Indefinite theta series and generalized 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 modular 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.