Chern Characters for Twisted Matrix Factorizations and the Vanishing of the Higher Herbrand Difference

We develop a theory of ``ad hoc Chern characters for twisted matrix factorizations associated to a scheme $X$, a line bundle ${mathcal L}$, and a regular global section $W in Gamma(X, {mathcal L})$. As an application, we establish the vanishing, in certain cases, of $h_c^R(M,N)$, the higher Herbrand difference, and, $eta_c^R(M,N)$, the higher codimensional analogue of Hochsters theta pairing, where $R$ is a complete intersection of codimension $c$ with isolated singularities and $M$ and $N$ are finitely generated $R$-modules. Specifically, we prove such vanishing if $R = Q/(f_1, dots, f_c)$ has only isolated singularities, $Q$ is a smooth $k$-algebra, $k$ is a field of characteristic $0$, the $f_i$s form a regular sequence, and $c geq 2$.