[Paper Review] Braided Subfactors, Spectral Measures, Planar algebras and Calabi-Yau algebras associated to SU(3) modular invariants

David Evans, Mathew Pugh|arXiv (Cornell University)|Oct 20, 2011

Algebraic structures and combinatorial models92 references3 citations

TL;DR

This paper establishes a framework linking braided subfactors, planar algebras, and almost Calabi-Yau algebras to $SU(3)$ modular invariants in conformal field theory. It constructs a non-degenerate bilinear form on an algebra $A$ associated with $SU(3)$ $\mathcal{ADE}$ graphs, proves the Nakayama automorphism equals the graph permutation $\nu$, and derives a 5-term resolution showing $A$ is an almost Calabi-Yau algebra of dimension 3, providing invariants for subfactors via Hochschild and cyclic homology.

ABSTRACT

Braided subfactors of von Neumann algebras provide a framework for studying two dimensional conformal field theories and their modular invariants. We review this in the context of SU(3) conformal field theories through corresponding SU(3) braided subfactors and various subfactor invariants including spectral measures for the nimrep graphs, A_2-planar algebras and almost Calabi-Yau algebras.

Motivation & Objective

• To understand the structure of $SU(3)$ modular invariants in rational conformal field theory using subfactor theory.
• To connect braided subfactors to spectral measures of nimrep graphs and planar algebras.
• To construct an almost Calabi-Yau algebra $A(\mathcal{G}, W)$ from a cell system $W$ on an $SU(3)$ $\mathcal{ADE}$ graph $\mathcal{G}$.
• To derive a projective resolution of $A$ as an $A$-$A$ bimodule to compute Hochschild and cyclic homology.
• To show that these homological invariants are intrinsic to the subfactor $N \subset M$ associated with the graph and cell system.

Proposed method

• Construct a non-degenerate bilinear form $f$ on the algebra $A$ by setting $f(u_{j\nu(j)}) = 1$ for top-degree generators and zero elsewhere.
• Define the Nakayama automorphism $\beta$ via the duality $(x,y) = (y, \beta(x))$, and prove $\beta = \nu$ using the graph permutation $\nu$ from the permutation matrix $P$.
• Derive a 5-term resolution of $A$ as an $A$-$A$ bimodule using tensor products over $R$ and $V$, with maps $\mu_0$ to $\mu_4$ defined via combinatorial sums over edges and cells.
• Use the resolution to show $A$ is an almost Calabi-Yau algebra of dimension 3, analogous to Calabi-Yau algebras from finite subgroups $\Gamma \subset SU(3)$.
• Relate the algebra $A(\mathcal{G}, W)$ to a subfactor $N \subset M$ via a nimrep construction, ensuring the graph $\mathcal{G}$ arises as the fusion graph.
• Establish that Hochschild (co)homology and cyclic homology of $A$ are invariants of the subfactor $N \subset M$, independent of the choice of cell system.

Experimental results

Research questions

• RQ1 How can braided subfactors be used to classify and construct $SU(3)$ modular invariants in conformal field theory?
• RQ2 What is the role of spectral measures and planar algebras in realizing the nimrep graphs of $SU(3)$ modular invariants?
• RQ3 How does the algebra $A(\mathcal{G}, W)$ associated with an $SU(3)$ $\mathcal{ADE}$ graph and cell system $W$ relate to Calabi-Yau structures?
• RQ4 What is the structure of the Hochschild and cyclic homology of $A(\mathcal{G}, W)$, and how do they reflect subfactor invariants?
• RQ5 Can the resolution of $A$ as an $A$-$A$ bimodule be used to define a Calabi-Yau-type condition for $SU(3)$-related algebras?

Key findings

• The Nakayama automorphism $\beta$ of the algebra $A(\mathcal{G}, W)$ is equal to the permutation $\nu$ of the graph vertices, as determined by the permutation matrix $P$ in the cell system.
• The algebra $A(\mathcal{G}, W)$ admits a 5-term resolution as an $A$-$A$ bimodule, with maps $\mu_0$ to $\mu_4$ defined via sums over edges and cells, confirming its structure as an almost Calabi-Yau algebra.
• The resolution (33) closely resembles the Calabi-Yau resolution of dimension 3, differing only in the first term, which is $ {}_{1}A_{\beta^{-1}} $ instead of $ A\otimes_R A $, indicating an almost Calabi-Yau structure.
• The Hochschild (co)homology and cyclic homology of $A(\mathcal{G}, W)$ are invariants of the subfactor $N \subset M$, as they depend only on the graph $\mathcal{G}$ and cell system $W$, which are encoded in the subfactor.
• The construction of $A(\mathcal{G}, W)$ from a subfactor $N \subset M$ ensures that the resulting algebra captures the topological and algebraic data of the modular invariant through its homological invariants.
• The work establishes a bridge between subfactor theory, conformal field theory, and noncommutative algebraic geometry by showing that $A(\mathcal{G}, W)$ is an almost Calabi-Yau algebra of dimension 3, extending known results for $SU(2)$ and finite subgroups of $SU(3)$.