Modular Nahm sums for symmetrizable matrices of indices $({2,\ldots, 2},1)$ and $({1,\ldots, 1},2)$

TL;DR

This paper introduces three new families of modular Nahm sums for symmetrizable matrices with specific index structures, extending previous work and constructing associated automorphic forms, including vector-valued modular functions for odd ranks.

Contribution

The paper develops three families of modular Nahm sums for arbitrary rank symmetrizable matrices with specific index patterns, expanding known cases and constructing related automorphic forms.

Findings

Constructed three families of modular Nahm sums for arbitrary rank.

Extended previous results for ranks 2 and 3 to higher ranks.

Built vector-valued automorphic forms, including modular functions for odd ranks.

Abstract

In this paper, we present three families of modular Nahm sums for symmetrizable matrices with arbitrary rank $r\geq 2$ of indices $({2,\ldots, 2},1)$ and $({1,\ldots, 1},2)$. Specifically, the cases corresponding to $r = 2$ and $r = 3$ of these families have been previously demonstrated by Mizuno, Warnaar, and B. Wang-L. Wang. Building upon these three families, we construct two vector-valued automorphic forms, one of which is a vector-valued modular function when $r$ is odd.