Spectral Triples Research Articles

Finite Two-Point Space Without Quantization on Noncommutative Space from a Generalized Fractional Integral Operator

We show in this communication that if the Mellin transform is replaced by a fractional generalized integral on a noncommutative space and using the basic axioms of spectral triples, the two-point space problem is finite without passing to quantization.

Spectral Triples in Particle Physics

We give an overview of the approach to the Standard Model of Particle Physics and its extensions based on the Noncommutative Geometry. The notion of spectral triples is introduced and their applications in particle physics are presented. We revisit known results based on different approaches within Noncommutative Geometry, list problems which appeared in these methods, propose possible solutions and indicate future directions of research.

Spectral triples for nested fractals

It is shown that, for nested fractals [31], the main structural data, such as the Hausdorff dimension and measure, the geodesic distance (when it exists) induced by the immersion in \mathbb R^n , and the self-similar energy can all be recovered by the description of the fractals in terms of the spectral triples considered in [18].

Boundary value problems with Atiyah–Patodi–Singer type conditions and spectral triples

We study realizations of pseudodifferential operators acting on sections of vector-bundles on a smooth, compact manifold with boundary, subject to conditions of Atiyah–Patodi–Singer type. Ellipticity and Fredholm property, compositions, adjoints and self-adjointness of such realizations are discussed. We construct regular spectral triples (\mathcal {A,H,D}) for manifolds with boundary of arbitrary dimension, where \mathcal H is the space of square integrable sections. Starting out from Dirac operators with APS-conditions, these triples are even in case of even dimensional manifolds; we show that the closure of \mathcal A in \mathcal L(\mathcal H) coincides with the continuous functions on the manifold being constant on each connected component of the boundary.

Derivations and Spectral Triples on Quantum Domains I: Quantum Disk

We study unbounded invariant and covariant derivations on the quantum disk. In particular we answer the question whether such derivations come from operators with compact parametrices and thus can be used to define spectral triples.

Loop groups and noncommutative geometry

We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level [Formula: see text] projective unitary positive-energy representations of any given loop group [Formula: see text]. The construction is based on certain supersymmetric conformal field theory models associated with [Formula: see text] in the setting of conformal nets. We then generalize the construction to many other rational chiral conformal field theory models including coset models and the moonshine conformal net.

Spectral triples from bimodule connections and Chern connections

We give a geometrical construction of Connes spectral triples or noncommutative Dirac operators \mathrlap{\,/}{D} starting with a bimodule connection on the proposed spinor bundle. The theory is applied to the example of M_2(\mathbb C) , and also applies to the standard q -sphere and the q -disk with the right classical limit and all properties holding except for \mathcal J now being a twisted isometry. We also describe a noncommutative Chern construction from holomorphic bundles which in the q -sphere case provides the relevant bimodule connection.

Smooth geometry of the noncommutative pillow, cones and lens spaces

This paper proposes a new notion of smoothness of algebras, termed differential smoothness , that combines the existence of a top form in a differential calculus over an algebra together with a strong version of the Poincaré duality realized as an isomorphism between complexes of differential and integral forms. The quantum two- and three-spheres, disc, plane and the noncommutative torus are all smooth in this sense. Noncommutative coordinate algebras of deformations of several examples of classical orbifolds such as the pillow orbifold, singular cones and lens spaces are also differentially smooth. Although surprising this is not fully unexpected as these algebras are known to be homologically smooth. The study of Riemannian aspects of the noncommutative pillow and Moyal deformations of cones leads to spectral triples that satisfy the orientability condition that is known to be broken for classical orbifolds.

Noncommutative geometry and the BV formalism: Application to a matrix model

We analyze a U(2)-matrix model derived from a finite spectral triple. By applying the BV formalism, we find a general solution to the classical master equation. To describe the BV formalism in the context of noncommutative geometry, we define two finite spectral triples: the BV spectral triple and the BV auxiliary spectral triple. These are constructed from the gauge fields, ghost fields and anti-fields that enter the BV construction. We show that their fermionic actions add up precisely to the BV action. This approach allows for a geometric description of the ghost fields and their properties in terms of the BV spectral triple.

Wieler solenoids, Cuntz–Pimsner algebras and -theory

We study irreducible Smale spaces with totally disconnected stable sets and their associated$K$-theoretic invariants. Such Smale spaces arise as Wieler solenoids, and we restrict to those arising from open surjections. The paper follows three converging tracks: one dynamical, one operator algebraic and one$K$-theoretic. Using Wieler’s theorem, we characterize the unstable set of a finite set of periodic points as a locally trivial fibre bundle with discrete fibres over a compact space. This characterization gives us the tools to analyse an explicit groupoid Morita equivalence between the groupoids of Deaconu–Renault and Putnam–Spielberg, extending results of Thomsen. The Deaconu–Renault groupoid and the explicit Morita equivalence lead to a Cuntz–Pimsner model for the stable Ruelle algebra. The$K$-theoretic invariants of Cuntz–Pimsner algebras are then studied using the Cuntz–Pimsner extension, for which we construct an unbounded representative. To elucidate the power of these constructions, we characterize the Kubo–Martin–Schwinger (KMS) weights on the stable Ruelle algebra of a Wieler solenoid. We conclude with several examples of Wieler solenoids, their associated algebras and spectral triples.

The graded product of real spectral triples

Forming the product of two geometric spaces is one of the most basic operations in geometry, but in the spectral-triple formulation of non-commutative geometry, the standard prescription for taking the product of two real spectral triples is problematic: among other drawbacks, it is non-commutative, non-associative, does not transform properly under unitaries, and often fails to define a proper spectral triple. In this paper, we explain that these various problems result from using the ungraded tensor product; by switching to the graded tensor product, we obtain a new prescription where all of the earlier problems are neatly resolved: in particular, the new product is commutative, associative, transforms correctly under unitaries, and always forms a well defined spectral triple.

Spectral Metric Spaces on Extensions of C*-Algebras

We construct spectral triples on C*-algebraic extensions of unital C*-algebras by stable ideals satisfying a certain Toeplitz type property using given spectral triples on the quotient and ideal. Our construction behaves well with respect to summability and produces new spectral quantum metric spaces out of given ones. Using our construction we find new spectral triples on the quantum 2- and 3-spheres giving a new perspective on these algebras in noncommutative geometry.

Spectral triples for noncommutative solenoidal spaces from self-coverings

Examples of noncommutative self-coverings are described, and spectral triples on the base space are extended to spectral triples on the inductive family of coverings, in such a way that the covering projections are locally isometric. Such triples are shown to converge, in a suitable sense, to a semifinite spectral triple on the direct limit of the tower of coverings, which we call noncommutative solenoidal space. Some of the self-coverings described here are given by the inclusion of the fixed point algebra in a C⁎-algebra acted upon by a finite abelian group. In all the examples treated here, the noncommutative solenoidal spaces have the same metric dimension and volume as on the base space, but are not quantum compact metric spaces, namely the pseudo-metric induced by the spectral triple does not produce the weak⁎ topology on the state space.

Deformations of spectral triples and their quantum isometry groups via monoidal equivalences

In this paper, we propose a new procedure to deform spectral triples and their quantum isometry groups. The deformation data are a spectral triple (mathcal {A},mathcal {H}, D), a compact quantum group {mathbb {G}} acting algebraically and by orientation-preserving isometries on (mathcal {A},mathcal {H},D) and a unitary fiber functor psi on {mathbb {G}}. The deformation procedure is a proper generalization of the cocycle deformation of Goswami and Joardar.

Fractal spectral triples on Kellendonk’sC∗-algebra of a substitution tiling

We introduce a new class of noncommutative spectral triples on Kellendonk's $C^*$-algebra associated with a nonperiodic substitution tiling. These spectral triples are constructed from fractal trees on tilings, which define a geodesic distance between any two tiles in the tiling. Since fractals typically have infinite Euclidean length, the geodesic distance is defined using Perron-Frobenius theory, and is self-similar with scaling factor given by the Perron-Frobenius eigenvalue. We show that each spectral triple is $\theta$-summable, and respects the hierarchy of the substitution system. To elucidate our results, we construct a fractal tree on the Penrose tiling, and explicitly show how it gives rise to a collection of spectral triples.

Examples of infinite direct sums of spectral triples

We study two ways of summing an infinite family of noncommutative spectral triples. First, we propose a definition of the integration of spectral triples and give an example using algebras of Toeplitz operators acting on weighted Bergman spaces over the unit ball of $\mathbb{C}^n$. Secondly, we construct a spectral triple associated to a general polygonal self-similar set in $\mathbb{C}$ using algebras of Toeplitz operators on Hardy spaces. In this case, we show that we can recover the Hausdorff dimension of the fractal set.

Shift–tail equivalence and an unbounded representative of the Cuntz–Pimsner extension

We show how the fine structure in shift–tail equivalence, appearing in the non-commutative geometry of Cuntz–Krieger algebras developed by the first two listed authors, has an analogue in a wide range of other Cuntz–Pimsner algebras. To illustrate this structure, and where it appears, we produce an unbounded representative of the defining extension of the Cuntz–Pimsner algebra constructed from a finitely generated projective bi-Hilbertian module, extending work by the third listed author with Robertson and Sims. As an application, our construction yields new spectral triples for Cuntz and Cuntz–Krieger algebras and for Cuntz–Pimsner algebras associated to vector bundles twisted by an equicontinuous$\ast$-automorphism.

Fubini theorem in noncommutative geometry

We discuss the Fubini formula in Alain Connes' noncommutative geometry. We present a sufficient condition on spectral triples for which a Fubini formula holds true. The condition is natural and related to heat semigroup asymptotics. We provide examples of spectral triples for which the Fubini formula fails.

A note on the perturbations of compact quantum metric spaces

In this short note, we consider the perturbation of compact quantum metric spaces. We first show that for two compact quantum metric spaces (A, P) and (B, Q) for which A and B are subspaces of an order-unit space C and P and Q are Lip-norms on A and B respectively, the quantum Gromov–Hausdorff distance between (A, P) and (B, Q) is small under certain conditions. Then some other perturbation results on compact quantum metric spaces derived from spectral triples are also given.

Ergodic actions and spectral triples

In this article, we give a general construction of spectral triples from certain Lie group actions on unital C ∗ -algebras. If the group G is compact and the action is ergodic, we actually obtain a real and finitely summable spectral triple which satisfies the first order condition of Connes’ axioms. This provides a link between the “algebraic” existence of ergodic action and the “analytic” finite summability property of the unbounded selfadjoint operator. More generally, for compact G we carefully establish that our (symmetric) unbounded operator is essentially selfadjoint. Our results are illustrated by a host of examples – including noncommutative tori and quantum Heisenberg manifolds.