<p>We study Appell functions associated with an arbitrary positive definite lattice <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> and a choice of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$M\le \textrm{dim}(\Lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>≤</mo> <mtext>dim</mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> linearly independent vectors <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$d_r\in \Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mi>r</mi> </msub> <mo>∈</mo> <mi mathvariant="normal">Λ</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$r=1,\dots ,M$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation>. These functions are instances of multi-variable quasi-elliptic functions, and specific examples have appeared at various places in mathematics and theoretical physics. For example, if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> is chosen to be one-dimensional, these functions reduce to the classical Appell function, which is a prominent example in the theory of mock modular forms. The Appell functions introduced here are examples of depth <i>M</i> mock modular forms. We derive an explicit structural formula for their modular completion. Motivated by partition functions in theoretical physics, we discuss the case where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> is the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$A_N$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi>N</mi> </msub> </math></EquationSource> </InlineEquation> root lattice in detail.</p>

Appell functions for general lattices

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2026年7月13日

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Aradhita Chattopadhyaya,
Jan Manschot

摘要

We study Appell functions associated with an arbitrary positive definite lattice $\Lambda $ $Λ$ and a choice of $M\le \textrm{dim}(\Lambda )$ $M \leq dim (Λ)$ linearly independent vectors $d_r\in \Lambda $ $d_{r} \in Λ$ , $r=1,\dots ,M$ $r = 1, \dots, M$ . 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 $\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.