Indefinite theta series and generalized error functions

TL;DR

This paper develops a higher-dimensional error function to construct modular completions of indefinite theta series with signature (2,n-2), advancing understanding of their modular properties beyond the Lorentzian case.

Contribution

It introduces a new higher-dimensional error function and applies it to establish modularity of certain indefinite theta series with signature (2,n-2).

Findings

Constructed modular completions for indefinite theta series with signature (2,n-2).

Determined modular properties of a generalized Appell-Lerch sum for lattice A_2.

Outlined extension of the method to higher signatures.

Abstract

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 $n_+\geq 2$. 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 $\mathbb{P}^2$. The extension of our method to $n_+>2$ is outlined.