  Theta series for indefinite quadratic lattices were introduced by Zwegers,
for signature (m-1,1), Alexandrov, Banerjee, Manschot and Pioline, for
signature (m-2,2), and Nazaroglu, for signature (m-q,q). These series are
modular modular completions, defined by means of generalized error functions,
of certain non-modular holomorphic generating series associated to lattice
vectors in positive cones. We show that these modular forms arise as integrals
of the theta forms, defined in work of Millson and the second author, over
certain singular q-cubes. We also give an explicit formula for the integrals of
such forms over singular q-simplices. The sign function occurring in the
holomorphic generating series arises as in intersection number of the singular
q-cube or q-simplex with a totally geodesic subsymmetric space of codimension
q. The cubical case for q=2 was treated in [11].
