Updating the holomorphic modular bootstrap

This paper updates the holomorphic modular bootstrap by incorporating exact S-matrix results to systematically classify admissible and tenable solutions for modular linear differential equations with up to six characters and effective central charge up to 24, ultimately identifying their corresponding conformal field theories and modular tensor categories.

The Gist

Imagine the universe is built from a giant, intricate Lego set. In the world of theoretical physics, Conformal Field Theories (CFTs) are like specific, stable structures you can build with these Legos. Some of these structures are "Rational" (RCFTs), meaning they are made of a finite, manageable number of unique Lego bricks (called "primaries").

For decades, physicists have tried to find every possible stable structure in this Lego set without actually building them in a lab. Instead, they use a mathematical "blueprint" called the Holomorphic Modular Bootstrap. Think of this blueprint as a set of strict rules that any valid Lego structure must follow. If a structure breaks a rule, it's not a real physical theory.

This paper, written by Suresh Govindarajan and Akhila Sadanandan, is an update to the instruction manual for finding these structures. Here is what they did, explained simply:

1. The Problem: Too Many Possibilities

The "blueprint" they use is a complex equation called a Modular Linear Differential Equation (MLDE). Solving this equation is like trying to find a specific needle in a haystack, but the haystack keeps growing.

• The "Needles": These are the valid theories (the stable Lego structures).
• The "Haystack": The mathematical solutions to the equation.
• The Catch: Most solutions to the equation are "junk." They might look like valid theories at first glance, but when you check the details, they have negative numbers or impossible physics. The goal is to filter out the junk and find the "admissible" (valid) ones.

2. The New Tools: Sharper Filters

The authors updated their method with two major improvements:

• Filtering the "Mod 1" Ambiguity: Imagine you are trying to guess a secret code. You know the numbers must be between 0 and 1, but you don't know the exact decimals. The authors used a clever mathematical trick (developed by Kaidi, Lin, and Parra-Martinez) to list all the possible decimal combinations that could work. This shrinks the haystack significantly.
• The "S-Matrix" Shortcut: Once they find a candidate solution, they need to check if the pieces fit together correctly. In physics, this is done using something called an S-matrix (which describes how particles interact). Previously, finding this matrix was like trying to solve a puzzle blindfolded. The authors incorporated a new, recent mathematical result that lets them calculate this matrix directly and quickly.

3. The Process: From "Admissible" to "Tenable"

The authors ran their updated filters through a specific range of Lego structures (those with up to 6 types of bricks and a complexity limit of 24).

• Step 1: Admissible Solutions. They found all the structures that passed the basic math checks (no negative numbers, integers only). They call these "admissible."
• Step 2: Tenable Solutions. Just because a structure passes the math check doesn't mean it makes sense physically. They then checked the "fusion rules" (how the pieces combine). If the pieces combine in a way that creates a logical, consistent universe, they call the solution "tenable."
  • Analogy: An "admissible" solution is a car that has an engine and wheels. A "tenable" solution is a car that actually drives down the road without falling apart.

4. The Results: A Complete Catalog

The paper provides a massive, organized list of these theories.

• They identified all the valid theories with up to 6 characters (types of bricks) where the complexity is low (effective central charge ≤ 24).
• For the "tenable" ones, they provided the S-matrix (the interaction map) and the fusion rules (how they combine).
• They even matched many of these new findings to known physical theories (like specific models of magnets or strings) or identified them as belonging to specific mathematical families (MTC classes).

5. Handling the "Ghost" Solutions

Sometimes, the math gives you a solution that looks real but has a hidden flaw, like a "ghost" brick that doesn't exist. The authors used three different detective methods to fix these:

1. GHM Duality: Checking if the theory is the "shadow" or "mirror image" of another known theory.
2. Hecke Operators: Using a mathematical transformation to see if the theory is related to a simpler, known one.
3. Involutive Dual: Checking if the theory is its own mirror image in a specific way.

Summary

In short, this paper is a comprehensive census of the "valid" Lego structures in a specific, complex corner of the physics universe. By updating the rules and using new shortcuts, the authors were able to:

1. Find every possible valid structure in their target range.
2. Filter out the ones that look good but are actually broken.
3. Provide the exact "instruction manuals" (S-matrices and fusion rules) for the ones that work.

They didn't invent new physics or apply this to medicine; they simply cleaned up the library of theoretical possibilities, ensuring that every book on the shelf is a valid, non-contradictory story about how the universe could be built.

Technical Summary

1. Problem Statement

The paper addresses the classification of Rational Conformal Field Theories (RCFTs) using the Holomorphic Modular Bootstrap program. The core challenge in this field is to identify all possible RCFTs characterized by a specific number of characters ($n$) and a Wronskian index ($\ell$).

The classification process involves three main hurdles:

1. Finding Admissible Solutions: Solving Modular Linear Differential Equations (MLDEs) to find $q$-series expansions with non-negative integer coefficients.
2. Resolving Ambiguities: Admissible solutions often come with normalization ambiguities (multiplicities of characters) that must be fixed to determine the correct physical theory.
3. Verifying Physical Consistency: Determining if an admissible solution is tenable, meaning its fusion rules (derived via the Verlinde formula) are non-negative and consistent with an RCFT.

Previous methods struggled with higher character counts ($n \ge 4$) and non-rigid MLDEs (those with accessory parameters) due to the complexity of solving Diophantine equations and the difficulty of extracting exact S-matrices.

2. Methodology

The authors propose an updated holomorphic modular bootstrap framework that integrates recent theoretical advances to streamline the classification process. The methodology consists of three distinct steps:

Step 1: Fixing Exponents via Modular Representation Theory

Instead of guessing exponents, the authors utilize the Kaidi-Lin-Parra-Martinez (KLP) method.

• They determine the allowed exponents modulo 1 by analyzing the representation theory of $PSL(2, \mathbb{Z}_N)$.
• This restricts the search space to a finite set of exponents for a given $n$ and central charge constraint ($c_{eff} \le 24$).
• The authors extend the KLP lists to include 6-character theories ($n=6$), which were previously incomplete.

Step 2: Solving MLDEs with Accessory Parameters

The paper focuses on non-rigid MLDEs (those with accessory parameters) for the following classes:

• $(4, 2)$, $(4, 4)$, $(5, 2)$, and $(6, 0)$.
• For these cases, the authors solve the Diophantine equations linking the accessory parameters to the Fourier coefficients of the characters.
• They impose constraints that the coefficients must be non-negative integers, effectively fixing the accessory parameters and the normalization of the characters.

Step 3: Determining S-Matrices and Tenable Solutions

A major innovation is the direct computation of the S-matrix from the MLDE without relying on group-theoretic monodromy analysis alone.

• Numerical Monodromy: The MLDE is rewritten in the coordinate $w = 1728/J(\tau)$. The connection matrix between the Frobenius bases at $w=0$ (T-matrix) and $w=1$ (S-matrix) is computed numerically.
• Exact Reconstruction: Using lattice basis reduction algorithms (Lenstra–Lenstra–Lovász, implemented in PARI/GP), the numerical S-matrix entries are converted into exact algebraic numbers in the ring $\mathbb{Z}(\zeta_{\bar{N}})$.
• Tenable Identification: The authors compute fusion rules using the Verlinde formula. A solution is deemed tenable if the fusion coefficients are non-negative integers. They also identify the identity operator (which may not be the zeroth character in non-unitary theories) to ensure consistency.

3. Key Contributions

1. Extension to $n=6$: The paper provides the first complete list of allowed exponents modulo 1 for 6-character theories, significantly expanding the search space beyond previous limits ($n \le 5$).
2. Direct S-Matrix Extraction: By incorporating the result from reference [21], the authors bypass the need for complex group-theoretic classification to find S-matrices, allowing for a systematic check of tenability for all admissible solutions.
3. Resolution of Normalization Ambiguities: The authors systematically resolve the ambiguity in character multiplicities ($y_i$) using three complementary methods:
   • GHM Duality: Mapping solutions to known Generalized Gaberdiel-Hampapura-Mukhi coset duals.
   • Hecke Operators: Using Hecke transforms to map known minimal models to new solutions.
   • Involutive Duality: Utilizing Bantay-Gannon duality to relate solutions of different Wronskian indices.
4. Comprehensive Classification: The paper classifies all admissible solutions for $n \in \{4, 5, 6\}$ with Wronskian index $\ell < 6$ and effective central charge $c_{eff} \le 24$.

4. Key Results

The authors present extensive tables (Appendix D) cataloging the results:

• Admissible Solutions: A complete list of solutions for $(4, 2)$, $(4, 4)$, $(5, 2)$, and $(6, 0)$ MLDEs.
• Tenable Solutions:
  • Many solutions are identified as known RCFTs (e.g., tensor products of minimal models like $M(5,2) \otimes A_{1,1}$, affine algebras like $E_{7,1}$, and non-unitary minimal models).
  • New tenable solutions are identified with specific Modular Tensor Category (MTC) labels.
  • The paper distinguishes between unitary and non-unitary tenable solutions.
• S-Matrices: Exact S-matrices are provided for all tenable solutions (except untenable ones), expressed in terms of roots of unity.
• Specific Findings:
  • For the $(4, 4)$ MLDE, they found that one accessory parameter must vanish for admissible solutions to exist, reducing it effectively to a one-parameter problem.
  • They identified theories related to the $E_{7}^{1/2}$ Intrinsically Non-Unitary VOA (IVOA) and confirmed their properties.
  • They noted the absence of $(6, 2)$ theories with $c_{eff} \le 24$ due to irreducibility constraints on exponents, though such theories exist for higher $c$.

5. Significance

This paper represents a significant update to the modular bootstrap program by:

• Automating the Classification: It moves the field from case-by-case analysis to a systematic, algorithmic approach capable of handling higher-rank theories.
• Bridging Theory and Data: By providing exact S-matrices and fusion rules, it connects abstract MLDE solutions directly to physical RCFT data and MTC classifications.
• Expanding the Landscape: The inclusion of $n=6$ and the rigorous check of tenability for $c_{eff} \le 24$ provides a definitive "dictionary" of possible 2D RCFTs in this regime, serving as a benchmark for future string theory and condensed matter applications.
• Methodological Robustness: The combination of KLP exponent fixing, numerical monodromy, and lattice reduction offers a robust pipeline for future classifications of theories with higher Wronskian indices or more characters.

In summary, the authors have successfully updated the holomorphic modular bootstrap to a state where it can systematically classify and validate RCFTs up to six characters, resolving long-standing ambiguities in normalization and providing a complete dataset of admissible and tenable solutions.