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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2510.10204 |
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| _version_ | 1866908587661459456 |
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| author | Chattopadhyaya, Aradhita Manschot, Jan |
| author_facet | Chattopadhyaya, Aradhita Manschot, Jan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10204 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Appell Functions for General Lattices Chattopadhyaya, Aradhita Manschot, Jan Number Theory High Energy Physics - Theory 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. |
| title | Appell Functions for General Lattices |
| topic | Number Theory High Energy Physics - Theory |
| url | https://arxiv.org/abs/2510.10204 |