Walker度量谱作用的Chamseddine-Connes猜想，证明非对易3-torus eta不变量的共形不变性，曲率公式的概念性理解（这是非常重要的，以了解在一个更概念的水平，惊人的取消，发生在我们的计算与非交换伪微分符号，以及为什么在年底数千项取消），建立高阶修正，我们的外尔定律为数控弯曲环面，（类似霍曼德的著名定理），发现新的例子，非交换爱因斯坦流形，研究量子杨米尔斯理论的非交换4-torus。

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.