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S-matrices in the holomorphic modular bootstrap approach

Suresh Govindarajan, Aditya Jain, Akhila Sadanandan, Abhiram Kidambi

Abstract

We numerically determine the S-matrix by using connection formulae in the modular linear differential equation (MLDE) approach to the holomorphic modular bootstrap. We then determine exact formulae using the fact that entries in the $S$-matrix are integer entries in a cyclotomic extension of the field of rational numbers. This provides a method that is intrinsic to the MLDE setup and does not require inputs outside this framework. The method is illustrated with a selection of examples.

S-matrices in the holomorphic modular bootstrap approach

Abstract

We numerically determine the S-matrix by using connection formulae in the modular linear differential equation (MLDE) approach to the holomorphic modular bootstrap. We then determine exact formulae using the fact that entries in the -matrix are integer entries in a cyclotomic extension of the field of rational numbers. This provides a method that is intrinsic to the MLDE setup and does not require inputs outside this framework. The method is illustrated with a selection of examples.
Paper Structure (18 sections, 4 theorems, 35 equations, 1 figure)

This paper contains 18 sections, 4 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Rational Conformal Field Theories
  4. Characters are solutions of an MLDE
  5. Galois symmetry in RCFT
  6. Non-unitary RCFTs
  7. Determining the S-matrix
  8. The strategy
  9. Obtaining exact expressions for the $S$-matrix
  10. A working example
  11. Another $(3,3)$ example
  12. Four-character examples
  13. Five-character examples
  14. Conclusion
  15. Some definitions
  16. ...and 3 more sections

Key Result

Proposition B.2

Let ${\bar{N}}$ (taken to be a finite positive integer) be the order of ${\bar{T}}$. For any $i, j \in \mathbb{Z}\cap [0,n-1]$, $\bar{S}_{ij}\in \mathbb{Z}[\zeta_{\bar{N}}]$, where $\mathbb{Z}[\zeta_{\bar{N}}]$ is the integer ring of the ${\bar{N}}$ dimensional cyclotomic extension over $\mathbb{Q}$

Figures (1)

  • Figure 1: The contours that determine the monodromy data of the MLDE

Theorems & Definitions (12)

  • Remark 3.1
  • Remark B.1
  • Proposition B.2
  • Definition B.3: Fusion ring
  • Lemma B.5: de Boer--Goeree, DeBoer:1990em
  • proof : Proof of \ref{['lemma:vf']}
  • Remark B.6
  • Theorem B.7: Ng--Schauenberg, Ng:2007
  • proof
  • Proposition B.8
  • ...and 2 more