Integrable WZW insertions on a punctured sphere

Fix a finite-dimensional complex simple $\mathfrak{g}$, its invariant form $B$ with long roots of square $2$, dual Coxeter number $h^{\vee }$, and highest root $\theta$. At level $k\in {\mathbb{Z}}_{\ge 0}$, an integrable WZW insertion of label $\lambda$ at a marked point is the affine module $L_k(\lambda )$ of lem-cft-wzw-integrable-affine-highest-weight-modules-exist-positive-energy, where $\lambda$ is dominant integral and $⟨\lambda ,{\theta }^{\vee }⟩\le k$. The affine bracket is the one in def-untwisted-affine-central-extension, with central generator acting by $k$, and $[d,x_m]=m{x}_m$. The highest vector has $d$-weight zero. The horizontal top is the degree-zero space $V_{\lambda }$, with its finite-dimensional $\mathfrak{g}$-action. A finite collection of such insertions uses one common level $k$ and the same normalized form.

The Sugawara denominator is $k+h^{\vee }$, which is nonzero under these hypotheses. The grading asserted here is the lower-bounded affine degree $-d$; restrictedness has the exact meaning in def-positive-energy-affine-module. It does not assert an integral Sugawara $L_0$-spectrum, unitarity, a finite-rank conformal-block bundle, or a full left-right CFT.

Restricted affine-current module

Integrable affine WZW highest-weight modules are graded and restricted

Untwisted affine central extension