Appell Functions for General Lattices

TL;DR

The paper presents a unified framework for Appell functions attached to positive-definite lattices, establishing that these multi-variable objects are depth-$M$ mock modular forms and deriving a structural formula for their modular completion. By embedding the Appell functions into an extended indefinite theta-series setting on $\underline{\Lambda}=\Lambda\oplus(-\Lambda_d)$, it expresses the non-holomorphic completion in terms of lower-depth Appell functions and non-holomorphic iterated integrals, ensuring modular and Jacobi covariance. The authors provide explicit specialization to root lattices $A_N$, with detailed analyses of $A_2$ and $A_3$ and a general discussion for $A_N$, linking the mathematics to partition functions in topologically twisted gauge theories and BPS index computations. These results illuminate holomorphic anomalies in gauge-theory partition functions and furnish concrete computational tools for constructing modular completions of multi-variable Appell-type functions.

Abstract

We study Appell functions associated to an arbitrary positive definite lattice $Λ$ and a choice of $M\leq {\rm dim}(Λ)$ linearly independent vectors $d_r\in Λ$, $r=1,\dots,M$. These functions are instances of multi-variable quasi-elliptic functions, and specific examples have appeared at various places in mathematics and theoretical physics. For example, if $Λ$ is chosen to be one-dimensional, these functions reduce to the classical Appell function, which is a prominent example in the theory of mock modular forms. The Appell functions introduced here are examples of depth $M$ mock modular forms. We derive a structural formula for their modular completion. Motivated by partition functions in theoretical physics, we discuss the case where $Λ$ is the $A_N$ root lattice in detail.