Weyl's Law and Connes' Trace Theorem for Noncommutative Two Tori
Farzad Fathizadeh, Masoud Khalkhali
备注：
17 pages. Derivation of small time heat kernel expansion and the construction of the Dixmier trace is expanded, one reference added, to appear in Letters in Mathematical Physics
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 $\mathbb{T}_θ^2$.
We consider the Dirac operator of a general metric in the canonical conformal class on the noncommutative two torus, twisted by an idempotent (representing the $K$-theory class of a general noncommutative vector bundle), and derive a local formula for the Fredholm index of the twisted Dirac operator. Our approach is based on the McKean-Singer index formula, and explicit heat expansion calculations by making use of Connes' pseudodifferential calculus. As a technical tool, a new rearrangement lemma is proved to handle challenges posed by the noncommutativity of the algebra and the presence of an…
We compute the Ricci curvature of a curved noncommutative three torus. The computation is done both for conformal and non-conformal perturbations of the flat metric. To perturb the flat metric, the standard volume form on the noncommutative three torus is perturbed and the corresponding perturbed Laplacian is analyzed. Using Connes' pseudodifferential calculus for the noncommutative tori, we explicitly compute the second term of the short time heat kernel expansion for the perturbed Laplacians on functions and on 1-forms. The Ricci curvature is defined by localizing heat traces suitably. Equiv…
We study a technical problem arising from the spectral geometry of noncommutative tori: the small time heat trace asymptotic associated to a general second order elliptic operator. We extend the rearrangement operators in the conformal case to the general setting using hypergeometric integrals over Grassmannians. The main result is the explicit formula of the second heat coefficient in terms of the coefficients. When specializing to examples in conformal case, we not only recover results in previous works but also obtain some extra functional relations whose validation provides experimental su…
For any number $h$ such that $\hbar:=h/2π$ is irrational and any skew-symmetric, non-degenerate bilinear form $σ:\mathbb{Z}^{2g}\times \mathbb{Z}^{2g} \to \mathbb{Z}$, let be $\mathcal{A}^h_{g,σ}$ be the twisted group $*$-algebra $\mathbb{C}[\mathbb{Z}^{2g}]$ and consider the ergodic group of $*$-automorphisms of $\mathcal{A}^h_{g,σ}$ induced by the action of the symplectic group Sp$(\mathbb{Z}^{2g},σ)$. We show that the only Sp$(\mathbb{Z}^{2g},σ)$-invariant state on $\mathcal{A}^h_{g,σ}$ is the trace state $τ$.
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 introduce and study several new topological operads that should be regarded as nonsymmetric analogues of the operads of little 2-disks, framed little 2-disks, and Deligne-Mumford compactifications of moduli spaces of genus zero curves with marked points. These operads exhibit all the remarkable algebraic and geometric features that their classical analogues possess; in particular, it is possible to define a noncommutative analogue of the notion of cohomological field theory with similar Givental-type symmetries. This relies on rich geometry of the analogues of the Deligne-Mumford spaces, co…
From quantum Ore extensions to quantum tori via noncommutative UFDs
所有产生于Kac-Moody代数的量子化泛包络代数的迭代斜多项式扩张是一个非常大的公理定义的代数类的特例，称为CGL扩张。为了在代数R上构造量子团簇代数结构的初始团簇，以及对R的自同构进行分类，需要将R嵌入到量子环面T中，且嵌入具有R包含相应的量子仿射空间代数A的性质.我们显式地构造这样一个嵌入A \subseteq R \subset T的每个CGL扩展R使用的方法的非交换诺特唯一因子分解域和运行一个Gelfand-Tsetlin型程序正常，而不是中心元素。一路上，我们分类的齐次素元的所有CGL扩展，我们证明了每个CGL扩展R有一个相关的极大环面，它涵盖了自同构的R对应的所有正常的元素。对于对称CGL扩张，我们描述了量子仿射空间代数A与Cauchon的量子仿射空间代数A之间的关系.
As the second part of the sequel, we investigate the variation of rearrangement operators (more precisely, the spectral functions behind) arising in the study of modular geometry on noncommutative (two) tori. We initiate a systematic approach by introducing transformations corresponding to basic operations in calculus, like differentiation and integration by parts. As for applications, we extend, in a uniform way, the Connes-Moscovici's functional relations on noncommutative two tori attached to the variation of second heat coefficients to noncommutative tori of arbitrary dimension. Moreover, …
We give a definition of noncommutative finite-dimensional Euclidean spaces $\mathbb R^n$. We then remind our definition of noncommutative products of Euclidean spaces $\mathbb R^{N_1}$ and $\mathbb R^{N_2}$ which produces noncommutative Euclidean spaces $\mathbb R^{N_1+N_2}$. We solve completely the conditions defining the noncommutative products of the Euclidean spaces $\mathbb R^{N_1}$ and $\mathbb R^{N_2}$ and prove that the corresponding noncommutative unit spheres $S^{N_1+N_2-1}$ are noncommutative spherical manifolds. We then apply these concepts to define "noncommutative" quaternionic p…
