$r$-Tuple Error Functions and Indefinite Theta Series of Higher-Depth

TL;DR

This paper introduces higher-dimensional $r$-tuple error functions to construct and complete indefinite theta series of higher signature lattices into modular forms, extending previous work on lower-dimensional cases.

Contribution

It generalizes the concept of double error functions to $r$-tuple error functions, enabling the construction of indefinite theta series for higher signature lattices with modular properties.

Findings

Defined properties of higher-dimensional $r$-tuple error functions.

Constructed indefinite theta series for signature $(r,n-r)$ lattices.

Showed these series can be completed to modular forms using $r$-tuple error functions.

Abstract

Theta functions for definite signature lattices constitute a rich source of modular forms. A natural question is then their generalization to indefinite signature lattices. One way to ensure a convergent theta series while keeping the holomorphicity property of definite signature theta series is to restrict the sum over lattice points to a proper subset. Although such series do not have the modular properties that a definite signature theta function has, as shown by Zwegers for signature $(1,n-1)$ lattices, they can be completed to a function that has these modular properties by compromising on the holomorphicity property in a certain way. This construction has recently been generalized to signature $(2,n-2)$ lattices by Alexandrov, Banerjee, Manschot, and Pioline. A crucial ingredient in this work is the notion of double error functions which naturally lends itself to generalizations to higher dimensions. In this work we study the properties of such higher dimensional error functions which we will call $r$-tuple error functions. We then construct an indefinite theta series for signature $(r,n-r)$ lattices and show they can be completed to modular forms by using these $r$-tuple error functions.