Modularity of Some Nahm Sums as Vector-valued Functions

TL;DR

This paper proves modular transformation formulas for Nahm sums viewed as vector-valued functions, establishing their modularity on certain congruence subgroups and connecting to identities like Kanade--Russell and Capparelli's.

Contribution

It develops modular transformation formulas for Nahm sums as vector-valued functions, confirming their modularity on specific congruence subgroups and relating to notable identities.

Findings

Nahm sums are modular on $\Gamma_0(N)$ for N=1,2,3,4.

Transformation formulas for Nahm sums related to Kanade--Russell and Capparelli's identities.

Vector-valued transformation formulas for certain theta series.

Abstract

Zagier observed that modular Nahm sums associated with the same matrix may form a vector-valued modular function on some congruence subgroup. We establish modular transformation formulas for several families of Nahm sums by viewing them as vector-valued functions, and thereby we show that they are indeed modular on the congruence subgroup $\Gamma_0(N)$ with $N=1,2,3,4$. In particular, we prove two transformation formulas discovered by Mizuno related to the Kanade--Russell mod 9 conjecture and Capparelli's identities. We also establish vector-valued transformation formulas for some theta series. As applications, we give modular transformation formulas for various families of Nahm sums involving those in the Andrews--Gordon identities and Bressoud's identities.