  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 idempotent
in the calculations in addition to a conformal factor.
