A spectral triple on the noncommutative torus

To define a spectral triple over a noncommutative algebra, we introduce the so-called noncommutative torus. In fact, there are many such tori, labelled by a dimension $n$ and by a family of parameters $\th_{ij}$ forming a real skewsymmetric matrix $\Th = - \Th^t\in M_n(\bR)$.

Fix an integer $n \in \{2,3,4,\dots\}$. In the algebra $A_0 := C(\bT^n)$, one can write down Fourier-series expansions:

$$f(\phi_1,\dots,\phi_n) \longleftrightarrow \sum_{r\in\bZ^n... ...t_{[0,1]^n} e^{-2\pi ir\cdot\phi} f(\phi) d^n\phi \in \bC,$$

where $r\cdot\phi := r_1\phi_1 +\cdots+ r_n\phi_n$, as usual. To ensure that this series converges uniformly and represents $f(\phi)$, we retreat to the dense subalgebra $\sA_0 := \Coo(\bT^n)$, in which the coefficients $c_r$ decrease rapidly to zero as $\vert r\vert \to \infty$. On the space of multisequences $\cc := \{c_r\}_{r\in\bZ^n}$, we introduce the seminorms

$$p_k(\cc) := \biggl( \sum_{r\in\bZ^n} (1 + r\cdot r)^k \vert c_r\vert^2 \biggr)^{1/2}, \word{for all} k \in \bN.$$

We say that ``$c_r \to 0$ rapidly'' if $p_k(\cc) < \infty$ for every $k$. Notice that $p_{k+1}(\cc) \geq p_k(\cc)$ for each $k$; these seminorms induce, on rapidly decreasing sequences, the topology of a Fréchet space, which indeed coincides with the usual Fréchet topology on $\Coo(\bT^n)$, i.e., the topology of uniform convergence of the functions and of all their derivatives.

We can think of $A_0$ as the $C^*$-algebra generated by $n$ commuting unitary elements, namely the functions $u_j$ defined by $u_j(\phi_1,\dots,\phi_n) := e^{2\pi i\phi_j}$, for $j = 1,\dots,n$.

Noncommutativity appears when we choose a real skewsymmetric matrix $\Th \in M_n(\bR)$, and introduce the (universal) $C^*$-algebra $A_\Th$ generated by unitary elements $u_1,\dots,u_n$ which no longer commute: instead, they satisfy the commutation relations

$$u_k u_j = e^{2\pi i\th_{jk}} u_j u_k, \word{for} j,k = 1,\dots,n.$$

(In quantum mechanics, these are called ``Weyl's form of the canonical commutation relations''.) To form polynomials with these generators, we introduce a Weyl system of unitary elements $\set{u^r : r \in \bZ^n}$ in $A_\Th$, by defining

$$u^r := \exp\bigl\{ \pi i\tsum_{j<k} r_j\th_{jk}r_k \bigr\} u_1^{r_1} u_2^{r_2} \dots u_n^{r_n}.$$

Notice that $\sg(r,\pm r) = 1$ by skewsymmetry of $\Th$.

We now define $\sA_\Th =: \Coo(\bT^n_\Th)$ to be the dense $*$-subalgebra of $A_\Th$ consisting of elements of the form

$$a = \sum_{r\in\bZ^n} a_r u^r$$

where $a_r \in \bC$ for each $r$, and $a_r \to 0$ rapidly.

There is an action of the abelian Lie group $\bT^n$ by $*$-automorphisms on the $C^*$-algebra $A_\Th$, given by

$$z \cdot u^r := z_1^{r_1} z_2^{r_2} \dots z_n^{r_n} u^r \word{for} r \in \bZ^n,$$

or, more simply, $z \cdot u_j = z_j u_j$, where $z = (z_1,\dots,z_n) \in \bT^n$. This action is generated by a set of $n$ commuting derivations $\dl_1,\dots,\dl_n$, namely,

$$\dl_j(a) := \ddto{t} e^{2\pi it\phi_j} \cdot a,$$

whose domain is the set of all $a \in A$ for which the map $t \mapsto e^{2\pi it\phi_j} \cdot a$ is differentiable.

The result of the previous exercise shows that $\sA_\Th$ is just the ``smooth subalgebra'' of the $C^*$-algebra $A_\Th$ with respect to the action of $\bT^n$. It is known that any such smooth subalgebra, under a continuous action of a compact Lie group on a $C^*$-algebra, is actually a pre-$C^*$-algebra.

We now define $\sH_\tau$ to be the completion of $A_\Th$ in the norm

$$\Vert a\Vert _2 := \sqrt{\tau(a^*a)}.$$

We remark that $\Vert a\Vert _2 \leq \Vert a\Vert$ for all $a$, so that the inclusion map $\eta_\tau \: A_\Th \to \sH_\tau$ is continuous. It is convenient to write $\ul{a} := \eta_\tau(a)$ to denote the element $a \in A_\Th$ regarded as a vector in $\sH_\tau$. It turns out that the trace $\tau$ is faithful, so that $\sH_\tau$ is just the Hilbert space of the ``GNS representation'' $\pi_\tau$ of $A_\Th$. This representation is defined --first on $\eta_\tau(A_\Th)$, then extended by continuity-- by

$$\pi_\tau(a) : \ul{b} \mapsto \ul{ab} : \sH_\tau \to \sH_\tau, \word{for each} a \in A_\Th.$$

The analogue of the $L^2$-spinor space for the noncommutative torus is just the tensor product $\sH := \sH_\tau \ox \bC^{2^m}$, where as usual, $n = 2m$ or $n = 2m + 1$ according as $n$ is even or odd. (In the commutative case $\Th = 0$, this means that we are using the spinor module for the untwisted spin structure on $\bT^n$.) Recall that we can regard $\bC^{2^m}$ as a Fock space $\La^\8 \bC^m$, carrying an irreducible represenation of the matrix algebra $B = \bCl(\bR^n)$ if $n$ is even, or $B = \bCl^0(\bR^n)$ if $n$ is odd. In the even case, there is a $\bZ_2$-grading operator $\Ga := 1_{\sH_\tau} \ox c(\ga)$, satisfying $\Ga^2 = 1$ and $\Ga^* = \Ga$.

The charge conjugation on $B$, that we have written $b \mapsto \chi(\bar b)$, is implemented by an antiunitary operator on $\bC^{2^m}$ of the form $C_0 K$, where $K$ is complex conjugation and $C_0$ is a certain $2^m \x 2^m$ matrix: this means that $(C_0K) b (C_0K)^{-1} = \chi(\bar b)$ as operators on $\bC^{2^m}$.

For instance, if $n = 2$ or $3$, then $C_0 = i \sg^2 = \twobytwo{0}{-1}{1}{0}$.

Now let $J := J_0 \ox C_0$. This is an antiunitary operator on $\sH$, such that $J^2 = \pm 1$ according as $C_0^2 = \pm 1$.

The closure of this operator, still denoted by $\ul{\dl}_j$, is then an unbounded skewadjoint operator on $\sH$.

Let $\ga^1,\dots,\ga^n$ be the generators of the action of the Clifford algebra $\bCl(\bR^n)$ on $\bC^{2^m}$: they are a set of $2^m \x 2^m$ matrices such that $\ga^j\ga^k + \ga^k\ga^j = 2\dl^{jk}$ for $j,k = 1,\dots,n$. The operator $C_0 K$ is determined by the relations

$$(C_0K) \ga^j (C_0K)^{-1} = - \ga^j \word{for} j = 1,\dots,n.$$

We can now define the Dirac operator on $\sH$ by

$$D := -i \sum_{j=1}^n \ul{\dl}_j \ox \ga^j.$$