Noncommutative residue, conformal invariants and lower dimensional volumes in Riemannian geometry

谱(0,4)-张量泛函与非交换留数

Spectral (0,4)-tensor functionals and the noncommutative residue

本文在无边偶数维紧致自旋流形上利用1-形式、Dirac算子和非交换留数导出谱(0,4)-张量泛函，并将其推广至一般谱三元组。

The spectral Einstein functional and the noncommutative residue for manifolds with boundary

In this paper, we define the spectral Einstein functional associated with the Dirac operator for manifolds with boundary. And we give the proof of Kastler-Kalau-Walze type theorem for the spectral Einstein functional associated with the Dirac operator on 4-dimensional manifolds with boundary.

Twisted Dirac Operators and the noncommutative residue for manifolds with boundary II

In this paper, we establish two kinds of Kastler-Kalau-Walze type theorems for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection on six-dimensional manifolds with boundary.

Laplace-Beltrami Operators on Noncommutative Tori

In this paper, we construct Laplace-Beltrami operators associated with arbitrary Riemannian metrics on noncommutative tori of any dimension. These operators enjoy the main properties of the Laplace-Beltrami operators on ordinary Riemannian manifolds. The construction takes into account the non-triviality of the group of modular automorphisms. On the way we introduce notions of Riemannian density and Riemannian volumes for noncommutative tori.

Adiabatic Limits, Vanishing Theorems and the Noncommutative Residue

In this paper, we compute the adiabatic limit of the scalar curvature and prove several vanishing theorems, we also derive a Kastler-Kalau-Walze type theorem for the noncommutative residue in the case of foliations.

Conformal Perturbations of Twisted Dirac Operators and Noncommutative residue

In this paper, we obtain two kinds of Kastler-Kalau-Walze type theorems for conformal perturbations of twisted Dirac operators and conformal perturbations of signature operators by a vector bundle with a non-unitary connection on six-dimensional manifolds with (respectively without)boundary.

重标度Dirac算子fDh与6维流形上的非交换留数

The Rescaled Dirac operator fDh and the noncommutative residue for 6-dimensional manifolds

本文计算了6维紧流形上重标度Dirac算子fDh的非交换留数，证明了有边界情形的Kastler-Kalau-Walze型定理，并给出若干重要特例。

Levi-Civita's Theorem for Noncommutative Tori

We show how to define Riemannian metrics and connections on a noncommutative torus in such a way that an analogue of Levi-Civita's theorem on the existence and uniqueness of a Riemannian connection holds. The major novelty is that we need to use two different notions of noncommutative vector field. Levi-Civita's theorem makes it possible to define Riemannian curvature using the usual formulas.

On the Noncommutative Residue for Projective Pseudodifferential Operators

A well known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a pseudodifferential projection vanishes. This statement is non-local and implies the regularity of the eta invariant at zero of Dirac type operators. We prove that in a filtered algebra the value of a projection under any residual trace depends only on the principal part of the projection. This general, purely algebraic statement applied to the algebra of projective pseudodifferential operators implies that the noncommutative residue factors to a map from the twisted K-theory of th…

子狄拉克算子、谱爱因斯坦泛函与非交换留数

Sub-Dirac operators, spectral Einstein functionals and the noncommutative residue

本文研究带边流形上子狄拉克算子的谱爱因斯坦泛函，并证明了四维带边情形下的Dabrowski-Sitarz-Zalecki型定理。

扭曲BCV空间的非交换留数与次黎曼极限

The noncommutative residue and sub-Riemannian limits for the twisted BCV spaces

本文推导了扭曲BCV空间的Kastler-Kalau-Walze型与Dabrowski-Sitarz-Zalecki型定理的次黎曼版本，并计算了扭曲乘积的Connes共形不变量及其次黎曼极限。

Differential Forms and the Noncommutative Residue for Manifolds with Boundary in the Non-product Case

In this paper, for an even dimensional compact manifold with boundary which has the non-product metric near the boundary, we use the noncommutative residue to define a conformal invariant pair. For a 4-dimensional manifold, we compute this conformal invariant pair under some conditions and point out the way of computations in the general.

Spectral Asymmetry, Zeta Functions and the Noncommutative Residue

In this paper we study the spectral asymmetry of (possibly nonselfadjoint) elliptic PsiDO's in terms of the difference of zeta functions coming from different cuttings. Refining previous formulas of Wodzicki in the case of odd class elliptic PsiDO's, our main results have several consequence concerning the local independence with respect to the cutting, the regularity at integer points of eta functions and a geometric expression for the spectral asymmetry of Dirac operators which, in particular, yields a new spectral interpretation of the Einstein-Hilbert action in gravity.

Noncommutative Residue and Dirac operators for Manifolds with the Conformal Robertson-Walker metric

In this paper, we prove a Kastler-Kalau-Walze type theorem for 4-dimensional and 6-dimensional spin manifolds with boundary associated with the conformal Robertson-Walker metric. And we give two kinds of operator theoretic explanations of the gravitational action for boundary in the case of 4-dimensional manifolds with flat boundary. In particular, for 6-dimensional spin manifolds with boundary with the conformal Robertson-Walker metric, we obtain the noncommutative residue of the composition of $π^+D^{-1}$ and $π^+D^{-3}$ is proportional to the Einstein-Hilbert action for manifolds with bound…

Riemannian manifolds in noncommutative geometry

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c manifolds; and conversely, in the presence of a spin^c structure. We also show how to obtain an analogue of Kasparov's fundamental class for a Riemannian manifold, and the associated notion of Poincaré duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.

The Dixmier trace and the noncommutative residue for multipliers on compact manifolds

In this paper we give formulae for the Dixmier trace and the noncommutative residue (also called Wodzicki's residue) of pseudo-differential operators by using the notion of global symbol. We consider both cases, compact manifolds with or without boundary. Our analysis on the Dixmier trace of invariant pseudo-differential operators on closed manifolds will be based on the Fourier analysis associated to every elliptic and positive operator and the quantization process developed by Delgado and Ruzhansky. In particular, for compact Lie groups this can be done by using the representation theory of …

带挠率的 Dirac 算子、谱 Einstein 泛函与非交换留数

Dirac operators with torsion, spectral Einstein functionals and the noncommutative residue

本文以非交换留数和带挠率 Dirac 算子为工具，将谱 Einstein 等泛函推广到带挠率的非交换情形，并建立其与带边界流形非交换留数的联系。

Gravity and the Noncommutative Residue for Manifolds with Boundary

We prove a Kastler-Kalau-Walze type theorem for the Dirac operator and the signature operator for $3,4$-dimensional manifolds with boundary. As a corollary, we give two kinds of operator theoretic explanations of the gravitational action in the case of 4-dimensional manifolds with flat boundary.

Some Conformal Invariants from the Noncommutative Residue for Manifolds with Boundary

We review previous work of Alain Connes, and its extension by the author, on some conformal invariants obtained from the noncommutative residue on even dimensional compact manifolds without boundary. Inspired by recent work of Yong Wang, we also address possible generalizations of these conformal invariants to the setting of compact manifolds with boundary.

Twisted Dirac Operators and the Noncommutative Residue for Manifolds with Boundary

In this paper, we give two Lichnerowicz type formulas for Dirac operators and signature operators twisted by a vector bundle with a non-unitary connection. We also prove two Kastler-Kalau-Walze type theorems for twisted Dirac operators and twisted signature operators on 4-dimensional manifolds with (resp. without) boundary.

Noncommutative residue invariants for CR and contact manifolds

In this paper we produce several new invariants for CR and contact manifolds by looking at the noncommutative residue traces of various geometric projections. In the CR setting these operators arise from the Kohn-Rossi complex and include the Szegö projections on forms. In the contact setting they stem from the generalized Szegö projections at arbitrary integer levels of Epstein-Melrose and from the contact complex of Rumin. In particular, we recover and extend recent results of Hirachi and Boutet de Monvel and answer a question of Fefferman.

Connes Trace Theorem for Curved Noncommutative Tori. Application to Scalar Curvature

In this paper we prove a version of Connes' trace theorem for noncommutative tori of any dimension~$n\geq 2$. This allows us to recover and improve earlier versions of this result in dimension $n=2$ and $n=4$ by Fathizadeh-Khalkhali. We also recover the Connes integration formula for flat noncommutative tori of McDonald-Sukochev-Zanin. As a further application we prove a curved version of this integration formula in terms of the Laplace-Beltrami operator defined by an arbitrary Riemannian metric. For the class of so-called self-compatible Riemannian metrics (including the conformally flat metr…

Riemannian geometry of noncommutative super surfaces

In this paper, a Riemannian geometry of noncommutative super surfaces is developed which generalizes [4] to the super case. The notions of metric and connections on such noncommutative super surfaces are introduced and it is shown that the connections are metric-compatible and have zero torsion when the super metric is symmetric, giving rise to the corresponding super Riemann curvature. The latter also satisfies the noncommutative super analogue of the first and second Bianchi identities. We also give some examples and study them in details.

Noncommutative spectral geometry of Riemannian foliations

We construct spectral triples in a sense of noncommutative differential geometry, associated with a Riemannian foliation on a compact manifold, and describe its dimension spectrum.

Noncommutative Residue and sub-Dirac Operators for Foliations

In this paper, we define lower dimensional volumes associated to sub-Dirac operators for foliations. In some cases, we compute these lower dimensional volumes. We also prove the Kastler-Kalau-Walze type theorems for foliations with or without boundary. As a corollary, we give an explanation of the gravitational action for the Robertson-Walker space $[a, b]\times_{f}M^{3}$.

Residue-Torsion and the Laplacian on Riemannian Manifolds

The Ray-Singer analytic torsion is the zeta-function trace of a certain sum of logarithm operators on the de Rham complex. In this note we examine the residue analytic torsion, defined using the residue-trace instead of the spectral zeta function quasi-trace.

Noncommutative spectral geometry of Riemannian foliations: some results and open problems

We review some applications of noncommutative geometry to the study of transverse geometry of Riemannian foliations and discuss open problems.

Thesis: Semi-Riemannian Noncommutative Geometry, Gauge Theory, and the Standard Model of Particle Physics

The subject of this PhD thesis is noncommutative geometry - more specifically spectral triples - and how it can be generalized to semi-Riemannian manifolds generally, and Lorentzian manifolds in particular. The first half of this thesis will thus be dedicated to the transition from Riemannian to semi-Riemannian manifolds. This entails a study of Clifford algebras for indefinite vector spaces and Spin structures on semi-Riemannian manifolds. An important consequence of this is the introduction of Krein spaces, which will enable us to generalize spectral triples to indefinite spectral triples. I…

The transversal wave equation and the noncommutative geodesic flow in Riemannian foliations

We study the transversal wave equation on a compact Riemannian foliated manifold. As applications, we get an Egorov's type theorem for transversally elliptic operators, state a relationship between the singularities of the Fourier transform of the spectrum distribution function of a transversally elliptic operator and the corresponding transversal bicharacteristic flow and give a description of the noncommutative geodesic flow in transversal geometry of Riemannian foliations.

On the noncommutative residue for pseudodifferential operators with log-polyhomogeneous symbols

We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion $a\sim \sum_{j=0}^\infty a_{m-j}, a_{m-j}(x,ξ)=\sum_{l=0}^k a_{m-j,l}(x,ξ) \log^l|ξ|,$ where $a_{m-j,l}$ is homogeneous in $ξ$ of degree $m-j$. We will explain why this algebra of pseudodifferential operators is natural. For a pseudodifferential operator in this class, $A$, and a classical elliptic pseudodifferential operator, $P$, we show that the generalized zeta-function $\Tr(AP^{-s})$ has a meromorphic continuation to the whole complex plane, however possibl…

Dirac Functional Determinants in Terms of the Eta Invariant and the Noncommutative Residue

The zeta and eta-functions associated with massless and massive Dirac operators, in a D-dimensional (D odd or even) manifold without boundary, are rigorously constructed. Several mathematical subtleties involved in this process are stressed, as the intrisic ambiguity present in the definition of the associated fermion functional determinant in the massless case and, also, the unavoidable presence (in some situations) of a multiplicative anomaly, that can be conveniently expressed in terms of the noncommutative residue. The ambiguity is here seen to disappear in the massive case, giving rise to…

Riemannian Geometry of Noncommutative Surfaces

A Riemannian geometry of noncommutative n-dimensional surfaces is developed as a first step towards the construction of a consistent noncommutative gravitational theory. Historically, as well, Riemannian geometry was recognized to be the underlying structure of Einstein's theory of general relativity and led to further developments of the latter. The notions of metric and connections on such noncommutative surfaces are introduced and it is shown that the connections are metric-compatible, giving rise to the corresponding Riemann curvature. The latter also satisfies the noncommutative analogue …

伪黎曼双曲空间的共振与留数算子

Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

本文研究实、复、四元数与八元数伪黎曼双曲空间上 Laplace–Beltrami 算子预解式的亚纯延拓，显式确定全部共振，并将留数像刻画为退化主序列的子表示。

Weyl's Law and Connes' Trace Theorem for Noncommutative Two Tori

We prove the analogue of Weyl's law for a noncommutative Riemannian manifold, namely the noncommutative two torus $\mathbb{T}_θ^2$ equipped with a general translation invariant conformal structure and a Weyl conformal factor. This is achieved by studying the asymptotic distribution of the eigenvalues of the perturbed Laplacian on $\mathbb{T}_θ^2$. We also prove the analogue of Connes' trace theorem by showing that the Dixmier trace and a noncommutative residue coincide on pseudodifferential operators of order -2 on $\mathbb{T}_θ^2$.

Residue Family Operators on Spinors and Spectral Theory of Dirac operator on Poincaré-Einstein Spaces

We study conformal $Spin$-subgeometry of submanifolds in a semi-Riemannian $Spin$-manifold, focusing on conformal $Spin$-manifolds $(M,[h])$ and their Poincaré-Einstein metrics $(X,g_+)$. Our approach is based on the spectral theory of Dirac operator in the ambient $Spin$-manifold, and associated spinor valued meromorphic family of distributions with residues given by the residue family operators $\slashed{D}_N^{res}(h;λ)$ on spinors. We develop basic aspects and properties of $\slashed{D}_N^{res}(h;λ)$ including conformal covariance, factorization properties by conformally covariant operators…

Scalar Curvature for Noncommutative Four-Tori

In this paper we study the curved geometry of noncommutative 4-tori $\mathbb{T}_θ^4$. We use a Weyl conformal factor to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information. We use Connes' pseudodifferential calculus to explicitly compute the terms in the small time heat kernel expansion of the perturbed Laplacian which correspond to the volume and scalar curvature of $\mathbb{T}_θ^4$. We establish the analogue of Weyl's law, define a noncommutative residue, prove the analogue of Connes' trace theorem, and find explicit formulas for the local f…

Dirac 算子的伪微分扰动与 Kastler-Kalau-Walze 型定理

The pseudo-differential perturbations of the Dirac operator and the Kastler-Kalau-Walze type theorems

本文在非交换几何框架内定义了 Dirac 算子的两类伪微分扰动，推导了其在 4 维紧流形上的非交换留数，并证明了带边界流形上的 Kastler-Kalau-Walze 型定理。

*-Compatible Connections in Noncommutative Riemannian Geometry

We develop the formalism for noncommutative differential geometry and Riemmannian geometry to take full account of the *-algebra structure on the (possibly noncommutative) coordinate ring and the bimodule structure on the differential forms. We show that *-compatible bimodule connections lead to braid operators $σ$ in some generality (going beyond the quantum group case) and we develop their role in the exterior algebra. We study metrics in the form of Hermitian structures on Hilbert *-modules and metric compatibility in both the usual and a cotorsion form. We show that the theory works well f…

Nonholonomic Clifford Structures and Noncommutative Riemann--Finsler Geometry

We survey the geometry of Lagrange and Finsler spaces and discuss the issues related to the definition of curvature of nonholonomic manifolds enabled with nonlinear connection structure. It is proved that any commutative Riemannian geometry (in general, any Riemann--Cartan space) defined by a generic off--diagonal metric structure (with an additional affine connection possessing nontrivial torsion) is equivalent to a generalized Lagrange, or Finsler, geometry modeled on nonholonomic manifolds. This results in the problem of constructing noncommutative geometries with local anisotropy, in parti…

Conformal Structures in Noncommutative Geometry

It is well-known that a compact Riemannian spin manifold can be reconstructed from its canonical spectral triple which consists of the algebra of smooth functions, the Hilbert space of square integrable spinors and the Dirac operator. It seems to be a folklore fact that the metric can be reconstructed up to conformal equivalence if one replaces the Dirac operator D by sign(D). We give a precise formulation and proof of this fact.

Singular Riemannian flows and characteristic numbers

Let $M$ be an even-dimensional, oriented closed manifold. We show that the restriction of a singular Riemannian flow on $M$ to a small tubular neighborhood of each connected component of its singular stratum is foliated-diffeomorphic to an isometric flow on the same neighborhood. We then prove a formula that computes characteristic numbers of $M$ as the sum of residues associated to the infinitesimal foliation at the components of the singular stratum of the flow.

Curvature in Noncommutative Geometry

Our understanding of the notion of curvature in a noncommutative setting has progressed substantially in the past ten years. This new episode in noncommutative geometry started when a Gauss-Bonnet theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. Ideas from spectral geometry and heat kernel asymptotic expansions suggest a general way of defining local curvature invariants for noncommutative Riemannian type spaces where the metric structure is encoded by a Dirac type operator. To carry explicit computations however one needs quite intriguing new ideas. We give an …

Scalar curvature in conformal geometry of Connes-Landi noncommutative manifolds

We first propose a conformal geometry for Connes-Landi noncommutative manifolds and study the associated scalar curvature. The new scalar curvature contains its Riemannian counterpart as the commutative limit. Similar to the results on noncommutative two tori, the quantum part of the curvature consists of actions of the modular derivation through two local curvature functions. Explicit expressions for those functions are obtained for all even dimensions (greater than two). In dimension four, the one variable function shows striking similarity to the analytic functions of the characteristic cla…

The Ricci Curvature in Noncommutative Geometry

Motivated by the local formulae for asymptotic expansion of heat kernels in spectral geometry, we propose a definition of Ricci curvature in noncommutative settings. The Ricci operator of an oriented closed Riemannian manifold can be realized as a spectral functional, namely the functional defined by the zeta function of the full Laplacian of the de Rham complex, localized by smooth endomorphisms of the cotangent bundle and their trace. We use this formulation to introduce the Ricci functional in a noncommutative setting and in particular for curved noncommutative tori. This Ricci functional u…

Nonassociative Riemannian Geometry by Twisting

Many quantum groups and quantum spaces of interest can be obtained by cochain (but not cocycle) twist from their corresponding classical object. This failure of the cocycle condition implies a hidden nonassociativity in the noncommutative geometry already known to be visible at the level of differential forms. We extend the cochain twist framework to connections and Riemannian structures and provide examples including twist of the $S^7$ coordinate algebra to a nonassociative hyperbolic geometry in the same category as that of the octonions.

From Monge to Higgs: a survey of distance computations in noncommutative geometry

This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, and the comparison of Connes distance with the minimal length that emerges in various models of quantum spacetime. Links with other areas of mathematics are studied, in particular the horizontal distance in sub-Riemannian geometry. The interpretation of Connes distance as a nonco…

通过Guillemin-Wodzicki留数研究三维空间形式中测地球的Steklov谱唯一性

Steklov Spectral Uniqueness of Geodesic Balls in 3-Dimensional Space Forms via Guillemin--Wodzicki Residues

本文通过Lee-Uhlmann全符号公式与Weyl不变量理论计算Dirichlet-to-Neumann算子的Guillemin-Wodzicki留数，进而证明三维空间形式中测地球及同心球壳区域的Steklov谱唯一性与刚性定理。

Noncommutative Ricci curvature and Dirac operator on $C_q[SL_2]$ at roots of unity

We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group $C_q[SL_2]$, using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for $q$ an odd $r$'th root of unity that its eigenvalues are given by $q$-integers $[m]_q$ for $m=0,1,...,r-1$ offset by the constant background curvature. We fully solve the Dirac equation for $r=3$.

黎曼度量丛

Riemannian Metric Bundle

本文提出并严格发展黎曼度量丛理论，研究其范畴结构及其与 K-理论和流形几何不变量的联系，为理解几何、拓扑与分析之间的相互作用提供新视角。

Feynman-Kac formula for perturbations of order $\leq 1$ and noncommutative geometry

Let $Q$ be a differential operator of order $\leq 1$ on a complex metric vector bundle $\mathscr{E}\to \mathscr{M}$ with metric connection $\nabla$ over a possibly noncompact Riemannian manifold $\mathscr{M}$. Under very mild regularity assumptions on $Q$ that guarantee that $\nabla^{\dagger}\nabla/2+Q$ generates a holomorphic semigroup $\mathrm{e}^{-zH^{\nabla}_{Q}}$ in $Γ_{L^2}(\mathscr{M},\mathscr{E})$ (where $z$ runs through a complex sector which contains $[0,\infty)$), we prove an explicit Feynman-Kac type formula for $\mathrm{e}^{-tH^{\nabla}_{Q}}$, $t>0$, generalizing the standard self…

形变量子化与内蕴非交换微分几何

Deformation quantization and intrinsic noncommutative differential geometry

本文提出非交换微分几何的内蕴表述，给出协变导数、曲率及Bianchi恒等式的几何定义，证明特定条件下Ricci曲率等价，并计算球对称等距嵌入的量子涨落与曲率，表明引力量子化可重整化，最后定义准连接并证明第一Bianchi恒等式。

量子化覆盖的非交换几何

Noncommutative Geometry of Quantized Coverings

本文提出一个统一的公理化理论，涵盖多种$C^*$-代数覆盖，并推广万有覆盖、基本群等几何构造，给出纯代数证明。