  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 Hochster's 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 $c \geq 2$.
