Nahm sum identities for Cartan matrices of type $D_k$

TL;DR

This paper presents a new proof of four Nahm sum identities related to type D_k Cartan matrices using Bailey pairs, extending previous conjectures and providing a parametric generalization with multiple proofs.

Contribution

It introduces a Bailey pairs-based proof for four Nahm sum identities and establishes a parametric generalization with two different proofs.

Findings

New Bailey pairs-based proofs for four Nahm sum identities.

A parametric generalization of two identities.

Two distinct proofs for the generalization.

Abstract

Around 2007, Warnaar proved four identities related to Nahm sums associated with twice the inverse of the Cartan matrix of type $D_k$. Three of these had been conjectured by Flohr, Grabow, and Koehn, while special cases of two of the identities were first conjectured in 1993 by Kedem, Klassen, McCoy, and Melzer. Warnaar's proof relies on a multi-sum identity from Andrews' proof of the Andrews-Gordon identities. We give a new proof of all four identities using the theory of Bailey pairs. Furthermore, we establish a parametric generalization of two of the identities and provide two distinct proofs of this generalization.