Coset Graphs in Bulk and Boundary Logarithmic Minimal Models

TL;DR

This paper explores the structure of logarithmic minimal models using coset graphs and W-projective representations, providing new formulas for boundary and bulk partition functions and revealing their underlying graph-theoretic structure.

Contribution

It introduces a framework connecting W-projective representations with coset graphs to compute partition functions in logarithmic minimal models.

Findings

W-projective representations form fundamental building blocks.

Derived Verlinde-like formulas involving coset graphs.

Proposed modular invariant partition functions using these structures.

Abstract

The logarithmic minimal models are not rational but, in the W-extended picture, they resemble rational conformal field theories. We argue that the W-projective representations are fundamental building blocks in both the boundary and bulk description of these theories. In the boundary theory, each W-projective representation arising from fundamental fusion is associated with a boundary condition. Multiplication in the associated Grothendieck ring leads to a Verlinde-like formula involving A-type twisted affine graphs A^{(2)}_{p} and their coset graphs A^{(2)}_{p,p'}=A^{(2)}_{p} x A^{(2)}_{p'}/Z_2. This provides compact formulas for the conformal partition functions with W-projective boundary conditions. On the torus, we propose modular invariant partition functions as sesquilinear forms in W-projective and rational minimal characters and observe that they are encoded by the same coset fusion graphs.