The Extended Algebra of the Minimal Models

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.

43 pages, 1 figure. Added reference, clarification to proof of Thm 5.1, and several paragraphs to Sec 3.2 addressing modules corresponding to simple current fixed points

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• The Verlinde formula in logarithmic CFT
• On the Percolation BCFT and the Crossing Probability of Watts
• The Extended Algebra of the SU(2) Wess-Zumino-Witten Models
• W-Algebras Extending Affine gl(1|1)
• $\mathcal{W}$-algebras with non-admissible levels and the Deligne exceptional series
• Paths and partitions: combinatorial descriptions of the parafermionic states
• Path representation of su(2)_k states I: Operators and particles for k=1,2