  Capparelli conjectured two modular identities for partitions whose parts
satisfy certain gap conditions, where were motivated by the calculation of
characters for the standard modules of certain affine Lie algebras and by
vertex operator theory. These identities were subsequently proved and refined
by Andrews, who related them to Jacobi theta functions, and also by
Alladi-Andrews-Gordon, Capparelli, and Tamba-Xie. In this paper we prove two
new companions to Capparelli's identities, where the evaluations are expressed
in terms of Jacobi theta functions and false theta functions.
