Dynkin diagrams, generalized Nahm sums and 2d CFTs

This paper extends the folklore conjecture on Nahm sums to generalized settings involving $ABCDEFGT$ type Dynkin diagrams, demonstrating that these sums correspond to characters of specific 2d rational conformal field theories, including supersymmetric Virasoro minimal models.

The Gist

Imagine the universe of mathematics and physics as a giant, intricate library. Inside this library, there are two very different kinds of books that, surprisingly, seem to be telling the exact same story.

The Story of the Paper

This paper, written by Kaiwen Sun and Haowu Wang, is like a detective story. The detectives are trying to prove a long-standing hunch (a "folklore conjecture") that two very different languages are actually dialects of the same tongue.

Here is the breakdown of the mystery, using simple analogies:

1. The Two Languages

• Language A: The "Nahm Sums" (The Recipe Book)
  Imagine a chef trying to bake a cake. They have a recipe that involves adding ingredients in a very specific, infinite pattern. In math, this is called a Nahm Sum. It's a complex formula that adds up infinite numbers.

  • The Mystery: Sometimes, when you follow this recipe perfectly, the result isn't just a messy number; it turns out to be a "Modular Function." Think of a modular function as a perfectly symmetrical snowflake. No matter how you rotate it or look at it, it fits together perfectly. These snowflakes are rare and special.
  • The Hunch: For a long time, mathematicians noticed that if you use a specific set of ingredients (based on shapes called Dynkin Diagrams, which look like stick-figure trees), the resulting "cake" (the sum) is always a perfect snowflake.

• Language B: The "2D CFTs" (The Physics Engine)
  On the other side of the library, physicists are studying 2D Conformal Field Theories (CFTs). Imagine these as the "operating systems" for tiny, two-dimensional universes. These systems have "characters" (like the stats of a video game character: strength, speed, magic).

  • The Connection: The paper suggests that the "perfect snowflakes" (Modular Functions) from the Recipe Book are actually just the "stats" (characters) of these tiny universes.

2. The Old Theory vs. The New Discovery

• The Old Theory (The "ADET" Club):
  Previously, it was believed that this magic only worked for a specific club of stick-figure trees called ADET (named after the letters A, D, E, T). If you used these trees, you got a perfect snowflake.
• The New Discovery (Expanding the Club):
  Sun and Wang asked: "What if we try the other trees? What about the ones with letters B, C, F, G, and T?"
  They found that the magic does work! But to make it work for these new trees, they had to tweak the recipe slightly. They introduced a "Generalized Nahm Sum."
  • The Analogy: Imagine the old recipe required you to use only standard flour. The new discovery says, "If you want to bake with whole wheat or rye (the new trees), you just need to add a little bit of baking powder (the diagonal matrix D) to the mix." Once you do that, the new trees also produce perfect snowflakes.

3. The "Folding" Trick

One of the coolest parts of the paper is a concept called Folding.

• Imagine you have a beautiful, symmetrical origami crane (a "Simply-Laced" tree like A or E).
• Now, imagine you fold that paper in half. You get a slightly different shape (a "Non-Simply-Laced" tree like B, C, or F).
• The authors discovered that the "physics" (the CFT) of the folded shape is deeply connected to the unfolded shape. It's like saying the music played by a folded accordion is just a remix of the music played by the open accordion. They proved that the "stats" of the folded system can be built by mixing the "stats" of the original system.

4. Why Does This Matter?

You might ask, "Who cares about infinite cake recipes and stick-figure trees?"

• It Unifies Math and Physics: It shows that the abstract patterns of number theory (how numbers add up) are the exact same patterns that govern the behavior of particles in the universe.
• It Solves Puzzles: There are many "puzzles" in math (identities) that people have been staring at for decades, wondering why they work. This paper says, "They work because they are describing a tiny universe!" It gives a physical reason for mathematical magic.
• It Predicts New Things: By proving that these new trees (B, C, F, G) work, the authors have generated a list of 35 new "recipes" that they believe will produce perfect snowflakes. They have already confirmed 28 of them and are hunting for the remaining 7.

Summary in a Nutshell

Sun and Wang are like explorers who found a hidden bridge between two islands.

• Island 1: A world of complex number recipes (Nahm sums).
• Island 2: A world of 2D physics universes (CFTs).

They proved that the bridge works not just for the main road (the old ADET trees), but for a whole new network of roads (the ABCDEFGT trees), provided you use a slightly different map (the generalized sum). This means the "language of the universe" is even more universal than we thought, connecting deep math to the fabric of reality in a beautiful, symmetrical way.

Technical Summary

1. Problem Statement

The paper addresses the classification and modularity of Nahm sums, which are $q$-hypergeometric series associated with specific algebraic data.

• Background: A well-known folklore conjecture in theoretical physics posits that Nahm sums associated with pairs of ADET Dynkin diagrams (where $T$ denotes the Tadpole diagram) are modular functions. These sums often correspond to the characters of rational 2-dimensional Conformal Field Theories (2D CFTs).
• The Gap: While the ADET case is well-studied, the behavior of Nahm sums for other Dynkin diagram types (specifically B, C, F, G) and the extension to generalized Nahm sums (involving non-simply-laced diagrams and symmetrizable matrices) remained an open problem.
• Core Question: Can the modularity conjecture be extended to pairs of Dynkin diagrams of types A, B, C, D, E, F, G, T within the framework of generalized Nahm sums? Furthermore, can these sums be explicitly identified with characters of known 2D CFTs?

2. Methodology

The authors employ a combination of algebraic number theory, representation theory, and computational verification:

• Generalized Nahm Sums: They utilize the definition of generalized Nahm sums introduced by Mizuno. For a quadruple $(A, B, C, D)$, where $A$ is a rational symmetric matrix, $B$ a vector, $C$ a scalar, and $D$ a diagonal matrix of positive integers (related to root lengths), the sum is defined as:
  $$f_{A,B,C,D}(\tau) = \sum_{n \in \mathbb{N}^r} \frac{q^{\frac{1}{2}n^t A D n + n^t B + C}}{(q^{d_1}; q^{d_1})_{n_1} \cdots (q^{d_r}; q^{d_r})_{n_r}}$$
• Conjectural Framework (Conjecture 1.1): The authors propose a specific construction for pairs of Dynkin diagrams $(X, Y)$:
  • Define $A(X, Y) = C(X) \otimes C(Y)^{-1}$ (Kronecker product of Cartan matrices).
  • Define $D(X, Y) = D(X) \otimes D(Y)$, where $D(X)$ encodes the ratios of squared lengths of simple roots.
  • Define the central charge $c(X, Y)$ based on the trace of $D(X, Y)$ and Coxeter numbers.
  • Hypothesis: The quadruple $(A(X, Y), 0, -c(X, Y)/24, D(X, Y))$ yields a modular generalized Nahm sum.
• Duality Analysis: They investigate the duality conjecture (Zagier/Mizuno) for these sums, noting that while general duality fails, it holds for their specific case where $B=0$, mapping the pair $(X, Y)$ to $(Y, X)$.
• CFT Identification: The authors compute the $q$-expansions of these sums to high orders and match them against known characters of rational 2D CFTs, including:
  • Virasoro minimal models ($M(p, q)$).
  • Supersymmetric minimal models ($SM(p, q)$).
  • Coset constructions (e.g., $(Y)_1 \times (Y)_1 / (Y)_2$).
  • Parafermion theories and free bosons ($U(1)_k$).
• Folding Conjecture: They propose that for non-simply-laced diagrams (folded from ADE), the associated CFT is a sub-theory (conformal embedding) of the CFT associated with the unfolded ADE diagram.

3. Key Contributions

A. Extension of the Modularity Conjecture

The paper generalizes the folklore conjecture from ADET diagrams to all Dynkin diagrams (A, B, C, D, E, F, G, T). They provide a precise formula for the matrix $A$, the diagonal matrix $D$, and the central charge $C$ for any pair $(X, Y)$.

B. Identification with 2D CFT Characters

The authors successfully identify 28 out of 35 generalized Nahm sums of rank $<4$ with characters of well-studied 2D CFTs. Key identifications include:

• Supersymmetric Models:
  • $(T_1, C_r) \to SM_{eff}(4r+6, 4)$ (Supersymmetric Virasoro minimal models).
  • $(T_1, D_r) \to SM_{eff}(8r+4, 2)$.
  • $(T_1, A_3) \to SM_{eff}(28, 2)$.
• Coset and Product Models:
  • $(A_1, C_r) \to U(1)_{4(r+1)}$ (Circle compact boson).
  • $(T_1, E_8) \to M_{eff}(11, 2)$.
  • $(E_8, T_1) \to (E_8)_1 / M_{eff}(11, 2)$ (Realized via Hecke operator action).
• Exotic Models:
  • $(A_2, A_1) \to D_2A$ (Related to the Monster group conjugacy class).
  • $(A_1, B_3) \to SM(8, 6)$.

C. Generalized Rogers-Ramanujan Identities

By equating the Nahm sums to CFT characters (which often have product representations), the authors derive new infinite product identities.

• Example: For $(T_1, D_r)$, they derive a generalized Rogers-Ramanujan identity involving products over specific congruence classes modulo $8r+4$.

D. The Folding Conjecture (Conjecture 3.1)

They propose that if $G'$ is a "folded" version of a simply-laced diagram $G$ (e.g., $F_4$ from $E_6$, $G_2$ from $D_4$), then the CFT associated with $(A_1, G')$ is a conformal sub-theory of the CFT associated with $(A_1, G)$.

• Evidence: They verify this for $\{E_6, F_4\}$, $\{D_4, G_2\}$, and $\{D_r, C_{r-1}\}$, showing that characters of the unfolded theory can be written as linear combinations of the folded theory's characters.

4. Results and Verification

• Modularity Verification: The authors confirm the modularity of 28 specific cases using existing literature (Andrews-Gordon, Bressoud, Warnaar identities) and recent proofs by Creutzig, Garner, and others.
• Open Cases: They identify 7 remaining pairs of rank $\le 3$ where modularity is not yet proven (e.g., $(T_1, G_2)$, $(G_2, T_1)$, $(B_3, A_1)$). They link these to conjectural identities by Kanade-Russell and Wang-Wang.
• Explicit Formulas: The paper provides explicit linear combinations of CFT characters for the Nahm sums, often involving differences or sums of Neveu-Schwarz (NS) and Ramond (R) sector characters.

5. Significance

• Unification: The work unifies disparate combinatorial identities (Rogers-Ramanujan type) under a single geometric framework based on Dynkin diagrams and Lie algebra root systems.
• Bridge between Math and Physics: It strengthens the connection between the algebraic theory of modular forms (Nahm's conjecture) and the physical theory of 2D CFTs, providing concrete examples of how fermionic sums (Nahm sums) map to bosonic/fermionic CFT characters.
• New Identities: The derivation of generalized Rogers-Ramanujan identities for non-simply-laced types expands the known landscape of $q$-series identities.
• Classification: By cataloging the CFTs associated with these sums, the paper offers a new classification tool for rational CFTs, particularly those involving supersymmetry and non-diagonal modular invariants.

In summary, Sun and Wang significantly advance the understanding of Nahm sums by extending the modularity conjecture to all Dynkin types, providing a dictionary between these sums and specific 2D CFTs, and proposing a structural relationship (folding) between the CFTs of simply-laced and non-simply-laced diagrams.