Theta integrals and generalized error functions

Recently  Alexandrov, Banerjee, Manschot and Pioline constructed generalizations of Zwegers theta functions for lattices of signature $(n-2,2)$.  They also suggested a generalization to the case of arbitrary signature $(n-q,q)$, and this case was subsequently proved by Nazaroglu. Their functions, which depend on certain collections $\mathcal{C}$ of negative vectors, are obtained by 'completing' a non-modular holomorphic generating series
by means of a non-holomorphic theta type series involving generalized error functions.

In joint work with Jens Funke, we show that their completed modular series arises as integrals of the $q$-form valued theta functions, defined in old joint work of the author and John Millson, over a certain singular $q$-cube determined by the data $\mathcal{C}$.  This gives an alternative construction of such series and a conceptual basis for their modularity. I will discuss the simplicial case and a curious 'convexity' problem for Grassmannians that arises in this context.