Appell Functions for General Lattices

2025/10/11 by Chattopadhyaya, Aradhita, Manschot, Jan
#FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2510.10204

Abstract

We study Appell functions associated to an arbitrary positive definite lattice Λ and a choice of M≤ \rm dim(Λ) linearly independent vectors d_r∈ Λ, r=1,…,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.

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Related

• On the discrete analogues of Appell function F₁
• Appell polynomials and their relatives
• On the explicit construction of higher deformations of partition statistics
• Dual n₁-Appell-like Systems in Infinite-Dimensional Analysis
• Short proofs of Ramanujan-like identities for the eighth order mock theta function V₀(q)