On the convergence of certain indefinite theta series

We study certain indefinite theta series associated to a finite set of vectors in an inner product space of signature $(n,2)$. In particular, under certain incidence conditions, we show in an elementary way the convergence of these theta series, thus proving a conjecture of Alexandrov, Banerjee, Manschot and Pioline. Moreover, we show that these conditions are necessary for the convergence.