Recently Alexandrov, Banerjee, Manschot and Pioline constructed generalizations of Zwegers theta functions for lattices of signature (n-2,2). Their functions, which depend on two pairs of time like vectors, are obtained by `completing' a non-modular holomorphic generating series by means of a non-holomorphic theta type series involving generalized error functions. We show that their completed modular series arises as integrals of the 2-form valued theta functions, defined in old joint work of the author and John Millson, over a surface S determined by the pairs of time like vectors. This gives an alternative construction of such series and a conceptual basis for their modularity.
If time permits, I will discuss current work with Jens Funke concerning the case of general signature (p,q).