  We investigate higher-dimensional analogues of the $bc$ systems of 2D RCFT.
When coupled to gauge fields and Beltrami differentials defining integrable
holomorphic structures the $bc$ partition functions can be explicitly evaluated
using anomalies and holomorphy. The resulting induced actions generalize the
chiral algebras of 2D RCFT to $2n$ dimensions. Moreover, $bc$ systems in four
and six dimensions are closely related to supersymmetric matter. In particular,
we show that $d=4, \mathcal{N}=2$ hypermultiplets induce a theory of self-dual
Yang-Mills fields coupled to self-dual gravity. In this way the $bc$ systems
fermionize both the algebraic sector of the $WZW_4$ theory, as defined by Losev
et. al., and the classical open $\mathcal{N}_{ws}=2$ string.
