Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rogers--Ramanujan Type Identities for Rank Two Partial Nahm Sums

Liuquan Wang, Wentao Zeng

TL;DR

The paper extends Nahm sums to lattice cosets f_{A,B,C,v+L}(q) and identifies 14 rank-two modular partial Nahm sums on three specific lattices, establishing modularity through Rogers–Ramanujan type identities and eta-product evaluations. By analyzing matrices A with negative, zero, and positive determinant, it develops a unified framework using F(u,v;q) and parity decompositions to convert complex double sums into modular product forms. It also provides extensive explicit identities, both in general α-parameter families and in numerous concrete parameter choices, illustrating how lattice coset restrictions yield modular objects even when A is not positive definite. The results contribute to the understanding of modularity in generalized Nahm sums and hint at deeper connections to Bailey pairs, conformal field theory, and Lie algebra representations. The work also highlights open directions, such as extending classifications beyond rank two, clarifying weight phenomena, and seeking algebraic interpretations of lattice-coset Nahm sums.

Abstract

Let $A$ be a $r\times r$ rational nonzero symmetric matrix, $B$ a rational column vector, $C$ a rational scalar. For any integer lattice $L$ and vector $v$ of $\mathbb{Z}^r$, we define Nahm sum on the lattice coset $v+L\in \mathbb{Z}^r/L$: \begin{align*}\label{eq-lattice-sum} f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^\mathrm{T} \in v+L} \frac{q^{\frac{1}{2}n^\mathrm{T} An+n^\mathrm{T} B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}}. \end{align*} If $L$ is a full rank lattice and a proper subset of $\mathbb{Z}^r$, then we call $f_{A,B,C,v+L}(q)$ a rank $r$ partial Nahm sum. When the rank $r=1$, we find eight modular partial Nahm sums using some known identities. When the rank $r=2$ and $L$ is one of the lattices $\mathbb{Z}(2,0)+\mathbb{Z}(0,1)$, $\mathbb{Z}(1,0)+\mathbb{Z}(0,2)$ or $\mathbb{Z}(2,0)+\mathbb{Z}(0,2)$, we find 14 types of symmetric matrices $A$ such that there exist vectors $B,v$ and scalars $C$ so that the partial Nahm sum $f_{A,B,C,v+L}(q)$ is modular. We establish Rogers--Ramanujan type identities for the corresponding partial Nahm sums which prove their modularity.

Rogers--Ramanujan Type Identities for Rank Two Partial Nahm Sums

TL;DR

The paper extends Nahm sums to lattice cosets f_{A,B,C,v+L}(q) and identifies 14 rank-two modular partial Nahm sums on three specific lattices, establishing modularity through Rogers–Ramanujan type identities and eta-product evaluations. By analyzing matrices A with negative, zero, and positive determinant, it develops a unified framework using F(u,v;q) and parity decompositions to convert complex double sums into modular product forms. It also provides extensive explicit identities, both in general α-parameter families and in numerous concrete parameter choices, illustrating how lattice coset restrictions yield modular objects even when A is not positive definite. The results contribute to the understanding of modularity in generalized Nahm sums and hint at deeper connections to Bailey pairs, conformal field theory, and Lie algebra representations. The work also highlights open directions, such as extending classifications beyond rank two, clarifying weight phenomena, and seeking algebraic interpretations of lattice-coset Nahm sums.

Abstract

Let be a rational nonzero symmetric matrix, a rational column vector, a rational scalar. For any integer lattice and vector of , we define Nahm sum on the lattice coset : \begin{align*}\label{eq-lattice-sum} f_{A,B,C,v+L}(q):=\sum_{n=(n_1,\dots,n_r)^\mathrm{T} \in v+L} \frac{q^{\frac{1}{2}n^\mathrm{T} An+n^\mathrm{T} B+C}}{(q;q)_{n_1}\cdots (q;q)_{n_r}}. \end{align*} If is a full rank lattice and a proper subset of , then we call a rank partial Nahm sum. When the rank , we find eight modular partial Nahm sums using some known identities. When the rank and is one of the lattices , or , we find 14 types of symmetric matrices such that there exist vectors and scalars so that the partial Nahm sum is modular. We establish Rogers--Ramanujan type identities for the corresponding partial Nahm sums which prove their modularity.

Paper Structure

This paper contains 20 sections, 30 theorems, 155 equations, 14 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nahm sums for matrix with negative determinant
  4. The cases $(\alpha/2,1,0)$ and $(0,1,\alpha/2)$ with $\alpha>0$
  5. The case $(0,1/2,0)$
  6. The cases $(1/2,1/2,0)$ and $(0,1/2,1/2)$
  7. The case $(0,1,0)$
  8. Nahm sums for matrix with zero determinant
  9. The case $(1,1,1)$
  10. The case $(1/2, 1/2, 1/2)$
  11. Nahm sums for matrix with positive determinant
  12. The cases $({3}/{4},-{1}/{2}, 1)$ and $(1,-1/2,3/4)$
  13. The cases $(1, -1, 2)$ and $(2,-1,1)$
  14. The case $(1, -1/2, 1)$
  15. The cases $(3/2,1,2)$ and $(2,1,3/2)$
  16. ...and 5 more sections

Key Result

Theorem 1.2

For $v\in \{0,1\}$, the partial Nahm sum $f_{A,B,C,v+L}(q)$ are modular for $(A,B,C,v+L)$ being

Theorems & Definitions (60)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • proof
  • ...and 50 more