Vertex operator algebras form a rigorous mathematical framework for two-dimensional conformal field theory and have deep connections with representation theory, low-dimensional topology, and quantum physics. A central goal in the field is to understand when a vertex operator algebra is strongly rational—a property ensuring that its category of representations forms a modular tensor category, a structure with important applications in topological quantum field theory and quantum computation.
Robert McRae, an associate professor in the Yau Mathematical Sciences Center (YMSC), has published a research article entitled “On rationality for C2-cofinite vertex operator algebras” in the Cambridge Journal of Mathematics. The paper establishes new structural results in the theory of vertex operator algebras (VOAs), resolves long-standing conjectures, and provides powerful new criteria for determining rationality.
Proof of a key identity for showing rigidity, using graphical calculus from braided tensor category theory
Establishing Structural Criteria for Rationality
In this work, McRae studies a broad class of VOAs known as C2-cofinite vertex operator algebras. While C2-cofiniteness guarantees certain finiteness and analytic properties, determining when such algebras are rational has remained a challenging problem.
The paper proves three main results. First, McRae shows that if the module category of a simple, self-contragredient, C2-cofinite vertex operator algebra is rigid as a tensor category, then it is automatically a factorizable finite ribbon category—that is, a (possibly non-semisimple) modular tensor category. This partially confirms a widely anticipated structure theorem in logarithmic conformal field theory.
Second, the paper establishes a modularity criterion for rigidity. If the modular S-transformation of the vacuum character can be expressed purely as a linear combination of ordinary characters (without pseudo-trace contributions), then the module category is rigid. This result connects modular invariance of characters with categorical duality.
Third, McRae proves that if the Zhu algebra of a C2-cofinite vertex operator algebra is semisimple, then the algebra itself is rational. This provides a practical and internally verifiable criterion for rationality, significantly simplifying the process of proving strong rationality in many cases.
Resolving a Long-Standing Conjecture on Affine W-Algebras
An important application of these results is the resolution of the Kac–Wakimoto–Arakawa conjecture concerning exceptional affine W-algebras. These algebras arise from quantum Drinfeld–Sokolov reduction of affine Lie algebra vertex operator algebras at admissible levels and play a central role in representation theory and conformal field theory.
Using recent results showing that the relevant Zhu algebras are semisimple, McRae’s theorem implies that all such C2-cofinite exceptional affine W-algebras are strongly rational. This completes the proof of the conjecture in full generality.
The paper also advances the long-standing coset rationality problem. Given a strongly rational vertex operator algebra containing a strongly rational subalgebra, it has been unclear when the corresponding coset (or commutant) algebra inherits rationality. McRae proves that if the coset algebra satisfies the C2-cofiniteness condition, then it is strongly rational. This reduces the problem to verifying C2-cofiniteness and clarifies the structural behavior of such subalgebras.
The work combines advanced techniques from tensor category theory with analytic methods involving genus-one correlation functions and modular transformations. By bridging analytic and categorical approaches, the paper unifies several strands of research in vertex operator algebras and provides new tools for the study of both rational and logarithmic conformal field theories.
Link to paper: https://dx.doi.org/10.4310/CJM.260225023743
Editor: Li Han
