Given a commuting $d$-tuple $\bar T=(T_1,\dots,T_d)$
of otherwise arbitrary nonnormal
operators on a Hilbert space, there is an
associated Dirac operator $D_{\bar T}$.  Significant attributes
of the $d$-tuple are best expressed in terms of $D_{\bar T}$,
including the Taylor spectrum and the notion of
Fredholmness.

In fact, {\it all}
properties of $\bar T$ derive from its Dirac operator.  We
introduce a general notion of Dirac operator (in
dimension $d=1,2,\dots$) that is appropriate
for multivariable operator theory.  We show that every abstract
Dirac operator is associated with a commuting $d$-tuple, and
that two Dirac operators are isomorphic iff their associated
operator $d$-tuples are unitarily equivalent.

By relating the curvature invariant introduced in a previous paper
to the index of a Dirac
operator, we establish a stability result for
the curvature invariant  for
pure $d$-contractions of finite rank.
It is shown that for the subcategory of all such $\bar T$
which are  a) Fredholm and and b) graded, the curvature
invariant $K(\bar T)$ is stable under compact perturbations.  We
do not know if this stability persists when
$\bar T$ is Fredholm but ungraded, though there is concrete evidence that
it does.