Generalised 4d Partition Functions and Modular Differential Equations

This paper establishes the equivalence between generalised Schur partition functions of 4d $\mathcal{N}=2$ $USp(2N)$ gauge theories and vector-valued modular forms by proving they satisfy specific modular linear differential equations, while also proposing extensions and conjectures linking these functions to quantum monodromy traces and 2d rational conformal field theory characters.

The Gist

The Big Picture: A Cosmic Rosetta Stone

Imagine the universe is built on two different languages.

1. Language A (4D Physics): This is the language of our everyday reality, but at a microscopic level. It describes particles, forces, and how they interact in four dimensions (three of space, one of time).
2. Language B (2D Math): This is a highly abstract, mathematical language used to describe patterns on a flat surface (two dimensions). It's full of complex numbers, symmetry, and "modular forms" (which are like perfect, repeating geometric patterns).

For a long time, physicists knew these two languages were somehow related, but the dictionary to translate between them was incomplete. This paper is like finding a new, perfect dictionary that translates a specific, complex sentence from the 4D world into a beautiful, solvable equation in the 2D world.

The Main Characters

To understand the paper, we need to meet three main characters:

1. The Superconformal Index (The "Fingerprint"):
   Imagine a 4D quantum system as a giant, complex machine. Even though the machine is chaotic, it has a "fingerprint" called the Superconformal Index. This fingerprint counts the stable, unchanging parts of the machine. It's a unique code that tells you exactly what kind of machine you are looking at.

2. The "Generalised Schur Partition Function" (The "Dial"):
   Usually, the fingerprint is fixed. But the authors of this paper discovered a "dial" (a parameter called $\alpha$) that they can turn.

   • When you turn the dial to $\alpha = 1$, you get the standard fingerprint.
   • When you turn it to $\alpha = 0$, you get a different, simpler version.
   • When you turn it to any number in between, you get a "hybrid" fingerprint.
     The paper asks: What happens if we turn this dial to weird, fractional numbers?

3. Modular Differential Equations (The "Recipe Book"):
   In the 2D mathematical world, there are "Recipe Books" (called Modular Linear Differential Equations or MLDEs). If you follow a recipe in this book, you get a specific pattern (a character of a Conformal Field Theory). These patterns are incredibly rigid and beautiful.

The Discovery: The Magic Connection

The authors focused on a specific type of 4D machine (called USp(2N) with specific ingredients). They took the "fingerprint" of this machine and started turning the $\alpha$ dial.

The Big Surprise:
They found that no matter what value they set the dial to, the resulting fingerprint always matched a specific entry in the 2D "Recipe Book."

• The Analogy: Imagine you have a 4D machine that can change its shape. You take a photo of it at different angles (different $\alpha$ values). You expect the photos to look random. Instead, you find that every single photo is a perfect, high-resolution print of a specific, famous painting from a 2D art gallery.
• The Proof: They didn't just guess this; they proved it mathematically. They showed that the complex integral (the math used to calculate the 4D fingerprint) can be reshaped into a contour integral (a path drawn on a map). This path is exactly the same path used to generate the 2D paintings.

Why Does This Matter?

This discovery is like finding a hidden bridge between two islands that were thought to be separate.

1. Solving the Unsolvable:
   Calculating the fingerprint of a 4D machine is usually incredibly hard, like trying to solve a 1,000-piece puzzle blindfolded. But because this paper shows the fingerprint is actually a 2D "painting" that follows a strict recipe (MLDE), physicists can now use the 2D recipe to solve the 4D puzzle instantly.

2. Finding New Art:
   By turning the dial ($\alpha$) to specific numbers, they found fingerprints that match Unitary 2D theories. In the 2D world, "Unitary" means the theory makes physical sense (probabilities add up to 1).

   • The Twist: The 4D machine they started with is "non-unitary" (a bit weird in 2D terms). But by tuning the dial, they accidentally discovered "perfect" 2D theories that were previously unknown or hard to find. It's like tuning a radio to a weird frequency and suddenly hearing a perfect symphony.

3. The "Monodromy" Mystery:
   The paper also touches on a concept called "Monodromy Traces" (think of this as counting how many times a particle loops around a hole in space). They found a connection between these loops and the "Recipe Book." They propose a new rule: If you count the loops in a specific way, you will always get a result that fits into a specific mathematical equation.

The "Two-Parameter" Extension

The authors didn't stop at one dial. They proposed a second dial (parameter $\beta$).

• The Analogy: Imagine the 4D machine has two knobs instead of one. They showed that if you turn both knobs, you can generate every possible painting in a whole section of the 2D art gallery.
• The Catch: They can prove the math works perfectly, but they don't yet know what physical "knob" in the real 4D universe corresponds to this second dial. It's like having a remote control that changes the TV channel to any show you want, but you don't know which button on the remote actually does it.

Summary in a Nutshell

• The Problem: 4D physics is hard; 2D math is rigid and structured.
• The Tool: A "Generalised Partition Function" with a tunable dial ($\alpha$).
• The Result: Tuning the dial turns the messy 4D physics into a clean, solvable 2D mathematical pattern.
• The Impact: This allows physicists to use 2D math to solve 4D problems, discover new "perfect" theories, and understand the deep, hidden geometry of the universe.

The paper essentially says: "If you look at this 4D system through the right lens (the $\alpha$ dial), it reveals itself to be a perfect 2D mathematical poem."

Technical Summary

1. Problem Statement

The paper addresses the structural relationship between 4d $\mathcal{N}=2$ superconformal field theories (SCFTs) and 2d Rational Conformal Field Theories (RCFTs). Specifically, it investigates the generalised Schur partition function, denoted $Z_G(q; \alpha)$, which is a one-parameter deformation of the standard Schur index.

• Context: The Schur index of a 4d $\mathcal{N}=2$ SCFT is known to coincide with the vacuum character of a non-unitary Vertex Operator Algebra (VOA) associated with the theory via the 4d/2d correspondence. For "quasi-lisse" VOAs, these characters satisfy a Modular Linear Differential Equation (MLDE) with a vanishing Wronskian index ($\ell=0$).
• The Gap: Previous empirical observations (e.g., in [18]) suggested that for specific rational values of the deformation parameter $\alpha$, the generalised Schur partition function $Z_G(q; \alpha)$ reproduces the Schur indices of different SCFTs or characters of unitary RCFTs. However, a rigorous analytical proof linking these functions to the underlying MLDE structure and contour integral representations was missing.
• Goal: To analytically prove that $Z_G(q; \alpha)$ for a specific family of theories satisfies an MLDE, map it to known contour integral representations of vector-valued modular forms (VVMF), and extend this to a two-parameter generalization.

2. Methodology

The authors employ a combination of gauge theory integral representations, elliptic function identities, and modular form theory.

1. Integral Representation: They start with the contour integral representation of the Schur index for the $USp(2N)$ SCFT with $2N+2$ fundamental hypermultiplets.
2. Variable Transformation: The core technical innovation involves a chain of variable changes transforming the gauge integration variables ($u_i$) into elliptic variables ($t_i$).
   • They utilize Jacobi theta functions and Jacobi elliptic functions ($sn, cn, dn$).
   • The transformation $t_i = \lambda \, \text{sn}^2(v_i, k)$ (where $\lambda$ is the modular lambda function) maps the gauge theory integrand into a form resembling the Dotsenko-Fateev or Selberg-type contour integrals used to solve MLDEs.
3. Contour Analysis: They analyze the integration contours in the complex plane, showing that the original unit circle integrals map to contours running from $0$ to $\lambda$ in the $t$-plane.
4. Matching with MLDE Solutions: They compare the resulting contour integrals with the general solution form for $\ell=0$ MLDEs proposed in [20]. By matching the leading $q$-expansion exponents (critical exponents) of the partition function with the indicial roots of the MLDE, they identify the specific parameters of the differential equation.

3. Key Contributions and Results

A. The $USp(2N)$ Case (Proposition 2)

The authors focus on the $USp(2N)$ SCFT with $2N+2$ fundamental hypermultiplets.

• Analytical Proof: They prove that the analytically continued generalised Schur partition function $Z_{USp(2N)}(q; \alpha)$ satisfies an order-$(N+1)$ MLDE with vanishing Wronskian index ($\ell=0$).
• Contour Integral Equivalence: They demonstrate that $Z_{USp(2N)}(q; \alpha)$ is equivalent to the contour integral $J_N$ (a specific case of the multi-variable integral in Eq. 2.40) with parameters $a = \alpha - 1/2$ and $b = 2\alpha$.
• Critical Exponents: They derive the critical exponents $\gamma_A$ of the MLDE in terms of $\alpha$:
  $$\gamma_0 = -\frac{N}{12}(2(N+2)\alpha - 1), \quad \gamma_A - \gamma_0 = \frac{A(A+1)}{2}\alpha$$
  This allows them to determine the central charge $c$ and conformal dimensions $h_A$ of the associated 2d VOA/RCFT.

B. Specific Cases and Unitary RCFTs

• $N=1$ (SU(2) with 4 hypers): They prove $Z_{SU(2)}(q; \alpha)$ satisfies the second-order MMS equation. They show that for specific $\alpha$, this function corresponds to:
  • The identity character of non-unitary VOAs (Deligne-Cvitanović series).
  • The non-identity character of unitary RCFTs (e.g., $SU(3)_1$ at $\alpha=1/3$).
• $N=2$ (USp(4)): They solve the third-order MLDE. They identify cases where the partition function matches the characters of unitary theories like the $D$-series invariant of $SU(2)_8$ (at $\alpha=1/5$).
• General $N$: They show that for $\alpha = 1/(2N+1)$, the partition function matches the characters of the $E_{N+2}$ series of unitary RCFTs (specifically $SU(2)_{4N}$ invariants).

C. Two-Parameter Generalization (Proposition 3)

Motivated by the fact that the contour integrals for $\ell=0$ MLDEs generally depend on two parameters ($a, b$), while the standard generalised Schur function depends on one ($\alpha$), the authors propose a two-parameter extension:
$$Z_{USp(2N)}(q; \alpha, \beta)$$

• Definition: They define this by raising the vector multiplet contribution to power $\alpha$ and the hypermultiplet contribution to power $\beta$ (specifically distinguishing between roots $\pm 2u_i$ and $\pm u_i \pm u_j$).
• Result: They prove that $Z_{USp(2N)}(q; \alpha, \beta)$ satisfies the full order-$(N+1)$ MLDE with vanishing Wronskian index, where the parameters $\alpha$ and $\beta$ map directly to the contour integral parameters $a$ and $b$.
• Significance: This extension allows for the complete classification of all unitary 3-character RCFTs with $\ell=0$ (e.g., Ising model, $SO(N)_1$ WZW models) to be generated from the USp(4) theory by tuning $\alpha$ and $\beta$.

D. Connection to Monodromy Traces (Conjecture 1)

The paper connects the parameter $\alpha$ to the trace of powers of the BPS monodromy operator $M$.

• Relation: Setting $\alpha = -k$ (where $k$ is an integer) relates $Z_G(q; \alpha)$ to the trace $\text{Tr } M^k$.
• Conjecture: The authors conjecture that for any 4d $\mathcal{N}=2$ SCFT, the trace $\text{Tr } M^k$ (regularized) satisfies an MLDE of the same order and Wronskian index as the Schur index ($k=-1$), but with a central charge $c(k)$ that depends linearly on $k$.
• Verification: They explicitly verify this central charge formula for the $USp(2N)$ case, showing consistency with the 4d/2d correspondence.

4. Significance and Implications

1. Unification of 4d and 2d Structures: The work provides a rigorous mathematical bridge showing that the "zoo" of solutions to MLDEs (including unitary RCFTs, non-unitary VOAs, and logarithmic CFTs) can be generated from a single 4d gauge theory by varying a continuous parameter $\alpha$.
2. Analytical Proof of Empirical Observations: It moves beyond numerical coincidences reported in previous literature (e.g., [18]) to provide an analytical derivation using contour integrals and elliptic functions.
3. New Tool for Classification: The two-parameter generalization offers a powerful new method to generate and classify unitary RCFT characters, potentially aiding in the search for new CFTs or understanding the landscape of 2d theories.
4. Insight into Monodromy: The conjecture linking monodromy traces to fixed-order MLDEs suggests a deep, universal algebraic structure underlying the BPS spectra of 4d SCFTs, independent of the specific theory's details (rank, matter content).
5. Technical Advancement: The successful mapping of the $USp(2N)$ gauge integral to the Dotsenko-Fateev type contour integrals demonstrates a novel application of elliptic function identities in the context of superconformal indices.

5. Limitations and Future Work

• Physical Origin of Two-Parameter Extension: While the two-parameter function $Z(\alpha, \beta)$ satisfies the MLDE, the authors note that it is not yet clear how to derive this specific limit from the full superconformal index $I(p,q,t)$ via a double scaling limit. The physical interpretation of the second parameter $\beta$ in 4d remains an open problem.
• General Gauge Groups: The proof relies on specific properties of $USp(2N)$ and its root system. Extending this to arbitrary gauge groups $G$ is technically challenging because the resulting integrals may not map to the standard $\ell=0$ contour integrals (Eq. 2.40), especially if the Schur index involves half-integer powers of $q$ or non-vanishing Wronskian indices.

In summary, the paper establishes a profound link between 4d SCFT partition functions and 2d modular differential equations, proving that a specific family of 4d theories generates a continuous family of 2d CFT characters, including unitary ones, through a deformation parameter that can be interpreted via monodromy traces.