Fusion algebra of critical percolation

TL;DR

This paper conjectures the explicit structure of the fusion algebra for critical percolation using Virasoro representations, revealing a quasi-rational, associative, and commutative algebra with indecomposable representations, supported by numerical lattice model studies.

Contribution

It introduces a detailed conjecture for the fusion algebra of critical percolation with novel indecomposable representations and compares these rules with recent theoretical results.

Findings

Fusion rules involve indecomposable Kac representations.

Fusion algebra exhibits sl(2) structure and quasi-rationality.

Numerical lattice model studies support the fusion rules.

Abstract

We present an explicit conjecture for the chiral fusion algebra of critical percolation considering Virasoro representations with no enlarged or extended symmetry algebra. The representations we take to generate fusion are countably infinite in number. The ensuing fusion rules are quasi-rational in the sense that the fusion of a finite number of these representations decomposes into a finite direct sum of these representations. The fusion rules are commutative, associative and exhibit an sl(2) structure. They involve representations which we call Kac representations of which some are reducible yet indecomposable representations of rank 1. In particular, the identity of the fusion algebra is a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the recent results of Eberle-Flohr and Read-Saleur. Notably, in agreement with Eberle-Flohr, we find the appearance of indecomposable representations of rank 3. Our fusion rules are supported by extensive numerical studies of an integrable lattice model of critical percolation. Details of our lattice findings and numerical results will be presented elsewhere.