Modular generalized Nahm sums with arbitrary rank $r$

TL;DR

This work extends Nahm's modularity program to generalized Nahm sums of arbitrary rank $r\ge 2$ with symmetrizers $D=\mathrm{diag}(2,\dots,2,1)$ and related indices. It develops Bailey-lemma based machinery to prove four families of multi-sum Rogers--Ramanujan type identities, which feed into modularity results for the associated generalized Nahm sums. The authors express these sums as generalized eta-quotients and verify modularity on explicit congruence subgroups, ultimately constructing vector-valued automorphic forms with Langlands-dual transformation properties. The results connect partition identities, affine algebras, and automorphic forms, providing new modular examples at arbitrary rank and revealing dualities among the associated vector-valued functions.

Abstract

In this paper, we construct two families of generalized Nahm sums of arbitrary rank $r\geq 2$ with the symmetrizers ${\rm diag} ({2,\ldots, 2},1)_{r\times r}$. Specifically, the cases corresponding to $r = 2$ and $r = 3$ of these two families have been previously demonstrated by Mizuno, Warnaar, and B. Wang-L. Wang. Additionally, we establish a family of Rogers-Ramanujan type identities associated with the index $({1,\ldots, 1},2)_{r\times r}$ for any rank $r\geq 2$. Building upon these three families, combined with another family of generalized Nahm sums (with the symmetrizers ${\rm diag} ({1,\ldots, 1},2)_{r\times r}$) established by B. Wang and L. Wang, we construct two vector-valued automorphic forms, one of which is a vector-valued modular function when $r$ is odd.

Theorems & Definitions (52)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 1.11