Grothendieck ring and Verlinde-like formula for the W-extended logarithmic minimal model WLM(1,p)

TL;DR

This paper explores the Grothendieck ring of the fusion algebra in W-extended logarithmic minimal models, deriving a Verlinde-like formula using modular data that is independent of the modular parameter.

Contribution

It introduces a new Verlinde-like formula for WLM(1,p) models based on the modular data, extending the understanding of fusion rings in logarithmic conformal field theories.

Findings

The regular representation matrices are mutually commuting but not diagonalizable.

A Jordan form is obtained via the modular S-matrix including pseudo-characters.

The Verlinde-like formula is independent of the modular parameter τ.

Abstract

We consider the Grothendieck ring of the fusion algebra of the W-extended logarithmic minimal model WLM(1,p). Informally, this is the fusion ring of W-irreducible characters so it is blind to the Jordan block structures associated with reducible yet indecomposable representations. As in the rational models, the Grothendieck ring is described by a simple graph fusion algebra. The 2p-dimensional matrices of the regular representation are mutually commuting but not diagonalizable. They are brought simultaneously to Jordan form by the modular data coming from the full (3p-1)-dimensional S-matrix which includes transformations of the p-1 pseudo-characters. The spectral decomposition yields a Verlinde-like formula that is manifestly independent of the modular parameter $\tau$ but is, in fact, equivalent to the Verlinde-like formula recently proposed by Gaberdiel and Runkel involving a $\tau$-dependent S-matrix.