⚛️ high-energy theory

S-matrices in the holomorphic modular bootstrap approach

This paper presents an intrinsic method within the holomorphic modular bootstrap framework to numerically determine and subsequently derive exact S-matrices by leveraging connection formulae in modular linear differential equations and the property that their entries are integers in a cyclotomic extension of the rational numbers.

Original authors: Suresh Govindarajan, Aditya Jain, Akhila Sadanandan, Abhiram Kidambi

Published 2026-02-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Suresh Govindarajan, Aditya Jain, Akhila Sadanandan, Abhiram Kidambi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a massive, cosmic jigsaw puzzle. The pieces of this puzzle represent the fundamental building blocks of a specific type of universe (called a Rational Conformal Field Theory, or RCFT).

For a long time, physicists have had a very good way to find the shapes of these puzzle pieces. They use a mathematical tool called a Modular Linear Differential Equation (MLDE). Think of this equation as a mold. If you pour the right "clay" (numbers) into this mold, it spits out the shapes of the pieces.

However, there was a major problem: We knew the shapes, but we didn't know how the pieces fit together.

In the world of these theories, the "fitting together" is governed by a special map called the S-matrix. This map tells you: "If you combine piece A and piece B, what new piece do you get?" Without this map, the puzzle is incomplete. You can't see the final picture.

Previously, to find this map, physicists had to use "cheat codes"—indirect methods that relied on knowing other theories or using complex duality tricks. This paper introduces a new, self-contained way to solve the puzzle from the inside out.

Here is how they did it, explained with simple analogies:

1. The Two Camps (The Strategy)

Imagine the puzzle pieces are living in a strange landscape.

Camp A is located at a point called $w=0$ . Here, the pieces behave in a very orderly, predictable way.
Camp B is located at a point called $w=1$ . Here, the pieces behave differently, but they are still orderly.

The S-matrix is essentially the "translation guide" that tells you how to convert the language of Camp A into the language of Camp B.

The authors' method is like sending two explorers out from Camp A and Camp B to meet in the middle (at a point called $w=0.5$ ).

They calculate exactly what the pieces look like as they approach the meeting point from the left (Camp A).
They calculate exactly what the pieces look like as they approach from the right (Camp B).
They match the two descriptions together.

By forcing these two descriptions to agree at the meeting point, they can mathematically derive the "translation guide" (the Connection Matrix). Once they have this, they can instantly calculate the S-matrix.

2. The "Fuzzy Photo" vs. The "Crystal Clear Image"

The first time they do this matching, it's like taking a photo with a slightly shaky hand. The numbers they get for the S-matrix are approximations. They look like:

$0.327882...$
$1.80397...$

These numbers are close, but they aren't the exact truth. In physics, we need the exact numbers, not the blurry ones.

3. The "Magic Ring" (The Secret Ingredient)

Here is where the paper gets clever. The authors know a secret rule about these puzzle pieces: The exact numbers in the S-matrix aren't just random decimals. They are special numbers that live in a "Magic Ring" called a Cyclotomic Extension.

Think of this ring like a special club. You can only be a member if your number is built from specific, whole-number ingredients (integers) mixed with a special "flavor" called a root of unity (a specific type of complex number).

Because they know the S-matrix must be a member of this club, they can take their blurry, approximate numbers (like $1.80397$) and ask: "What is the simplest, most exact number in this Magic Ring that looks like 1.80397?"

The answer pops out instantly. It turns out that $1.80397$ is actually exactly equal to a specific combination of integers and roots of unity. Suddenly, the blurry photo snaps into a crystal-clear, exact image.

4. Why This Matters

Before this paper, if you found a new puzzle piece using the MLDE mold, you were stuck. You had the shape, but you didn't know how it connected to the rest of the universe. You had to go find an external expert (using indirect methods) to tell you how it fit.

Now, the authors have built a machine that:

Takes the shape from the mold.
Sends explorers to the middle to find the connection.
Uses the "Magic Ring" rule to snap the blurry numbers into exact, perfect integers.

The Result: They can now fully describe these universes without needing any outside help. They have turned a "Diophantine problem" (a notoriously hard math puzzle) into a solvable, step-by-step recipe.

Summary

The Problem: We could find the shapes of cosmic puzzle pieces but didn't know how they fit together (the S-matrix).
The Method: They matched two different mathematical descriptions of the pieces at a meeting point to find the connection.
The Trick: They used a known mathematical rule (that the answers must be "integers in a special ring") to turn rough approximations into exact, perfect answers.
The Impact: This gives physicists a complete, self-contained toolkit to understand these theories, making the "Holomorphic Modular Bootstrap" a much more powerful tool for exploring the universe.

1. Problem Statement

The Holomorphic Modular Bootstrap is a framework used to classify Rational Conformal Field Theories (RCFTs) by treating their Virasoro characters as independent solutions to a Modular Linear Differential Equation (MLDE).

The Challenge: While the MLDE approach successfully identifies "admissible" solutions (those with non-negative integer $q$ -series coefficients), it often fails to provide the complete modular data required to verify if a solution corresponds to a physical CFT.
Specific Gap: For a given admissible solution, the S-matrix (which encodes the fusion rules via the Verlinde formula) is generally unknown unless the MLDE is of the hypergeometric type. Without the S-matrix, one cannot determine the fusion algebra or confirm unitarity.
Existing Limitations: Previous methods to recover the S-matrix rely on indirect constructions (e.g., generalized cosets, Hecke operators, or vector-valued modular forms) that often require external inputs or knowledge of dual theories, rather than deriving the S-matrix intrinsically from the MLDE itself.

2. Methodology

The authors propose a novel, intrinsic method to compute the S-matrix directly from the MLDE framework using connection formulae and Galois symmetry.

A. Numerical Determination via Connection Matrices

Variable Transformation: The MLDE is analyzed in the variable $z = J(\tau)/1728$ $z = J (τ) /1728$ , where $J$ $J$ is the Klein invariant. The singular points are $z=0, 1, \infty$ $z = 0, 1, \infty$ .
- $z=0$ corresponds to the T-matrix (monodromy).
- $z=1$ corresponds to the S-matrix (monodromy).
- $z=\infty$ corresponds to the U-matrix.
Local Solutions: The authors switch variables to $w = 1/z$ to align the T-matrix with the monodromy at $w=0$ and the S-matrix with the monodromy at $w=1$ .
Frobenius Series: They construct local basis solutions, $B_0(w)$ around $w=0$ and $B_1(u)$ around $u=1-w=0$ , using the Frobenius power series method.
Matching: The two bases are matched at an intermediate regular point $w_0 = 1/2 + \epsilon$ . This yields a linear system to solve for the Connection Matrix $C$ , which relates the two bases:
$B_1(1-w_0) = C \cdot B_0(w_0)$
S-Matrix Extraction: Once $C$ is determined numerically, the S-matrix is computed via the similarity transformation:
$S = C^{-1} \cdot M_1 \cdot C$
where $M_1$ is the diagonal monodromy matrix at $w=1$ .

B. Exact Reconstruction via Cyclotomic Integrality

The numerical estimates of the S-matrix entries are converted into exact algebraic numbers using number-theoretic properties of RCFTs:

Galois Symmetry: The entries of the normalized S-matrix ( $\bar{S} = S/S_{00}$ ) lie in the ring of integers of a cyclotomic extension, $\mathbb{Z}[\zeta_{\bar{N}}]$ , where $\bar{N}$ is the order of the normalized T-matrix.
Integer Basis: The authors express each entry $\bar{S}_{ij}$ as a linear combination of the basis elements of $\mathbb{Z}[\zeta_{\bar{N}}]$ with integer coefficients.
Optimization: By minimizing the difference between the numerical estimate and the algebraic form over a range of integers, they uniquely identify the exact integer coefficients, thereby recovering the exact S-matrix.

3. Key Contributions

Intrinsic S-Matrix Calculation: The paper provides the first method to compute the S-matrix entirely within the MLDE framework without relying on external dualities or known parent theories.
Extension Beyond Hypergeometric Cases: Unlike previous methods restricted to hypergeometric MLDEs, this approach applies to non-hypergeometric equations (e.g., those with accessory parameters).
Resolution of Degeneracies: The method can sometimes fix the degeneracies (multiplicities) of characters, which are otherwise ambiguous in the MLDE approach.
Handling Non-Unitary Theories: The authors address the complications arising in non-unitary CFTs (where the lowest dimension operator is not the identity) by utilizing Galois symmetries (Galois shuffles) to relate the physical identity to the lowest dimension operator.
Appendix Proof: Abhiram Kidambi provides a concise proof confirming that S-matrix entries belong to the cyclotomic ring $\mathbb{Z}[\zeta_{\bar{N}}]$ , justifying the exact reconstruction step.

4. Results and Examples

The authors validate their method on several classes of theories:

(3, 3) Theories: They re-derived the S-matrix for a known non-hypergeometric theory ( $c=48/7$ ) and a new case ( $c=5$ ) where $S_{00}=0$ . In the latter case, they successfully applied the Coste-Gannon result (entries in $\mathbb{Q}(\zeta_N)$ ) despite the theory potentially lacking a corresponding CFT.
(4, 0) and (4, 2) Theories:
- Recovered the S-matrix for a non-unitary theory ( $c_{eff}=6/5$ ) associated with the Lee-Yang minimal model tensor product.
- Determined the S-matrix for a new $(4, 2)$ theory ( $c=8/3$ ), identifying it as a six-primary CFT.
(5, 0) and (5, 2) Theories: They analyzed a pair of GHM-dual theories related to the Thompson group and the Monster group (conjugacy class 3C). The numerical method yielded identical S-matrices for both duals, confirming the consistency of the approach. The entries were successfully identified as elements of $\mathbb{Z}[\zeta_{11}]$ .

5. Significance and Future Directions

Completing the Bootstrap: This method removes a major bottleneck in the holomorphic modular bootstrap, allowing researchers to fully characterize admissible solutions (including their fusion rules) without external inputs.
New Physics: It enables the systematic search for new RCFTs and Modular Tensor Categories (MTCs) by filtering admissible solutions based on their derived S-matrices.
Theoretical Open Questions: The authors highlight that while they assume Galois symmetry holds for all admissible solutions (not just physical RCFTs), a rigorous proof using Differential Galois Theory for MLDEs remains an open problem.
F and R Symbols: The paper suggests that these connection-matching techniques could potentially be extended to compute F-symbols (associativity of conformal blocks) and R-symbols, which are currently unknown for many higher-rank theories.

In summary, this work bridges the gap between the numerical solutions of MLDEs and the exact algebraic structure of RCFTs, providing a robust, self-contained algorithm for determining the full modular data of rational conformal field theories.