Let $P\colon E \to F$ be an elliptic pseudodifferential operator over $M$. Assuming that $P$ defines a finitely summable Fredholm module, we may apply the Chern-Connes character to it to get a cyclic cocycle and this may then be mapped into the homology of $M$, i.e., we have $ch_\ast(P) \in H_\ast(M)$.
On the other hand, we may do the clutching construction using the symbol $\sigma(P)$ of $P$ to get an element $[\sigma_P] \in K_{cpt}^0(TM)$, apply the usual cohomological Chern character, integrate over the fiber, and finally multiply with the Todd genus of $M$, to get the index class $ind(P) = \pi_! ch^\ast [\sigma_P] \wedge Td(M) \in H^\ast(M)$. (I think I forgot here some sign $(-1)^?$.)
If $P$ is a graded operator (so that $E$ and $F$ are the positive and negative parts, respectively, of the vector bundle on which $P$ acts) we have $ch_\ast(P) \in H_{ev}(M)$ and Connes and Moscovici proved in Cyclic Cohomology, The Novikov Conjecture And Hyperbolic Groups that this is the Poincare dual of $ind(P)$ (up to some constants that only depend on the degree $q$ in which we compare these classes).
But if $P$ is an ungraded operator (in this case we would have $E=F$) we get $ch_\ast(P) \in H_{odd}(M)$ since now $P$ itself defines an odd $K$-homology class. But the Poincare dual of $ind(P)$ still lives in $H_{ev}(M)$, i.e., we can not compare them.
What is the correct formulation of the local index formula in the case of ungraded operators? Is it in the literature somewhere?
I expect somehow a small fix to the construction of the index class $ind(P)$ such that it lives then in the correct degrees (maybe something like taking a suitable product with $S^1$ similarly as the trick used by Connes and Moscovici in their above cited paper where they reduced from the case of odd-dimensional manifolds to even-dimensional ones).
