W-Extended Fusion Algebra of Critical Percolation

TL;DR

This paper analyzes the fusion algebra of critical percolation in two dimensions, revealing a complex structure of 26 W-indecomposable representations and confirming the extended W-symmetry through lattice methods.

Contribution

It provides the first detailed construction and classification of the W-extended fusion algebra for critical percolation, including explicit fusion rules and representation content.

Findings

Identified 26 W-indecomposable representations with various ranks.

Constructed explicit fusion rules and Cayley table for these representations.

Confirmed the closure of the fusion algebra under the extended W-symmetry.

Abstract

Two-dimensional critical percolation is the member LM(2,3) of the infinite series of Yang-Baxter integrable logarithmic minimal models LM(p,p'). We consider the continuum scaling limit of this lattice model as a `rational' logarithmic conformal field theory with extended W=W_{2,3} symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture. We find that the representation content of the ensuing closed fusion algebra contains 26 W-indecomposable representations with 8 rank-1 representations, 14 rank-2 representations and 4 rank-3 representations. We identify these representations with suitable limits of Yang-Baxter integrable boundary conditions on the lattice and obtain their associated W-extended characters. The latter decompose as finite non-negative sums of W-irreducible characters of which 13 are required. Implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit Cayley table. The closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry.