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Remarks on Nahm sums for symmetrizable matrices

Yuma Mizuno

TL;DR

This work extends Nahm sums to symmetrizable matrices by introducing $f_{A,b,c,d}(q)$ with $AD$ symmetric positive definite and analyzes their modularity via a Bloch-group torsion criterion. It develops a comprehensive asymptotic framework at roots of unity, employing the Nahm data, Rogers dilogarithm, and cyclic quantum dilogarithm to derive an explicit expansion that underpins the modularity analysis. The main result establishes one direction of Nahm's conjecture in the symmetrizable setting: if the Nahm sum is modular, the associated Bloch-group element $\xi_{A,d}$ is torsion, using the injectivity of the $R_\zeta$ map. Complementing the theory, the paper reports numerical searches for modular candidates in rank 2 and 3, uncovering Langlands-dual patterns and connecting to known identities such as Kanade–Russell and Capparelli, highlighting both the reach and the complexity of modularity in this generalized context.

Abstract

Nahm sums are specific $q$-hypergeometric series associated with symmetric positive definite matrices. In this paper we study Nahm sums associated with symmetrizable matrices. We show that one direction of Nahm's conjecture, which was proven by Calegari, Garoufalidis, and Zagier for the symmetric case, also holds for the symmetrizable case. This asserts that the modularity of a Nahm sum implies that a certain element in a Bloch group associated with the Nahm sum is a torsion element. During the proof, we investigate the radial asymptotics of Nahm sums. Finally, we provide lists of candidates of modular Nahm sums for symmetrizable matrices based on numerical experiments.

Remarks on Nahm sums for symmetrizable matrices

TL;DR

This work extends Nahm sums to symmetrizable matrices by introducing with symmetric positive definite and analyzes their modularity via a Bloch-group torsion criterion. It develops a comprehensive asymptotic framework at roots of unity, employing the Nahm data, Rogers dilogarithm, and cyclic quantum dilogarithm to derive an explicit expansion that underpins the modularity analysis. The main result establishes one direction of Nahm's conjecture in the symmetrizable setting: if the Nahm sum is modular, the associated Bloch-group element is torsion, using the injectivity of the map. Complementing the theory, the paper reports numerical searches for modular candidates in rank 2 and 3, uncovering Langlands-dual patterns and connecting to known identities such as Kanade–Russell and Capparelli, highlighting both the reach and the complexity of modularity in this generalized context.

Abstract

Nahm sums are specific -hypergeometric series associated with symmetric positive definite matrices. In this paper we study Nahm sums associated with symmetrizable matrices. We show that one direction of Nahm's conjecture, which was proven by Calegari, Garoufalidis, and Zagier for the symmetric case, also holds for the symmetrizable case. This asserts that the modularity of a Nahm sum implies that a certain element in a Bloch group associated with the Nahm sum is a torsion element. During the proof, we investigate the radial asymptotics of Nahm sums. Finally, we provide lists of candidates of modular Nahm sums for symmetrizable matrices based on numerical experiments.
Paper Structure (8 sections, 5 theorems, 65 equations, 1 table)

This paper contains 8 sections, 5 theorems, 65 equations, 1 table.

Table of Contents

  1. Introduction
  2. Asymptotics of Nahm sums
  3. Asymptotics at roots of unity
  4. Nahm sums and Bloch groups
  5. Bloch groups
  6. Nahm's conjecture for symmetrizable case
  7. Experiments on modular Nahm sums
  8. Cyclic quantum dilogarithm function

Key Result

Theorem 2.1

Let $Q = (A, b, c, d)$ be a quadruple as above, and $\delta$ be a strong denominator of $Q$. Suppose that $\alpha$ is a rational number whose denominator $m$ is prime to $\delta$ and $d_1, \dots, d_N$. Let $\zeta = \mathbf{e}(\alpha)$ be the corresponding $m$th root of unity. Let $\theta_i$ be the $ as $\varepsilon$ tends to $0$ from right within the real line. Here $\chi(d, \alpha) = \prod_{i=1}^

Theorems & Definitions (11)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Conjecture 4.1
  • Proposition A.1
  • ...and 1 more