Appell functions for general lattices

Source abstract

Abstract We study Appell functions associated with an arbitrary positive definite lattice $$\Lambda$$ Λ and a choice of $$M\le \textrm{dim}(\Lambda )$$ M ≤ dim ( Λ ) linearly independent vectors $$d_r\in \Lambda$$ d r ∈ Λ , $$r=1,\dots ,M$$ 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 $$\Lambda$$ 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 an explicit structural formula for their modular completion. Motivated by partition functions in theoretical physics, we discuss the case where $$\Lambda$$ Λ is the $$A_N$$ A N root lattice in detail.