Geometrization conjecture for holomorphic conformal field theories

Let $\overline{\mathcal{M}}_\infty$ be the stable moduli space of curves and let $\lambda_\infty$ be its stable Hodge line bundle. For a positive integer $k$, consider the space $H^0(\overline{\mathcal{M}}_\infty,\lambda_\infty^{\otimes k})$ of stable Teichmüller modular forms. A group-like element is an element with the multiplicative compatibility property under the relevant stable degeneration or clutching operations. A vacuum section is the stable partition-function section associated with a holomorphic vertex operator algebra.

Geometrization conjecture. If $k$ is a positive integer divisible by $4$, every group-like element of

$$H^0(\overline{\mathcal{M}}_\infty,\lambda_\infty^{\otimes k})$$

is the vacuum section of a holomorphic vertex operator algebra.

The conjecture is motivated by the Friedan–Shenker geometrization program for two-dimensional conformal field theories. The source gives no resolution, and the terminology and precise correspondence between group-like elements and vacuum sections should be checked against the paper's definitions.