Modularity of Nahm Sums for the Tadpole Diagram

TL;DR

This paper proves Rogers-Ramanujan type identities for Nahm sums related to the tadpole Cartan matrix, establishing their modularity and confirming a conjecture, with implications for vector-valued modular functions.

Contribution

It demonstrates the modularity of Nahm sums for the tadpole diagram and confirms a specific conjecture, extending understanding of their mathematical structure.

Findings

Proved Rogers-Ramanujan identities for rank 3 Nahm sums.

Established the modularity of these Nahm sums.

Proposed conjectures for general rank cases.

Abstract

We prove Rogers-Ramanujan type identities for the Nahm sums associated with the tadpole Cartan matrix of rank $3$. These identities reveal the modularity of these sums, and thereby we confirm a conjecture of Penn, Calinescu and the first author in this case. We show that these Nahm sums together with some shifted sums can be combined into a vector-valued modular function on the full modular group. We also present some conjectures for a general rank.