Modular Nahm sums for symmetrizable matrices of indices $({2,\ldots, 2},1)$ and $({1,\ldots, 1},2)$

This paper presents three families of modular Nahm sums for symmetrizable matrices of indices $({2,\ldots, 2},1)$ and $({1,\ldots, 1},2)$ with arbitrary rank $r\geq 2$, extending previous results for low ranks and utilizing these families to construct two vector-valued automorphic forms.

The Gist

Imagine you are a master chef trying to bake the perfect cake. In the world of mathematics, specifically in a field called Number Theory, there is a famous recipe called a Nahm Sum.

Think of a Nahm Sum as a complex mathematical recipe that mixes numbers together in a specific way to create a "flavor" (a function). For a long time, mathematicians have been asking a very tricky question: "Which specific recipes (combinations of numbers) produce a flavor that is perfectly symmetrical and repeats itself in a beautiful, predictable pattern?"

When a recipe produces this perfect, repeating pattern, mathematicians call it a Modular Function. It's like a song that, no matter how you shift the key or speed it up, always sounds harmonious and fits perfectly into a larger musical structure.

The Problem: A Puzzle with Missing Pieces

For decades, mathematicians knew a few of these perfect recipes. They were like finding a few golden tickets in a factory. But they didn't know the full list, especially for complex recipes involving many ingredients (high "rank").

In this paper, the authors (Julia Du, Kathy Ji, Erin Shen, and Clara Xu) act like a team of master chefs who have discovered three new families of perfect recipes.

The Ingredients: Symmetrizable Matrices

To understand their discovery, imagine you have a grid of numbers (a matrix) that acts as the "instruction manual" for your recipe.

• Some of these manuals are perfectly balanced (symmetric).
• Others are slightly lopsided but can be "balanced out" by a special tool called a symmetrizer.

The authors focused on two specific types of lopsided manuals:

1. The "Heavy-End" Manual: Most of the weights are on one side, with a single light weight at the end (Indices: 2, 2, ..., 2, 1).
2. The "Light-End" Manual: Most weights are light, with one heavy weight at the end (Indices: 1, 1, ..., 1, 2).

They figured out that no matter how big the manual gets (as long as it has at least 2 ingredients), they can write down the exact recipe that creates a perfect, symmetrical flavor.

The Magic Trick: The "Langlands Dual" Mirror

One of the most exciting parts of their discovery is a concept they call "Langlands Duality."

Imagine you have a mirror. If you stand in front of it, you see a reflection. In math, some recipes have a "mirror twin."

• Recipe A might look complicated and messy.
• Recipe B (its twin) looks different but is secretly the same thing, just viewed from the other side of the mirror.

The authors found that for their new families of recipes, if you take the "Heavy-End" version and look at it through the mirror, it transforms perfectly into the "Light-End" version. This connection is like finding out that two different languages are actually just different ways of saying the exact same sentence.

The Result: A Vector-Valued Orchestra

The authors didn't just find single recipes; they found families of them. They bundled these recipes together into a "vector-valued automorphic form."

Think of this as an orchestra:

• Each individual recipe is a musician playing a single note.
• When played together, they form a chord.
• The authors proved that this entire orchestra plays in perfect harmony. Even if you change the tempo or the key (mathematical transformations), the whole group stays in sync.

Why Does This Matter?

You might ask, "Who cares about mathematical cake recipes?"

1. Physics Connection: These mathematical patterns appear in the real world, specifically in Quantum Physics. They describe how particles behave in certain states (like in string theory or conformal field theory). Finding new "modular" recipes helps physicists understand the fundamental laws of the universe.
2. Solving Old Mysteries: This paper solves a puzzle that was only partially solved for small numbers (rank 2 and 3) by previous mathematicians. The authors have now solved it for any size (rank $r \ge 2$).
3. New Tools: They provided a "key" (the Rogers-Ramanujan identities) that unlocks these complex sums, turning them into simpler, known products. It's like finding a shortcut through a dense forest that everyone else was walking around.

In a Nutshell

The authors of this paper have:

1. Discovered three infinite families of perfect mathematical recipes.
2. Proven that these recipes create beautiful, repeating patterns (modular functions).
3. Shown that these recipes come in "mirror twin" pairs that transform into each other.
4. Created a grand orchestra of these functions that stays in perfect harmony under any transformation.

They have expanded the map of the mathematical universe, showing us that even in the most complex, high-dimensional landscapes, there is a hidden, beautiful order waiting to be discovered.

Technical Summary

Here is a detailed technical summary of the paper "Modular Nahm Sums for Symmetrizable Matrices of Indices (2, . . . , 2, 1) and (1, . . . , 1, 2)" by Du, Ji, Shen, and Xu.

1. Problem Statement and Context

The paper addresses Nahm's problem, a famous conjecture proposed by Nahm (1979) and refined by Zagier (2007). The problem seeks to classify all triples $(A, b, c)$ of rational numbers such that the associated $q$-hypergeometric series (Nahm sum)
$$f_{A,b,c}(q) = \sum_{n \in \mathbb{N}^r} \frac{q^{\frac{1}{2}n^T A n + n^T b + c}}{(q;q)_{n_1} \cdots (q;q)_{n_r}}$$
is a modular function.

While significant progress has been made for rank $r=1$ (where exactly 7 modular triples exist) and specific cases of $r=2, 3$, a general solution for arbitrary rank $r \ge 2$ remains elusive. Furthermore, the problem has been extended to generalized Nahm sums involving symmetrizable matrices. A matrix $A$ is symmetrizable if there exists a diagonal matrix $D = \text{diag}(d_1, \dots, d_r)$ such that $AD$ is symmetric and positive definite. The generalized sum is defined as:
$$\tilde{f}_{A,b,c,d}(q) = \sum_{n \in \mathbb{N}^r} \frac{q^{\frac{1}{2}n^T AD n + n^T b + c}}{(q^{d_1}; q^{d_1})_{n_1} \cdots (q^{d_r}; q^{d_r})_{n_r}}.$$

Specific Goal: The authors aim to construct three infinite families of modular Nahm sums for symmetrizable matrices of arbitrary rank $r \ge 2$ with specific index vectors:

1. $d = (2, 2, \dots, 2, 1)$
2. $d = (1, 1, \dots, 1, 2)$

These families generalize previous results by Mizuno, Warnaar, and Wang–Wang for low ranks ($r=2, 3$).

2. Methodology

The authors employ a multi-step methodology combining combinatorial identities, the Bailey machinery, and the theory of modular forms.

A. Bailey Machinery and Rogers–Ramanujan Type Identities

The core of the proof relies on establishing multi-sum Rogers–Ramanujan type identities. The authors utilize the Bailey pair technique:

1. Selection of Base Pairs: They start with specific Bailey pairs from Slater's list (Groups C and G) and recent extensions by Wang–Wang.
2. Iterative Application: They apply Bailey's Lemma iteratively ($r-1$ or $r-2$ times) to generate new Bailey pairs with higher rank.
3. Limiting Processes: By taking limits and applying the Jacobi Triple Product Identity, they transform the resulting sums into infinite products (eta-quotients).
4. Derivation of Identities: This process yields three families of multi-sum identities (Theorems 2.1, 2.2, and 2.3) that equate complex $q$-series sums to products of generalized eta functions.

B. Connection to Modular Forms

Once the Nahm sums are expressed as sums of generalized eta-quotients, the authors apply Robins' criterion (Lemma 4.2) to verify modularity.

• They calculate the weight ($w$), the order at infinity ($\text{Ord}_\infty$), and the order at zero ($\text{Ord}_0$) for the eta-quotients.
• They determine specific scaling factors $t$ and subgroups $\Gamma_1(N)$ such that the scaled functions become modular.

C. Vector-Valued Automorphic Forms

To address the structure of the solution space, the authors construct vector-valued functions $G_{4r-2}(\tau)$ and $H_{4r-4}(\tau)$ by grouping the Nahm sums for different parameters $b_j, c_j$.

• They establish modular transformation formulas (S-transformations) between "Langlands dual" pairs of these vector functions.
• They utilize properties of Weber modular functions and theta series to derive the transformation matrices (involving cosine sums and roots of unity).

3. Key Contributions and Results

A. Three Families of Modular Nahm Sums

The paper proves that for any rank $r \ge 2$, the following families of generalized Nahm sums are modular functions for specific congruence subgroups:

1. Family 1 (Index $d=(2, \dots, 2, 1)$, Matrix $A$):

   • The sums $\tilde{f}_{A, b_j, c_j, d}(q^{32r-8})$ are modular for $\Gamma_1(128(4r-1)^2)$.
   • The matrix $A$ is a specific tridiagonal-like symmetrizable matrix.
   • This generalizes previous $r=2, 3$ cases conjectured by Mizuno.

2. Family 2 (Index $d=(2, \dots, 2, 1)$, Matrix $B$):

   • The sums $\tilde{f}_{B, b_j, c_j, d}(q^{32r-24})$ are modular for $\Gamma_1(64(4r-3)^2)$.
   • Matrix $B$ differs from $A$ in its bottom-right entry.

3. Family 3 (Index $d=(1, \dots, 1, 2)$, Matrix $A^\vee$):

   • The sums $\tilde{f}_{A^\vee, b_j, c_j, d^\vee}(q^{32r-8})$ are modular for $\Gamma_1(128(4r-1)^2)$.
   • $A^\vee$ is the transpose of $A$ (in a symmetrizable sense), representing the "Langlands dual" case.

B. Vector-Valued Modular Functions

The authors construct two vector-valued functions:

• $G_{4r-2}(\tau)$: A vector of length $4r-2$ combining the sums from Families 1 and 3.
  • Result: For odd $r$, $G_{4r-2}(\tau)$ is a vector-valued modular function for the congruence subgroup $\Gamma_0(2)$.
  • For general $r$, it is a vector-valued automorphic form for a group generated by $\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0 \\ 4 & 1 \end{pmatrix}$.
• $H_{4r-4}(\tau)$: A vector of length $4r-4$ combining sums from Family 2 and its dual.
  • Result: It is a vector-valued automorphic form for the group generated by $\begin{pmatrix} 1 & 4 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 1 & 0 \\ 8 & 1 \end{pmatrix}$.

C. New Identities

The paper provides explicit Rogers–Ramanujan type identities (Theorem 1.5) equating the multi-sums to sums of products of eta functions. These identities are non-trivial generalizations of known results and serve as the bridge between the combinatorial sums and modular forms.

4. Significance

• Resolution of Conjectures: The work confirms and generalizes several conjectures by Mizuno regarding modular Nahm sums for symmetrizable matrices of rank $r \ge 4$.
• Generalization of Rank: It moves beyond the sporadic low-rank examples ($r=2,3$) to provide infinite families of solutions for arbitrary rank, significantly expanding the known landscape of Nahm's problem.
• Symmetrizable Matrices: It advances the theory of Nahm sums from symmetric matrices to symmetrizable matrices, which are crucial in the representation theory of affine Lie algebras and partition identities (e.g., Capparelli's identities).
• Langlands Duality: The explicit construction of transformation formulas between dual pairs ($A$ and $A^\vee$) provides concrete evidence of the "Langlands dual" phenomenon in the context of Nahm sums, linking different modular structures via S-transformations.
• Vector-Valued Modularity: The construction of vector-valued modular functions for $\Gamma_0(2)$ (specifically for odd ranks) offers new examples of such forms, which are of interest in conformal field theory and the study of vertex operator algebras.

In summary, this paper provides a rigorous framework for generating infinite families of modular Nahm sums using the Bailey machinery, establishes their modularity via eta-quotients, and elucidates the vector-valued structure of these solutions, thereby deepening the understanding of the interplay between $q$-series, combinatorics, and modular forms.