Geometric conditions on spectral triples

We begin by listing a set of requirements on a spectral triple $(\sA, \sH, D)$, whose algebra $\sA$ is unital but not necessarily commutative, such that $(\sA, \sH, D)$ provides a ``spin geometry'' generalization of our ``standard commutative example'' $(\Coo(M), L^2(M,S), \Dslash)$. Again we shall assume, for convenience, that $D$ is invertible.

In many examples, including the noncommutative examples we shall meet in the next two sections, one can often take $b_j^0 = 1$, so that $\cc$ may be replaced, for convenience, by the cycle $\sum_j a_j^0 \ox a_j^1 \oxyox a_j^n \in Z_n(\sA, \sA)$. In the commutative case, where $\sA^\opp = \sA$, this identification may be justified: the product map $m\: \sA \ox \sA \to \sA$ is a homomorphism.

The data set $(\sA, \sH, D; \Ga \text{ or } 1, J, \cc)$ satisfying these six conditions constitute a ``noncommutative spin geometry''. In the fundamental paper where these conditions were first laid out [Con2], Connes added one more nondegeneracy condition (Poincaré duality in $K$-theory) as a requirement. We shall not go into this matter here.

To understand the orientation condition in the standard commutative example, we show that $\cc$ arises from a volume form on the oriented compact manifold $M$. Choose a metric $g$ on $M$ and let $\nu_g$ be the corresponding Riemannian volume form. Furthermore, let $\{(U_j,a_j)\}$ be a finite atlas of charts on $M$, where $a_j\: U_j \to \bR^n$, and let $\{f_j\}$ be a partition of unity subordinate to the open cover $\{U_j\}$; then for $r = 1,\dots,n$, each $f_j a_j^r$ lies in $\Coo(M)$ with $\supp(f_j a_j^r) \subset U_j$. Over each $U_j$, let $\{\th_j^1,\dots,\th_j^n\}$ be a local orthonormal basis of $1$-forms (with respect to the metric $g$). Then

$$\nu_g \bigr\vert _{U_j} = \th_j^1 \wyw \th_j^n = h_j da_j^1 \wyw da_j^n,$$

for some smooth functions $h_j \: U_j \to \bC$. We write $a_j^0 := (-i)^m f_j h_j \in \Coo(M)$, where as usual, $n = 2m$ or $n = 2m + 1$. With that notation, we get

$$(-i)^m \nu_g = (-i)^m \tsum_j f_j \bigl( \nu_g \bigr\vert _{U_j} \bigr) = \tsum_j a_j^0 da_j^1 \wyw da_j^n.$$

Now we define

$$\cc := \inv{n!} \sum_{\sg\in S_n} (-1)^\sg \sum_j a_j^0 \ox a_j^{\sg(1)} \oxyox a_j^{\sg(n)}.$$ (26)

Therefore, $\cc$ is a Hochschild $n$-cycle in $Z_n(\sA,\sA)$, for $\sA = \Coo(M)$. Its representative as a bounded operator on $\sH$ is
$$\begin{align*} \inv{n!} \sum_{\sg\in S_n} (-1)^\sg \sum_j a_j^0 [\Dslash, a_j^... ...^m c(\th_j^1) \dots c(\th_j^n) \\ &= c(\ga) = \Ga \text{ or } 1, \end{align*}$$
since $c(\ga) = \Ga$ for $n = 2m$, and $c(\ga) = 1$ for $n = 2m + 1$.

This calculation shows that the elements $a_j^1,\dots,a_j^n$ occurring in the cycle $c$ are local coordinate functions for $M$. An alternative approach would be to embed $M$ in some $\bR^N$ and take the $a_j^r$ to be some of the cartesian coordinates of $\bR^N$, regarded as functions on $M$. This is illustrated in the following example.

Consider the following element of $M_2(\sA)$, with $\sA = \Coo(\bS^2)$:

$$p := \frac{1}{2} \twobytwo{1 + z}{x + iy}{x - iy}{1 - z}.$$ (27)

Note that $\tr(p - \half) = 0$, where $\tr(a) := a_{11} + a_{22}$ means the matrix trace $\tr \: M_2(\sA) \to \sA$.

If we replace $- \frac{i}{2} \nu$ by the Hochschild $2$-cycle $\cc$, the same calculation that solves the previous exercise also shows that $\pi_D(\cc) = \Ga$.

This computation has a deeper significance. One can show that the left $\sA$-module $M_2(\sA) p$ is isomorphic to $\sE_1$ in our classification of $\sA$-modules of sections of line bundles over $\bS^2$; and we have seen in Section  that $\sE_1 \isom \Ga(\bS^2,L)$ where $L \to \bS^2$ is the tautological line bundle. The first Chern class $c_1(L)$ equals (a standard multiple of) $[\nu] \in H_\dR^2(\bS^2)$. One can trace a parallel relation between spin$^c$ structures on $\bS^2$ defined, via the principal $U(1)$-bundle $SU(2) \to \bS^2$, on associated line bundles, and the Chern classes of each such line bundle. For that, we refer to [BHMS].