C₂-Cofiniteness, Rationality, and Regularity

$C_2$ -cofiniteness is a finite-dimensional quotient condition, rationality is semisimplicity for admissible modules, and regularity is semisimplicity for all weak modules. They are not interchangeable definitions. For a VOA of CFT type, regularity is equivalent to rationality together with $C_2$ -cofiniteness; deleting rationality is invalid, as the logarithmic triplet algebras are $C_2$ -cofinite but nonsemisimple.

Required background. Vertex Operator Algebras: Axioms, Grading, and Locality supplies modes and gradings. Modules, Intertwining Operators, and Tensor Categories distinguishes weak, admissible, ordinary, and generalized modules. Minimal Models and Fusion Rules supplies the Ising model used below.

Helpful background. Modular Crossing and Spectral Bounds explains why finite modular data matter physically.

Three distinct finiteness conditions

For a VOA $V$ , define

C_2(V)=\operatorname{span}_{\mathbb C}\{u_{-2}v:u,v\in V\}.

The VOA is $C_2$ -cofinite if $V/C_2(V)$ is finite-dimensional. This quotient is a commutative Poisson algebra: the products induced by $u_{-1}v$ and $u_0v$ retain a finite shadow of the operator products. Cofiniteness strongly constrains spanning sets, characters, and differential equations, but says nothing by itself about splitting extensions of modules.

Fix the following conventions. “Rational” means every admissible, $\mathbb Z_{\geq0}$ -gradable module is a direct sum of irreducible admissible modules. “Regular” means every weak module is a direct sum of irreducible ordinary modules. A VOA is of CFT type when $V_n=0$ for $n<0$ and $V_0=\mathbb C\mathbf1$ . With these definitions, Abe, Buhl, and Dong prove:

Theorem. If $V$ is of CFT type, then $V$ is regular if and only if it is both rational and $C_2$ -cofinite Abe, Buhl, and Dong 2004, Theorem 4.5, pp. 3397–3398.

The forward direction includes both semisimplicity and finite spanning control. For the reverse direction, cofiniteness forces irreducible weak modules to be ordinary and supplies finite mode-spanning sets; rationality then splits admissible subquotients. The proof is not the assertion that a finite-dimensional quotient makes every module semisimple. That false step would confuse finite representation type with vanishing extension groups.

Consequences also require their own hypotheses. Under $C_2$ -cofiniteness, irreducible weak modules are ordinary and fusion rules are finite in the setting of the cited theorem. A finite set of irreducibles plus modularly behaved trace functions still does not automatically yield rigidity or a modular tensor category; those need the tensor-category theorems and their self-duality and semisimplicity assumptions.

The Ising $C_2$ quotient

The first QFT application is the minimal model treated in Minimal Models and Fusion Rules. Let $V=L(\tfrac12,0)$ be generated by the conformal vector $\omega=L_{-2}\mathbf1$ . Modulo $C_2(V)$ , every state containing a Virasoro mode $L_{-n}$ with $n\geq3$ vanishes after rewriting it as a derivative or an element $u_{-2}v$ . Therefore the quotient is generated by $x=[\omega]$ .

The vacuum singular vector at level six has a nonzero $L_{-2}^3\mathbf1$ term; every other term contains either $L_{-n}$ with $n\geq3$ or derivatives that vanish in the quotient. Hence its image gives $x^3=0$ . The lower classes $1,x,x^2$ survive, so

V/C_2(V)\cong\mathbb C[x]/(x^3).

This explicit three-dimensional quotient proves $C_2$ -cofiniteness for the Ising VOA. Separately, minimal-model representation theory proves rationality. The theorem above then yields regularity. Notice that the quotient dimension happens here to equal the number of irreducible ordinary modules, but no general theorem identifies those two integers; the Zhu algebra, not the $C_2$ quotient, controls lowest-weight module data.

An independent check comes from the graded spanning set. Since $x^3=0$ , the quotient has representatives only at weights $0,2,4$ . Direct inspection of the vacuum module shows nonzero classes at those weights and no additional generator, confirming dimension three.

Counterexample: the triplet algebra

For $p\geq2$ , the triplet VOA $\mathcal W(p)$ has central charge

c=1-\frac{6(p-1)^2}{p}.

Adamović and Milas prove that $\mathcal W(p)$ is $C_2$ -cofinite and has $2p$ inequivalent irreducible modules, but is irrational; nonsplit indecomposable and logarithmic modules occur Adamović and Milas 2008, Theorems 2.1, 3.12, and Proposition 4.2, pp. 2674–2687. Thus

C_2\text{-cofinite}\nRightarrow\text{rational or regular}.

This is the adversarial test: any argument that infers semisimplicity merely from a finite $C_2$ quotient must misclassify $\mathcal W(p)$ . What survives is finite mode-spanning control and a finite irreducible set, together with logarithmic modular phenomena; what fails is decomposition of every module into simples.

The counterexample is structural rather than numerical: its indecomposable modules retain extension data that neither the quotient dimension nor the irreducible count can detect.

Exercises

Explain why regularity implies rationality under the definitions above.

Solution

Every admissible module is in particular a weak module. Regularity decomposes it into irreducible ordinary modules, and ordinary modules are admissible after shifting the lowest conformal weight. Hence every admissible module is completely reducible.

Why does finite-dimensionality of $V/C_2(V)$ not rule out an $L_0$ Jordan block?