The Extended Algebra of the Minimal Models

TL;DR

This paper explores the extended algebra structure of minimal models in conformal field theory, proving irreducibility of representations and providing recursive methods for operator product expansions, with detailed analysis of simplest models.

Contribution

It introduces a detailed investigation of the extended algebra of minimal models, proving irreducibility of all highest weight representations and developing recursive techniques for operator product expansions.

Findings

All highest weight representations are irreducible.

No singular vectors exist in the extended theory.

Recursive methods for operator product expansion are established.

Abstract

The minimal models M(p',p) with p' > 2 have a unique (non-trivial) simple current of conformal dimension h = (p' - 2) (p - 2) / 4. The representation theory of the extended algebra defined by this simple current is investigated in detail. All highest weight representations are proved to be irreducible: There are thus no singular vectors in the extended theory. This has interesting structural consequences. In particular, it leads to a recursive method for computing the various terms appearing in the operator product expansion of the simple current with itself. The simplest extended models are analysed in detail and the question of equivalence of conformal field theories is carefully examined.