Summary: Recently, Alexandrov, Banerjee, Manschot and Pioline [S. Alexandrov et al., Sel. Math., New Ser. 24, No. 5, 3927–3972 (2018; Zbl 1420.11078)] 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. The holomorphic generating series is interpreted as the series of intersection numbers of the surface \(S\) with complex divisors associated to positive lattice vectors.