Noncommutative Torus

Operator Algebra · Noncommutative Geometry · Poisson Geometry

Also called quantum torus. We’ll use the word quantum torus since it is shorter.

Idea

A $n$-dimensional quantum tori can be seen as

Speculations

  1. An $n$-dimensional noncommutative torus $A_\theta$, view as a twisted group $C^\ast$-algebra, twisted by a group 2-cocycle, is directly related to the projective representations of $\mathbb{Z}^n$.
  2. Now the problem of the classification of all irreducible representations of a Lie group can, in some sense, be reduced to the problem of the classification of all irreducible projective representations of free abelian groups. Note $\mathbb{Z}^n$ is a free abelian group, and every finitely generated free abelian group is of this form. (see doi.org/10.24033/asens.1444 )
  3. It has been proved that two quantum torus are in a same $SO(n,n;\mathbb{Z})$-orbit iff they are completely Morita equivalent. (see arXiv:math/0311502 . there are many notions of Moritq equivalence for $C^\ast$ algebras, see [[Morita equivalence]])
  4. We have a embedding $\mathfrak{so}(n,\mathbb{Z})\rightarrow O(n,n;\mathbb{Z})$, whose action on a quantum torus is an isomorphism. (also see arXiv:math/0311502 though this should be obvious)
  5. If we can construct some sort of functor $F:{\text{certain antisymmetric matrces}}\rightarrow {\text{quantum tori}}:\theta\mapsto A_\theta$, then this functor can be seen as some sort of modular form on $O(n,n;\mathbb{Z})$. - Hence it should be seen as a section of some (maybe higher, circle) bundle over some moduli stack (of antisymmetric matrices?)
  6. The embedding of $\mathfrak{so}(n,\mathbb{Z})$ to $O(n,n;\mathbb{Z})$ gives ‘periodity’ for functor $F$, namely, we have $F(e^B\theta) = (e^B)^\ast F(\theta) = F(\theta)$. This reminds one of the periodicity of a modular form that gives its Fourier development.
  7. $O(n,n;\mathbb{Z})$ is also the T-duality group. - The $\mathfrak{so}(n,\mathbb{Z})$-action is particularly simple in the T-duality picture. (see arXiv:1804.00677 )
  8. The isomorphisms of twisted $K$-theories in $T$-duality is Baum-Connes assembly.
  9. T-duality can be seen as some sort of quantization. A quantum tori is a quantization of a ordinary torus, of which the Poisson structure is given by the same antisymmetric matrice $\theta$. In T-duality it is the Mackey obstruction of $\mathbb{Z}^n = \pi_1(\mathbb{T}^n)$ comming from the twist $\delta\in H^3(E,\mathbb{Z})$ of the underlying torus bundle.

Strict deformation quantization

Let $\mathbb{T}^n$ be equipped with a constant Poisson structure $\Pi$, s.t. $\Pi_{ij} = \{ \theta_i,\theta_j \}$ where $\theta_i$ are coordinates.

Via Fourier transform one can identify $C^\infty(\mathbb{T}^n)$ with the Schwartz space $\mathcal{S}(\mathbb{Z}^n)$ of complex-valued functions on $\mathbb{Z}^n$ with rapid decay at $\infty$. The pointwise product of functions becomes the convolution $$ \hat{f}\ast \hat{g} (n) = \sum_{k\in \mathbb{Z}^n} \hat{f}(n)\hat{g}(n-k). $$

Now use the Poisson structure $\Gamma$, which is a skew-symmetric real matrix, to twist the convolution, and define on $\mathcal{S}(\mathbb{Z}^n)$ $$ \hat{f}\ast_\hbar\hat{g} (n_ = \sum_{k\in\mathbb{Z}^n}\hat{f}(n)\hat{g}(n-k)e^{-i\hbar\pi \Pi(k,n-k)}. $$ This can be pulled back to $C^\infty(\mathbb{T}^n)$ and yield a new product. Put $\hbar = 1$, this defines the algebra $A^\infty_\Pi$. A suitable completion of $A^\infty_\Pi$ yields a $C^\ast$-algebra $A_{\Pi}$, the quantum torus.

Let $SO(n,n;\mathbb{Z})$ be the subgroup of the split orthogonal group consisting of those with determinant $1$. We have a theorem about the Morita class of $A_{\Gamma}$.

Theorem. If $\Pi$ is a $(n\times n)$ skew-symmetric, $g\in SO(n,n;\mathbb{Z})$ and the action by fractional linear transformation $g\cdot \Pi$ is defined, then $A_\Pi$ and $A_{g\cdot\Pi}$ are strongly Morita equivalent.

The converse holds for $n=2$ but not in general. For smooth quantum tori, the theorem and its converse hold w.r.t. a refined notion of Morita equivalence called complete Morita equivalence.

A proof that is worth looking into: Quantization and Morita equivalence for constant Dirac structures on tori that uses Dirac structures on $\mathbb{T}^n$, and is related to foliations.

More

Harmonic analysis and number theory: