Noncommutative Torus Research Articles

Cohesive self-dual Yang-Mills modules over four manifolds

This research is motivated by Atiyah, Hitchin, and Singers paper Self-duality in Four-Dimensional Riemannian Geometry , which introduced a relationship between self-dual Yang-Mills fields on smooth manifolds and holomorphic vector bundles on their twistor spaces. Here, self-duality is a specific structure in 4-dimensional manifolds and Yang-Mills fields are gauge fields that satisfy Yang-Mills equations in 4-dimensions and are corresponded to the holomorphic bundles on twistor spaces. In this paper, we extend the relationship from vector bundles to a generalization of the Yang-Mills fields. To achieve this purpose, we apply Atiyah, Hitchin, and Singers theorem to cohesive modules, which was originally introduced by Block in in studying coherent sheaves over complex manifolds and the relations between homomorphic torus and its dual non-commutative torus. We introduce the notion of cohesive self-dual Yang-Mills modules and show that the twistor correspondence actually induces the equivalence between the dg category of cohesive self-dual Yang Mills modules P_(A_SD ) and the dg category of holomorphic cohesive modules P_(A_Hol ) on the twistor spaces.

Spectral triples, Coulhon-Varopoulos dimension and heat kernel estimates

We investigate the relations between the (completely bounded) local Coulhon-Varopoulos dimension and the spectral dimension of spectral triples associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical (or noncommutative) Lp-spaces associated to finite measure spaces. More precisely, we prove that the completely bounded local Coulhon-Varopoulos dimension d exceeds the spectral dimension, i.e. that the associated Hodge-Dirac operator is d+-summable. We explore different settings to compare these two values: compact Riemannian manifolds, compact Lie groups, sublaplacians, metric measure spaces, noncommutative tori and quantum groups. Specifically, we prove that, while very often equal in smooth compact settings, these dimensions can diverge. Finally, we show that the existence of a symmetric sub-Markovian semigroup on a von Neumann algebra with finite completely bounded local Coulhon-Varopoulos dimension implies that the von Neumann algebra is necessarily injective.

Local invariants of noncommutative tori

The notion of a generic curved noncommutative torus is considered, which extends the notion of a conformally deformed noncommutative torus introduced by Connes and Tretkoff. For this manifold, an asymptotic expansion is established for the heat semigroup generated by the Laplace–Beltrami operator (in fact, for an arbitrary selfadjoint positive elliptic differential operator of order 2 2 ) and an algorithm is provided to compute the local invariants that arize as the coefficients in the expansion. This allows one to extend a series of previous results by several authors beyond the conformal case and/or for multidimensional tori.

Symmetrized non-commutative tori revisited

For the flip action of \mathbb{Z}_{2} on an n -dimensional noncommutative torus A_{\theta} , using an exact sequence by Natsume, we compute the K-theory groups of A_{\theta}\rtimes\mathbb{Z}_{2} . The novelty of our method is that it also provides an explicit basis of \operatorname{K}_{0}(A_{\theta}\rtimes\mathbb{Z}_{2}) , for any \theta . As an application, for a simple n -dimensional torus A_{\theta} , using classification techniques, we determine the isomorphism class of A_{\theta}\rtimes\mathbb{Z}_{2} in terms of the isomorphism class of A_{\theta} .

Subhomogeneous Operator Systems and Classification of Operator Systems Generated by Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}-Commuting Unitaries

A unital C∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^*$$\\end{document}-algebra is called N-subhomogeneous if its irreducible representations are finite dimensional with dimension at most N. We extend this notion to operator systems, replacing irreducible representations by boundary representations. This is done by considering UCP(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {UCP }(\\mathcal {S})$$\\end{document} which is the matrix state space associated with an operator system S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {S}$$\\end{document} and identifying the boundary representations as absolute matrix extreme points. We show that two N-subhomogeneous operator systems are completely order equivalent if and only if they are N-order equivalent. Moreover, we show that a unital N-positive map into a finite dimensional N-subhomogeneous operator system is completely positive. We apply these tools to classify pairs of q-commuting unitaries up to ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}-isomorphism. Similar results are obtained for operator systems related to higher dimensional non-commutative tori.

Spectral Torsion

We introduce a trilinear functional of differential one-forms for a finitely summable regular spectral triple with a noncommutative residue. We demonstrate that for a canonical spectral triple over a closed spin manifold it recovers the torsion of the linear connection. We examine several spectral triples, including Hodge-de Rham, Einstein-Yang-Mills, almost-commutative two-sheeted space, conformally rescaled noncommutative tori, and quantum SU(2) group, showing that the third one has a nonvanishing torsion if nontrivially coupled.

Noncommutative Riemannian geometry of Kronecker algebras

We study aspects of noncommutative Riemannian geometry of the path algebra arising from the Kronecker quiver with N arrows. To start with, the framework of derivation based differential calculi is recalled together with a discussion on metrics and bimodule connections compatible with the ⁎-structure of the algebra. As an illustration, these concepts are applied to the noncommutative torus where examples of torsion free and metric (Levi-Civita) connections are given.In the main part of the paper, noncommutative geometric aspects of (generalized) Kronecker algebras are considered. The structure of derivations and differential calculi is explored, and torsion free bimodule connections are studied together with their compatibility with hermitian forms, playing the role of metrics on the module of differential forms. Moreover, for several different choices of Lie algebras of derivations, non-trivial Levi-Civita connections are constructed.

Spectral geometry of functional metrics on noncommutative tori

Spectral geometry of functional metrics on noncommutative tori

L-functions of noncommutative tori

L-functions of noncommutative tori

Asymptotics of singular values for quantised derivatives on noncommutative tori

We obtain Weyl type asymptotics for pseudodifferential operators on quantum torus Tθd of the form TIθ−1. Here T stands for classical pseudodifferential operators on quantum torus of order 0, and Iθ−1 is the Riesz potential of order −1. As an application, we obtain Weyl type asymptotics for the quantised derivative ▪ of an operator x from the homogeneous Sobolev space W˙d1(Tθd). The asymptotic coefficient is equivalent to the norm of ▪ in the principal ideal Ld,∞, as well as the norm of x in W˙d1(Tθd). We precise and rectify earlier results in our previous paper (Comm. Math. Phys. 371 (2019), no. 3, 1231–1260) on quantum integration formula for ▪.

Tracing projective modules over noncommutative orbifolds

For an action of a finite cyclic group F on an n -dimensional noncommutative torus A_\theta , we give sufficient conditions when the fundamental projective modules over A_\theta , which determine the range of the canonical trace on A_\theta , extend to projective modules over the crossed product C^* -algebra A_\theta \rtimes F . Our results allow us to understand the range of the canonical trace on A_\theta \rtimes F , and determine it completely for several examples including the crossed products of 2-dimensional noncommutative tori with finite cyclic groups and the flip action of \mathbb Z_2 on any n -dimensional noncommutative torus. As an application, for the flip action of \mathbb Z_2 on a simple n -dimensional torus A_\theta , we determine the Morita equivalence class of A_\theta \rtimes \mathbb Z_2, in terms of the Morita equivalence class of $A_\theta.

Elliptic curves and continued fractions

It is proved that the rank of an elliptic curve is one less the arithmetic complexity of the corresponding non-commutative torus. As an illustration, we consider a family of elliptic curves with complex multiplication.

Gap labeling theorem for multilayer thin film heterostructures

Quasiperiodic systems show a universal gap structure due to quasiperiodicity which is analogous to gap openings at the Brillouin zone boundary in periodic systems. The integrated density of states (IDoS) below those energy gaps are characterized by a few integers, which is known as the ``gap labeling theorem'' (GLT) for quasiperiodic systems. In this study, focusing on multilayer thin film systems such as twisted bilayer graphene and stacked transition metal dichalcogenides, we extend the GLT for multilayer systems of arbitrary dimensions and number of layers, using an approach based on the algebra called ``a noncommutative torus''. We find that the energy gaps and the associated IDoS are generally characterized by $_{DN}C_D$ integer labels in $N$ layer systems in the $D$ dimensions, when the effect of the interlayer coupling can be approximated by a quasiperiodic intralayer coupling for each layer. We demonstrate that the generalized GLT holds for quasiperiodic 1D tight binding models by numerical simulations.

Arithmetic complexity revisited

The arithmetic complexity counts the number of algebraically independent entries in the periodic continued fraction $\theta=[b_1,\dots, b_N, \overline{a_1,\dots,a_k}]$. If $\mathscr{A}_{\theta}$ is a noncommutative torus corresponding to the rational elliptic curve $\mathscr{E}(K)$, then the rank of $\mathscr{E}(K)$ is given by a simple formula $r(\mathscr{E}(K))= c(\mathscr{A}_{\theta})-1$, where $c(\mathscr{A}_{\theta})$ is the arithmetic complexity of $\theta$. We prove that $c(\mathscr{A}_{\theta})$ is equal to the dimension of the Brock-Elkies-Jordan variety of $\theta$ introduced in [1]. Following Zagier and Lemmermeyer, we evaluate the Shafarevich-Tate group of $\mathscr{E}(K)$.

Dixmier trace formulas and negative eigenvalues of Schrödinger operators on curved noncommutative tori

In a previous paper we established Cwikel-type estimates on noncommutative tori and used them to get analogues in this setting of the Cwikel-Lieb-Rozenblum (CLR) and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators. In this paper, we focus on “curved” NC tori, where the role of the usual Laplacian is played by Laplace-Beltrami operators associated with arbitrary Riemannian metrics. The Cwikel-type estimates of our previous paper are extended to pseudodifferential operators and powers of Laplace-Beltrami operators. There are several applications of these estimates. First, we get Lp-versions of the usual formula for the trace of ΨDOs on NC tori, i.e., for combinations of ΨDOs with Lp-position operators. Next, we get Lp-versions of the analogues for NC tori Connes' trace theorem and Connes' integration formula. They give formulas for the NC integrals (a.k.a. Dixmier traces) of products of Lp-position operators with ΨDOs or powers of the Laplace-Beltrami operators. Moreover, by combining our Cwikel-type estimates with suitable versions of the Birman-Schwinger principle we get versions of the CLR and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators associated with powers of Laplace-Beltrami operators and Lp-potentials. As in the original Euclidean case the Lieb-Thirring inequalities imply a dual Sobolev inequality for orthonormal families. Finally, we discuss spectral asymptotics and semiclassical Weyl's laws for our classes of operators on curved NC tori. This supersedes a previous conjecture of [63].

Quantum geodesics on quantum Minkowski spacetime

We apply a recent formalism of quantum geodesics to the well-known quantum Minkowski spacetime [x i , t] = ıλ p x i with its flat quantum metric as a model of quantum gravity effects, with λ p the Planck scale. As examples, quantum geodesic flow of a plane wave gets an order λ p frequency dependent correction to the classical geodesic velocity. A quantum geodesic flow with classical velocity v of a Gaussian with width initially centred at the origin changes its shape but its centre of mass moves with , an order correction. This implies, at least within perturbation theory, that a ‘point particle’ cannot be modelled as an infinitely sharp Gaussian due to quantum gravity corrections. For contrast, we also look at quantum geodesics on the noncommutative torus with a 2D curved weak quantum Levi-Civita connection.

A calculus for magnetic pseudodifferential super operators

This work develops a magnetic pseudodifferential calculus for super operators OpA(F); these map operators onto operators (as opposed to Lp functions onto Lq functions). Here, F could be a tempered distribution or a Hörmander symbol. An important example is Liouville super operators L̂=−iopA(h),⋅ defined in terms of a magnetic pseudodifferential operator opA(h). Our work combines ideas from the magnetic Weyl calculus developed by Măntoiu and Purice [J. Math. Phys. 45, 1394–1417 (2004)]; Iftimie, Măntoiu, and Purice [Publ. Res. Inst. Math. Sci. 43, 585–623 (2007)]; and Lein (Ph.D. thesis, 2011) and the pseudodifferential calculus on the non-commutative torus from the work of Ha, Lee, and Ponge [Int. J. Math. 30, 1950033 (2019)]. Thus, our calculus is inherently gauge-covariant, which means that all essential properties of OpA(F) are determined by properties of the magnetic field B = dA rather than the vector potential A. There are conceptual differences to ordinary pseudodifferential theory. For example, in addition to an analog of the (magnetic) Weyl product that emulates the composition of two magnetic pseudodifferential super operators on the level of functions, the so-called semi-super product describes the action of a pseudodifferential super operator on a pseudodifferential operator.

Lie bi-algebras on the non-commutative torus

We use the canonical trace on the non-commutative torus to construct a Lie bi-algebra splitting of the algebra of its smooth functions. In the special case of rational parameter (the algebra is then generated by clock and shift matrices) this Lie bi-algebra is , corresponding to unitary and upper triangular matrices. The Lie bi-algebra has a remnant in the classical limit N → ∞: the elements of tend to real functions while tends to a space of complex analytic functions. The limit results into an infinite dimensional Lie bi-algebra for the smooth functions on the commutative torus.

Appendix to: C. Farsi and N. Watling, symmetrized noncommutative tori. Math. Ann. 296 (1993), no. 4, 739–741

Appendix to: C. Farsi and N. Watling, symmetrized noncommutative tori. Math. Ann. 296 (1993), no. 4, 739–741

Local index formulae on noncommutative orbifolds and equivariant zeta functions for the affine metaplectic group

We consider the algebra A of bounded operators on L2(Rn) generated by quantizations of isometric affine canonical transformations. The algebra A includes as subalgebras noncommutative tori of all dimensions and toric orbifolds. We define the spectral triple (A,H,D) with H=L2(Rn,Λ(Rn)) 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 function ζB(z)=Tr(B|D|−2z) has a meromorphic extension to C with at most simple poles. Our main result then is an explicit algebraic expression for the Connes-Moscovici cocycle. As a corollary we obtain local index formulae for noncommutative tori and toric orbifolds.