Dynkin diagrams, generalized Nahm sums and 2d CFTs

TL;DR

This paper extends the conjecture that Nahm sums from Dynkin diagrams are modular functions to a broader class, linking them to characters of 2d rational conformal field theories and identifying specific cases with known models.

Contribution

It generalizes the folklore conjecture to new Dynkin diagram types and connects generalized Nahm sums to well-studied 2d CFT characters, providing explicit examples.

Findings

Generalized Nahm sums for certain Dynkin diagram pairs match characters of specific 2d CFTs.

Identified cases where Nahm sums correspond to supersymmetric Virasoro minimal models.

Extended the modularity conjecture to a wider class of diagrams.

Abstract

A folklore conjecture states that the Nahm sum associated with a pair of Dynkin diagrams of type $ADET$ is a modular function. In this paper, we extend this conjecture to Dynkin diagrams of type $ABCDEFGT$ in the context of generalized Nahm sums. The modular Nahm sums are closely related to the characters of 2d rational conformal field theories. In this work, we identify many specific generalized Nahm sums with characters of some well-studied 2d CFTs. For example, we find that the generalized Nahm sums associated with $(T_1, C_r)$ and $(T_1,D_r)$ correspond to the supersymmetric Virasoro minimal models $\mathrm{SM}(4r+6, 4)$ and $\mathrm{SM}(8r+4, 2)$, respectively.