Sugawara modes and their normal ordering

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra as in Loop algebra of a simple Lie algebra, with normalized invariant form $B$ from Untwisted affine central extension. Let $\rho$ be a restricted affine-current module of level $k$ as in Restricted affine-current module, and assume $k+h^{\vee }\mathrm{\ne }0$ for the number in Dual Coxeter number for the normalized invariant form. Choose any $B$-dual bases $(x_a)$ and $(x^a)$ of $\mathfrak{g}$. Write $J_x(n)=\rho (x\otimes t^n)$.

For modes, define $:J_{x_a}(r)J_{x^a}(s):=\{\begin{array}{ll}{J}_{x_a}(r)J_{x^a}(s),& r<0,\\ J_{x^a}(s)J_{x_a}(r),& r\ge 0.\end{array}$ The Sugawara mode of index $m\in \mathbb{Z}$ is $L_m=\frac{1}{2(k+h^{\vee })}{\sum }_a{\sum }_{r\in \mathbb{Z}}:J_{x_a}(r)J_{x^a}(m-r):\mathrm{.}$ If a compact real form $\mathfrak{k}$ of Compact real form of a complex semisimple Lie algebra is additionally supplied, put $Q=-B{\mathrm{\mid }}_{\mathfrak{k}}$; it is positive by the normalized form in Residue two cocycle on a loop algebra. If $(X_i)$ is $Q$-orthonormal, then $(-X_i)$ is its $B$-dual, so the same formula reads $L_m=-\frac{1}{2(k+h^{\vee })}{\sum }_i{\sum }_{r\in \mathbb{Z}}:J_{X_i}(r)J_{X_i}(m-r):\mathrm{.}$ By Restricted Sugawara quadratic mode sums are well defined, the mode sum is locally finite on every vector and independent of the chosen dual bases. At the critical level $k=-h^{\vee }$ the scalar denominator is zero and this normalized formula is not defined; no operator is asserted there.