Logarithmic and quasi-lisse modularity have distinct scopes

For a $C_2$-cofinite vertex operator algebra $V$ with no nonzero negative-weight states, Huang's genus-one modular-invariance theorem concerns spaces of analytic extensions of pseudo-$q$-traces, shifted by $-c\mathrm{/}24$, of products of geometrically modified intertwining operators among grading-restricted generalized modules; the intertwining operators may be logarithmic. Its inputs include a finite-dimensional associative algebra $P$ with a symmetric linear functional; a grading-restricted generalized $V$-$P$ bimodule that is projective as a right $P$-module; and $P$-compatible intertwining operators. These are pseudo-trace correlation functions, not ordinary characters obtained from an ordinary trace on a semisimple $L(0)$-grading.

The 2026 quasi-lisse result has different data. It starts with a charged conformal vertex algebra $(V,\omega ,h)$, where $Y(h,z)={\sum }_{n\in \mathbb{Z}}{h}_n{z}^{-n-1}$, $h\in V_1$, $h_0$ acts semisimply on $V$ with integer eigenvalues, $L(1)h=0$, and $h_{n}h=0$ for $n\ge 2$ while $h_{1}h=2\kappa \mathbf{1}$ for some $\kappa \in \mathbb{C}$, and assumes finite strong generation (generation by finitely many states using derivatives and normally ordered products) and stable quasi-lissity. The charge grading defines the charged conformal-block sheaf for a degree-zero line bundle $L$ on an elliptic curve $E$. Here the Zhu Poisson algebra is $R_V=V\mathrm{/}{C}_2(V)$ with its induced commutative product and Poisson bracket; the charge-zero quotient is obtained by killing the ideal generated by nonzero $h_0$-charge subspaces. Stable quasi-lisse means the spectrum of that quotient has finitely many symplectic leaves. The theorem gives a holonomic D-module with regular singularities along a lattice $N\alpha \in \mathbb{Z}+\mathbb{Z}\tau$ and an integrable connection away from it. Stable rationality means that the category of finitely generated stable $V$-modules is semisimple; under this additional condition, trace functions of irreducible stable modules span the charged blocks and are flat on $0<-N\mathrm{Im}(\alpha )<\mathrm{Im}(\tau )$, with covariance under the Jacobi group $S{L}_2(\mathbb{Z})⋉{\mathbb{Z}}^2$.

These are not the finite semisimple character/Verlinde theorem for a strongly rational VOA. Strong rationality here means CFT type, $C_2$-cofiniteness, rationality, and self-contragredience (with simplicity in the workspace convention). The hypotheses and outputs above do not identify a logarithmic pseudo-trace space or a charged elliptic family with that finite semisimple modular-functor statement.

C2-cofiniteness

Logarithmic generalized module

The source statements were checked in the pinned full texts. The Huang theorem is a 2025-version arXiv preprint and the charged quasi-lisse theorem is arXiv:2605.29921v1; the cutoff for this source-status observation is 2026-10-26. No completeness claim about all later or adjacent work is made.