[Paper Review] Ocneanu Cells and Boltzmann Weights for the SU(3) ADE Graphs

David Evans, Mathew Pugh|ORCA Online Research @Cardiff|Jun 23, 2009

Algebraic structures and combinatorial models35 references17 citations

TL;DR

This paper computes Ocneanu cells and Boltzmann weights for SU(3) ADE graphs, excluding the exceptional $\mathcal{E}_4^{(12)}$ case, enabling the construction of integrable statistical mechanical models and providing a foundation for SU(3) planar algebras and braided subfactors. The results realize all SU(3) modular invariants through $\alpha$-induction and Hecke algebra representations on nimrep graphs.

ABSTRACT

We determine the cells, whose existence has been announced by Ocneanu, on all the candidate nimrep graphs except $\mathcal{E}_4^{(12)}$ proposed by di Francesco and Zuber for the SU(3) modular invariants classified by Gannon. This enables the Boltzmann weights to be computed for the corresponding integrable statistical mechanical models and provide the framework for studying corresponding braided subfactors to realise all the SU(3) modular invariants as well as a framework for a new SU(3) planar algebra theory.

Motivation & Objective

• To determine Ocneanu cells for all SU(3) ADE graphs except $\mathcal{E}_4^{(12)}$, as announced by Ocneanu.
• To compute Boltzmann weights for integrable statistical mechanical models associated with these graphs.
• To provide a framework for realizing all SU(3) modular invariants via braided subfactors and $\alpha$-induction.
• To lay the groundwork for a new SU(3) planar algebra theory and its modules.
• To resolve discrepancies in prior Hecke algebra representations, particularly for $\mathcal{E}^{(24)}$, by identifying typographical errors in earlier work.

Proposed method

• Computes numerical values of Ocneanu cells using the representation theory of the SU(3) Hecke algebra.
• Constructs matrix representations $U^{(i,j)}$ of the Hecke algebra on the nimrep graphs, encoding trivalent vertex weights.
• Uses the quantum integer notation $[n] = \frac{q^n - q^{-n}}{q - q^{-1}}$ to express matrix entries in terms of quantum dimensions.
• Applies the $\alpha$-induction procedure to link subfactors to modular invariants via the Verlinde algebra.
• Compares computed representations with those in Sochen [41], identifying inconsistencies and attributing them to typographical errors.
• Employs spectral measures and graph spectra to validate the correspondence between nimrep graphs and modular invariants.

Experimental results

Research questions

• RQ1 What are the explicit numerical values of the Ocneanu cells for the SU(3) ADE graphs, excluding $\mathcal{E}_4^{(12)}$?
• RQ2 How can the Boltzmann weights for integrable SU(3) statistical models be derived from these cells?
• RQ3 Are the Hecke algebra representations on the nimrep graphs unique, and do they match those in prior literature?
• RQ4 Can all SU(3) modular invariants be realized through braided subfactors using this framework?
• RQ5 What is the role of the $\mathcal{E}_4^{(12)}$ graph in the SU(3) ADE classification, and why is it excluded from this computation?

Key findings

• The Ocneanu cells have been computed for all SU(3) ADE graphs except $\mathcal{E}_4^{(12)}$, providing a complete set of numerical weights for trivalent vertices.
• Boltzmann weights for integrable SU(3) statistical models are derived from the computed cells, enabling the construction of these models.
• Matrix representations $U^{(i,j)}$ of the SU(3) Hecke algebra are explicitly determined, with entries expressed in terms of quantum integers $[n]$.
• Discrepancies in the Hecke representation for $\mathcal{E}^{(24)}$ reported by Sochen [41] are attributed to typographical errors, as the computed weights differ in absolute value.
• The framework supports the realization of all SU(3) modular invariants via $\alpha$-induction and braided subfactors, with the exception of $\mathcal{E}_4^{(12)}$.
• The results provide a consistent foundation for developing a new SU(3) planar algebra theory and its module categories.