Indefinite theta series and generalised error functions

Theta series for lattices with indefinite signature ( n + ,n ? ) arise in many areas of mathematics including representation theory and enumerative algebraic geometry. Their mod- ular properties are well understood in the Lorentzian case ( n + = 1), but have remained obscure when n + ? 2. Using a higher-dimensional generalization of the usual (complementary) error function, discovered in an independent physics project, we construct the modular completion of a class of ?conformal? holomorphic theta series ( n + = 2). As an application, we determine the modular properties of a generalized Appell-Lerch sum attached to the lattice A 2 , which arose in the study of rank 3 vector bundles on P 2 . The extension of our method to n + > 2 is outlined