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The kernel of the modular representation and the Galois action in RCFT

P. Bantay

TL;DR

This work establishes that the kernel of the RCFT modular representation is a congruence subgroup whose level equals the Dehn twist order $N$, and it provides an explicit description of the kernel through a Galois-action framework augmented by permutation orbifolds. By showing the modular data generate a finite cyclotomic field and that the action factors through $SL_2(N)$ via a $D$-representation, the authors derive a practical criterion for kernel membership and derive strong bounds on the conductor in terms of the number of primaries. The combination of Galois theory and orbifold covariance yields new constraints on the spectrum of the Dehn twist and on the arithmetic properties of modular matrix elements, with concrete minimal-model examples illustrating the theory. These results advance RCFT classification by enabling systematic enumeration of possible modular representations consistent with arithmetic constraints and provide a bridge between number theory and conformal field theory.

Abstract

It is shown that for the modular representations associated to Rational Conformal Field Theories, the kernel is a congruence subgroup whose level equals the order of the Dehn-twist. An explicit algebraic characterization of the kernel is given. It is also shown that the conductor, i.e. the order of the Dehn-twist is bounded by a function of the number of primary fields, allowing for a systematic enumeration of the modular representations coming from RCFTs. Restrictions on the spectrum of the Dehn-twist and arithmetic properties of modular matrix elements are presented.

The kernel of the modular representation and the Galois action in RCFT

TL;DR

This work establishes that the kernel of the RCFT modular representation is a congruence subgroup whose level equals the Dehn twist order , and it provides an explicit description of the kernel through a Galois-action framework augmented by permutation orbifolds. By showing the modular data generate a finite cyclotomic field and that the action factors through via a -representation, the authors derive a practical criterion for kernel membership and derive strong bounds on the conductor in terms of the number of primaries. The combination of Galois theory and orbifold covariance yields new constraints on the spectrum of the Dehn twist and on the arithmetic properties of modular matrix elements, with concrete minimal-model examples illustrating the theory. These results advance RCFT classification by enabling systematic enumeration of possible modular representations consistent with arithmetic constraints and provide a bridge between number theory and conformal field theory.

Abstract

It is shown that for the modular representations associated to Rational Conformal Field Theories, the kernel is a congruence subgroup whose level equals the order of the Dehn-twist. An explicit algebraic characterization of the kernel is given. It is also shown that the conductor, i.e. the order of the Dehn-twist is bounded by a function of the number of primary fields, allowing for a systematic enumeration of the modular representations coming from RCFTs. Restrictions on the spectrum of the Dehn-twist and arithmetic properties of modular matrix elements are presented.

Paper Structure

This paper contains 8 sections, 19 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. The Galois action
  3. Galois action on $\Lambda$ matrices
  4. The Galois action revisited
  5. The kernel of the modular representation
  6. Examples
  7. Summary
  8. Appendix

Key Result

Lemma 1

The set $\mathcal{J}$ is invariant under the permutations $\tilde{\pi }_{l}$, i.e. $\tilde{\pi }_{l}\left( \mathcal{J}\right) =\mathcal{J}$. For a primary $\left[ p,n,k\right] \in \mathcal{J}_{0}$ one has for some function $\tilde{k}$ of $l,p,n$ and $k$, and

Theorems & Definitions (43)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • proof
  • ...and 33 more