Quasimodularity of q-series associated to root lattices
Abstract
Motivated by conjectures arising in part from the study of class-š¯’® vertex algebras associated with four-dimensional supersymmetric field theories, we study multiple q-zeta functions attached to finite-dimensional simple Lie algebras. We prove their conjectured quasimodularity for all classical Lie algebras of types A, B, C, and D, as well as a more general Weyl-symmetrized version in which the exponents attached to the positive roots are allowed to vary, under a suitable parity condition. Our proof is based on a substantially more general result. We introduce new combinatorial objects, called sets of level one and pairs of level two, which generalize the configurations arising from classical root systems. To these we associate multivariable lattice sums and prove that, under a natural admissibility condition, they are Fourier expansions of polar quasi-Jacobi forms of level one and level two, respectively. Quasimodularity of the Lie-theoretic q-series follows by specialization of the elliptic variables. The positive roots of type A form a set of level one, and those of types B, C, and D are described by pairs of level two. The proof combines a structural analysis of these new combinatorial configurations with reduction procedures and induction on the rank. The resulting framework applies to a much wider class of lattice sums than those coming from root systems and supports more general quasimodularity conjectures for q-series associated with integral hyperplane arrangements.
Related
- The Borisov--Nuer conjecture and root decompositions in even unimodular Lorentzian lattices
- Petersson-Rigid Lattices in a Census of 100 Rank-Three Root Bases
- Boundary and Interior of Root Sets of Power Series
- Qin's quasimodularity conjecture for Hilbert schemes of points
- Root of the generic cubic as a power series in the discriminant
BibTeX
@misc{milas2026quasimodularity,
title = {{Quasimodularity of $q$-series associated to root lattices}},
author = {Antun Milas and Jan-Willem M. van Ittersum},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40034},
eprint = {2610.40034},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40034},
}
RIS
TY - UNPB
TI - Quasimodularity of q-series associated to root lattices
AU - Antun Milas
AU - Jan-Willem M. van Ittersum
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40034
UR - https://arxiv.org/abs/2610.40034
AB - Motivated by conjectures arising in part from the study of class-š¯’® vertex algebras associated with four-dimensional supersymmetric field theories, we study multiple q-zeta functions attached to finite-dimensional simple Lie algebras. We prove their conjectured quasimodularity for all classical Lie algebras of types A, B, C, and D, as well as a more general Weyl-symmetrized version in which the exponents attached to the positive roots are allowed to vary, under a suitable parity condition. Our proof is based on a substantially more general result. We introduce new combinatorial objects, called sets of level one and pairs of level two, which generalize the configurations arising from classical root systems. To these we associate multivariable lattice sums and prove that, under a natural admissibility condition, they are Fourier expansions of polar quasi-Jacobi forms of level one and level two, respectively. Quasimodularity of the Lie-theoretic q-series follows by specialization of the elliptic variables. The positive roots of type A form a set of level one, and those of types B, C, and D are described by pairs of level two. The proof combines a structural analysis of these new combinatorial configurations with reduction procedures and induction on the rank. The resulting framework applies to a much wider class of lattice sums than those coming from root systems and supports more general quasimodularity conjectures for q-series associated with integral hyperplane arrangements.
ER -