Positive-energy unitary module for an affine current algebra

Let $\mathfrak{k}$ be the compact real form of the complex simple algebra underlying Restricted affine-current module. A positive-energy unitary module consists of a complex inner-product space (not assumed complete) with positive-definite inner product $H={⨁}_{r\in {\mathbb{Z}}_{\ge 0}}{H}_r,$ where the displayed direct sum is orthogonal. Each $H_r$ is finite-dimensional and $H_0\mathrm{\ne }0$. The representation of the restricted affine-current algebra has central level an integer $k\ge 0$, satisfies $X(n)H_r\subseteq H_{r-n}$ for every $X\in \mathfrak{k}$ (with $H_s=0$ for $s<0$), and satisfies $X(n{)}^{\ast }=-X(-n)$. In particular zero modes are skew-adjoint on each energy space. This definition selects the nonnegative integral level branch; it does not assert that every positive-energy unitary representation has integral level. The module is irreducible if it has no nonzero proper subspace invariant under all current modes.

Define the energy operator on the graded space by $E{\mathrm{\mid }}_{H_r}=r\mathrm{id}$. The stated mode shift gives $[E,X(n)]=-nX(n)$. This is the convention corresponding to $E=-d$ when adjoining the degree derivation $d$ of Degree derivation and full untwisted affine algebra, which satisfies $[d,x_n]=n{x}_n$. The derivation $d$ is not part of the current central extension used in the Sugawara mode formula.