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PreprintCC BY 4.0arXiv

Iterated integrals for generalized Appell functions

  • Aradhita Chattopadhyaya
  • Jan Manschot
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Abstract

Appell functions are a large class of multi-variable quasi-elliptic functions, which are also instances of higher depth mock modular forms. A central aspect of these functions is their non-holomorphic modular completion. Our previous work developed Appell functions for a general positive definite lattice Λ, and expressed the non-holomorphic completion in terms of the generalized error functions MP, which are integrals over a P-dimensional hyperplane in Λ⊗ ℂ. In this sequel, we express MP as a P-dimensional iterated integral of variables wj∈ ℍ, such that the completion of the Appell function takes the form of an iterated Eichler integral of modular forms. We apply this to the case of the root lattice of the AN Lie algebra, which is of particular interest because of their appearance in partition functions of topological twisted Yang-Mills theories.

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Cite

BibTeXRIS
BibTeX
@misc{chattopadhyaya2026iterated,
  title = {{Iterated integrals for generalized Appell functions}},
  author = {Aradhita Chattopadhyaya and Jan Manschot},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26438},
  eprint = {2609.26438},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26438},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Iterated integrals for generalized Appell functions
AU  - Aradhita Chattopadhyaya
AU  - Jan Manschot
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26438
UR  - https://arxiv.org/abs/2609.26438
AB  - Appell functions are a large class of multi-variable quasi-elliptic functions, which are also instances of higher depth mock modular forms. A central aspect of these functions is their non-holomorphic modular completion. Our previous work developed Appell functions for a general positive definite lattice Λ, and expressed the non-holomorphic completion in terms of the generalized error functions M_P, which are integrals over a P-dimensional hyperplane in Λ⊗ ℂ. In this sequel, we express M_P as a P-dimensional iterated integral of variables w_j∈ ℍ, such that the completion of the Appell function takes the form of an iterated Eichler integral of modular forms. We apply this to the case of the root lattice of the A_N Lie algebra, which is of particular interest because of their appearance in partition functions of topological twisted Yang-Mills theories.
ER  -