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

几何学是关于形状的测量，以及一般的空间测量。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，以及给定空间在所有维度上的曲率。如何定义和计算非对易空间的曲率？非对易空间是经典空间的一种更为复杂的现代类比。这些新型空间的混沌和模糊特性，特别是缺乏经典的点，使得几乎所有的经典方法都无用。来自谱几何和量子力学的想法，即关于一个空间的信息，编码在它的自然几何算子的谱中，如狄拉克和拉普拉斯，给出了如何在NC情况下进行的线索。著名的关于闭黎曼流形的Hodge-de Rham Laplace算子的本征值关于其体积的渐近分布的Weyl定律是此类结果的第一个结果。一般来说，热核迹的短时渐近展开给出了一个无限的谱不变量序列。借用马克·卡克的话：人们可以听到尺寸、体积和标量曲率。这种情况的本质是由Alain Connes在非对易几何(NCG)中在谱三元组的概念下公理化的。在足够的正则性条件下，谱三元组可视为非对易自旋黎曼流形。在2010年前，曲面NC空间中的曲率的实际计算是未知的或似乎是可行的。在进入NCG 30多年后，在过去的4年里，我们(以及独立和同时的Connes和Moscovici)第一次能够获得弯曲的NC二维环面的曲率公式。我们(Fathizadeh-Khalkhali)也得到了NC曲线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的曲率公式来获得。这是通过评估S=0时的谱Zeta泛函\Zeta_a(S)：=Trace(a\Delta-S)的值来实现的。这里的一个新的纯非对易特征是，在曲率的最终公式中，从第三类von Neumann因子理论和量子统计力学中出现了模自同构群。一个副产品是Nc二维环面的Gauss-Bonnet定理。我们还得到了诸如Connes迹定理、非对易Wodzicki留数等工具。现在，一个全新的、未知的领域打开了，有如此多开放和有趣的基本问题等待研究。我计划在过去四年中取得的突破的基础上，继续我的研究，了解非对易空间的曲线几何。这包括：构造新的非对易黎曼流形，找到4维的NC Gauss-Bonnet密度，计算非对易环面的标量曲率和Gauss-Bonnet定理，将我的Riemann-Roch定理推广到非对易2-环面上的所有全纯线丛，验证Robertson-Walker度规的谱作用的Chamseddine-Connes猜想，证明非对易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(在更概念性的水平上理解在我们的计算中发生的惊人的取消，以及为什么在数千个取消项的末尾，这是非常重要的)，建立了NC弯曲环面的Weyl定律(类似于Hormander著名定理)的高阶修正，发现了非对易爱因斯坦流形的新例子，研究了非对易4-环面上的量子Yang-Mills理论。

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.