Mario Paschke, Andrzej Sitarz
We present examples of equivariant noncommutative Lorentzian spectral geometries. The equivariance with respect to a compact isometry group (or quantum group) allows to construct the algebraic data of a version of spectral triple geometry adapted to the situation of an indefinite metric. The spectrum of the equivariant Dirac operator is calculated.
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A Lorentzian Equivariant Index Theorem
本文提出一种洛伦兹时空下扭曲狄拉克算子的等价指标公式，结合等价和非等价情况的简单技术，推导出洛伦兹“指标=谱流”公式的等价形式。
We study a noncommutative analogue of a spacetime foliated by spacelike hypersurfaces, in both Riemannian and Lorentzian signatures. First, in the classical commutative case, we show that the canonical Dirac operator on the total spacetime can be reconstructed from the family of Dirac operators on the hypersurfaces. Second, in the noncommutative case, the same construction continues to make sense for an abstract family of spectral triples. In the case of Riemannian signature, we prove that the construction yields in fact a spectral triple, which we call a product spectral triple. In the case o…
We show how the twisting of spectral triples induces a transition from an euclidean to a lorentzian noncommutative geometry, at the level of the fermionic action. More specifically, we compute the fermionic action for the twisting of a closed euclidean manifold, then that of a two-sheet euclidean manifold, and finally the twisting of the spectral triple of electrodynamics in euclidean signature. We obtain the Weyl and the Dirac equations in lorentzian signature (and in the temporal gauge). The twisted fermionic action is then shown to be invariant under an action of the Lorentz group. This per…
We present the notion of temporal Lorentzian spectral triple which is an extension of the notion of pseudo-Riemannian spectral triple with a way to ensure that the signature of the metric is Lorentzian. A temporal Lorentzian spectral triple corresponds to a specific 3+1 decomposition of a possibly noncommutative Lorentzian space. This structure introduces a notion of global time in noncommutative geometry. As an example, we construct a temporal Lorentzian spectral triple over a Moyal--Minkowski spacetime. We show that, when time is commutative, the algebra can be extended to unbounded elements…
Emergence of Time from a Twisted Spectral Triple in Almost-Commutative Geometry
本文探讨了在几乎交换框架内，扭曲谱三元体如何产生伪黎曼结构，并提出了一种统一的代数机制解决洛伦兹签名问题，通过连接扭曲与伪黎曼谱三元体的映射概念，提供了替代温克旋转的方法。
In general, Morita equivalence of spectral triples need not be a symmetric relation. In this paper, we show that Morita equivalence of spectral triples is an equivalence relation for equivariant torus spectral triples.
Emergence of Lorentz symmetry from an almost-commutative twisted spectral triple
本文在近交换扭曲谱三元组框架下，提出一种不引入复数的维克转动替代方案，从自伴狄拉克算子出发实现从黎曼到伪黎曼谱三元组的转变，为洛伦兹号差问题提供新视角。
This is a review of recent results regarding the application of Connes' noncommutative geometry to the Standard Model, and beyond. By twisting (in the sense of Connes-Moscovici) the spectral triple of the Standard Model, one does not only get an extra scalar field which stabilises the electroweak vacuum, but also an unexpected 1-form field. By computing the fermionic action, we show how this field induces a transition from the Euclidean to the Lorentzian signature. Hints on a twisted version of the spectral action are also briefly mentioned.
This thesis concerns the research on a Lorentzian generalization of Alain Connes' noncommutative geometry. In the first chapter, we present an introduction to noncommutative geometry within the context of unification theories. The second chapter is dedicated to the basic elements of noncommutative geometry as the noncommutative integral, the Riemannian distance function and spectral triples. In the last chapter, we investigate the problem of the generalization to Lorentzian manifolds. We present a first step of generalization of the distance function with the use of a global timelike eikonal c…
We present a new example of a finite-dimensional noncommutative manifold, namely the noncommutative cylinder. It is obtained by isospectral deformation of the canonical triple associated to the Euclidean cylinder. We discuss Connes' character formula for the cylinder. In the second part, we discuss noncommutative Lorentzian manifolds. Here, the definition of spectral triples involves Krein spaces and operators on Krein spaces. A central role is played by the admissible fundamental symmetries on the Krein space of square integrable sections of a spin bundle over a Lorentzian manifold. Finally, …
Given a spectral triple on a unital $C^{*}$-algebra $A$ and an equicontinuous action of a discrete group $G$ on $A$, a spectral triple on the reduced crossed product $C^{*}$-algebra $A\rtimes_r G$ was constructed by Hawkins, Skalski, White and Zacharias in [On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262-291], extending the construction by Belissard, Marcolli and Reihani in [Dynamical systems on spectral metric spaces, preprint (2010), arXiv:1008.4617], by using the Kasparov product to make an ansatz for the Dirac operator. Supposing …
We study the "quantized calculus" corresponding to the algebraic ideas related to "twisted cyclic cohomology" introduced in [KMT]. With very similar definitions and techniques as those used in [jlo], we define and study "twisted entire cyclic cohomology" and the "twisted Chern character" associated with an appropriate operator-theoretic data called "twisted spectral data", which consists of a spectral triple in the conventional sense of noncommutative geometry ([Con]) and an additional positive operator having specified properties. Furthermore, it is shown that given a spectral triple (in the …
Motivated by the space of spinors on a Lorentzian manifold, we define Krein spectral triples, which generalise spectral triples from Hilbert spaces to Krein spaces. This Krein space approach allows for an improved formulation of the fermionic action for almost-commutative manifolds. We show by explicit calculation that this action functional recovers the correct Lagrangians for the cases of electrodynamics, the electro-weak theory, and the Standard Model. The description of these examples does not require a real structure, unless one includes Majorana masses, in which case the internal spaces …
We show how twisting the spectral triple of the Standard Model of elementary particles naturally yields the Krein space associated with the Lorentzian signature of spacetime. We discuss the associated spectral action, both for fermions and bosons. What emerges is a tight link between twist and Wick rotation.
We consider the wave operator on static, Lorentzian manifolds with timelike boundary and we discuss the existence of advanced and retarded fundamental solutions in terms of boundary conditions. By means of spectral calculus we prove that answering this question is equivalent to studying the self-adjoint extensions of an associated elliptic operator on a Riemannian manifold with boundary $(M,g)$. The latter is diffeomorphic to any, constant time hypersurface of the underlying background. In turn, assuming that $(M,g)$ is of bounded geometry, this problem can be tackled within the framework of b…
We classify and construct all real spectral triples over noncommutative Bieberbach manifolds, which are restrictions of irreducible real equivariant spectral triple over the noncommutative three-torus. We show that in the classical case the constructed geometries correspond exactly to spin structures over Bieberbach manifolds and the Dirac operators constructed for a flat metric.
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…
Torsion and Lorentz symmetry from Twisted Spectral Triples
本文通过扭曲黎曼自旋流形的谱三元组，从无挠 Dirac 算子生成保持正交和测地线的挠率，揭示扭曲酉群生成挠率且洛伦兹群为其正规子群，并探讨了相关谱作用。
A critical survey of twisted spectral triples beyond the Standard Model
本文综述了扭曲谱三元组在标准模型中的应用，指出其最初旨在生成标量场以稳定电弱真空并拟合Higgs质量，但最终发现其核心价值在于引入了一个与欧几里得到洛伦兹签名转换相关的新1-形式场。
We consider the algebra $A$ of bounded operators on $L^2(\mathbb{R}^n)$ generated by quantizations of isometric affine canonical transformations. The algebra $A$ includes as subalgebras all noncommutative tori and toric orbifolds. We define the spectral triple $(A, H, D)$ with $H=L^2(\mathbb R^n, Λ(\mathbb R^n))$ and the Euler operator $D$, a first order differential operator of index $1$. We show that this spectral triple has simple dimension spectrum: For every operator $B$ in the algebra $Ψ(A,H,D)$ generated by the Shubin type pseudodifferential operators and the elements of $A$, the zeta f…
We investigate examples of quasi-spectral triples over two-dimensional commutative sphere, which are obtained by modifying the order-one condition. We find equivariant quasi-Dirac operators and prove that they are in a topologically distinct sector than the standard Dirac operator.
Quantum causality in $κ$-Minkowski and related constraints
本研究针对1+1维$κ$-Minkowski时空的量子因果结构展开，基于扭导数构建洛伦兹谱三元组，揭示因果函数类与量子约束条件，其中包含光速极限的量子对应形式。
If $G$ is a compact Lie group, $T$ a maximal torus in $G$ (with Lie algebras $\mathfrak{g}$ and $\mathfrak{t}$ respectively) and $W$ the corresponding Weyl group, then the Berry-Robbins problem for $G$, as formulated by Sir Michael Atiyah and Roger Bielawski, asks whether there exists a continuous $SU(2) \times W$ equivariant map from the space of regular Cartan triples (an open subset of $\mathfrak{t} \otimes \mathbb{R}^3$) to $G/T$, where $SU(2)$ acts via a regular Lie group homomorphism $SU(2) \to G$. This was settled positively by Atiyah and Bielawski, and their maps are even smooth, but t…
With the bare essentials of noncommutative geometry (defined by a spectral triple), we first describe how it naturally gives rise to gauge theories. Then, we quickly review the notion of twisting (in particular, minimally) noncommutative geometries and how it induces a Wick rotation, that is, a transition of the metric signature from euclidean to Lorentzian. We focus on comparatively more tractable examples of spectral triples; such as the ones corresponding to a closed Riemannian spin manifold, $U(1)$ gauge theory, and electrodynamics. By minimally twisting these examples and computing their …
The spectral action on the equivariant real spectral triple over \A(SU_q(2)) is computed explicitly. Properties of the differential calculus arising from the Dirac operator are studied and the results are compared to the commutative case of the sphere S^3.
Asymmetric noncommutative torus has vanishing Einstein tensor
研究通过部分共形缩放计算非交换 torus 上谱三元组的谱度量，发现其 torsion 和 Einstein 张量均消失。
A reconstruction theorem for Connes-Landi deformations of commutative spectral triples
本文提出了一种关于Connes-Landi变形的谱三元组的抽象定义，并证明其在进一步变形下的良好行为，同时展示了其在量化和去量化过程中的应用。
We propose an algebraic formulation of the notion of causality for spectral triples corresponding to globally hyperbolic manifolds with a well defined noncommutative generalization. The causality is given by a specific cone of Hermitian elements respecting an algebraic condition based on the Dirac operator and a fundamental symmetry. We prove that in the commutative case the usual notion of causality is recovered. We show that, when the dimension of the manifold is even, the result can be extended in order to have an algebraic constraint suitable for a Lorentzian distance formula.
We study the spectral geometry of the quantum projective plane CP^2_q, a deformation of the complex projective plane CP^2, the simplest example of a spin^c manifold which is not spin. In particular, we construct a Dirac operator D which gives a 0^+ summable spectral triple, equivariant under U_q(su(3)). The square of D is a central element for which left and right actions on spinors coincide, a fact that is exploited to compute explicitly its spectrum.
We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly noncommutative) manifold satisfying certain regularity assumptions. In fact, we identify the quantum isometry group with the u…
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