Periodic $Y$-Systems and Nahm Sums: The Rank 2 Case

TL;DR

This paper classifies rank 2 periodic $Y$-systems with the symplectic property, explores their quantum dilogarithm identities, and connects associated Nahm sums to modular functions, confirming recent results.

Contribution

It provides a complete classification of rank 2 periodic $Y$-systems with symplectic property and links their properties to Nahm sums and modularity.

Findings

Six such $Y$-systems identified

Periodicity linked to reddening sequences and quantum dilogarithm identities

Nahm sums associated with these systems are modular functions

Abstract

We classify periodic $Y$-systems of rank 2 satisfying the symplectic property. We find that there are six such $Y$-systems. In all cases, the periodicity follows from the existence of two reddening sequences associated with the time evolution of the $Y$-systems in positive and negative directions, which gives rise to quantum dilogarithm identities associated with Donaldson-Thomas invariants. We also consider $q$-series called the Nahm sums associated with these $Y$-systems. We see that they are included in Zagier's list of rank 2 Nahm sums that are likely to be modular functions. It was recently shown by Wang that they are indeed modular functions.