Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Moonshine Anomaly

Theo Johnson-Freyd

TL;DR

This work determines the Moonshine anomaly $\\omega^{\\nat}$ for the Monster group acting on its moonshine module $V^{\\natural}$ by combining orbifold/anomaly theory for holomorphic CFTs with a finite-group form of T-duality. The main result shows $\\omega^{\\nat}$ has exact order $24$ and is not realized as a Chern class or a fractional Pontryagin class, with a prime-by-prime analysis supported by T-duality between Monster centralizers and Leech lattice CFT centralizers. The proof strategically uses a Leech-lattice-based CFT, LHS spectral sequences, and McKay-type subgroup analyses to bound the 2- and 3-parts and to rule out Chern-class realization, while Remarkably connecting deep moonshine data to a higher-categorical TFT framework. The findings illuminate how anomalies in holomorphic CFTs encode moonshine phenomena, and suggest (via conjectures discussed) that the third cohomology $\\mathrm{H}^3(\\mathds{M}, \mathrm{U}(1))$ may be isomorphic to $\\mathbb{Z}_{24}$, reinforcing the unity between algebraic, geometric, and topological perspectives in moonshine. The approach provides a robust method to compute finite-group anomalies in holomorphic theories and links orbifold theory to the Leech lattice CFT through finite-group T-duality.

Abstract

The anomaly for the Monster group $\mathbb{M}$ acting on its natural (aka moonshine) representation $V^\natural$ is a particular cohomology class $ω^\natural \in \mathrm{H}^3(\mathbb{M},\mathrm{U}(1))$ that arises as a conformal field theoretic generalization of the second Chern class of a representation. This paper shows that $ω^\natural$ has order exactly $24$ and is not a Chern class. In order to perform this computation, this paper introduces a finite-group version of T-duality, which is used to relate $ω^\natural$ to the anomaly for the Leech lattice CFT.

The Moonshine Anomaly

TL;DR

This work determines the Moonshine anomaly for the Monster group acting on its moonshine module by combining orbifold/anomaly theory for holomorphic CFTs with a finite-group form of T-duality. The main result shows has exact order and is not realized as a Chern class or a fractional Pontryagin class, with a prime-by-prime analysis supported by T-duality between Monster centralizers and Leech lattice CFT centralizers. The proof strategically uses a Leech-lattice-based CFT, LHS spectral sequences, and McKay-type subgroup analyses to bound the 2- and 3-parts and to rule out Chern-class realization, while Remarkably connecting deep moonshine data to a higher-categorical TFT framework. The findings illuminate how anomalies in holomorphic CFTs encode moonshine phenomena, and suggest (via conjectures discussed) that the third cohomology may be isomorphic to , reinforcing the unity between algebraic, geometric, and topological perspectives in moonshine. The approach provides a robust method to compute finite-group anomalies in holomorphic theories and links orbifold theory to the Leech lattice CFT through finite-group T-duality.

Abstract

The anomaly for the Monster group acting on its natural (aka moonshine) representation is a particular cohomology class that arises as a conformal field theoretic generalization of the second Chern class of a representation. This paper shows that has order exactly and is not a Chern class. In order to perform this computation, this paper introduces a finite-group version of T-duality, which is used to relate to the anomaly for the Leech lattice CFT.

Paper Structure

This paper contains 15 sections, 18 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Acknowledgments
  4. Orbifolds of conformal field theories
  5. Physical picture
  6. Mathematical foundations
  7. T-duality for finite groups
  8. Super case
  9. Proof of Theorem \ref{['moonshine theorem']}
  10. The "large" primes p = 11, 17, 19, 23, 29, 31, 41, 47, 59, 71
  11. The "small" primes p = 3, 5, 7, 13
  12. The prime p = 2
  13. $\omega^\natural$ is not a Chern class
  14. Further remarks on the value of $\omega^\natural$
  15. Soicher's presentation for $\mathrm{Co}_1$

Key Result

Theorem 1

The order of $\omega^\natural \in \mathop{\mathrm{H}}\nolimits^3(\mathds M,\mathrm U(1))$ is exactly $24$. Although Chern and fractional Pontryagin classes can arise as anomalies, $\omega^\natural$ is neither a Chern nor a fractional Pontryagin class of any representation of $\mathds M$.

Theorems & Definitions (37)

  • Theorem 1
  • Theorem 2: JFT
  • Conjecture 1
  • Conjecture 2
  • Conjecture 3
  • Example 2.1.1
  • Example 2.1.2
  • Proposition 2.2.1: MR2730815
  • proof
  • Proposition 2.3.1
  • ...and 27 more