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The normalized Sugawara formula is undefined at critical level

Statement refuted

For ${\hat{s l}}_{2}$ , the normalized Sugawara formula defines operators at the critical level $k = - 2$ .

Facts & Assumptions

Given: ${s l}_{2}$ with basis $e, f, h$ , brackets $[h, e] = 2 e$ , $[h, f] = - 2 f$ , $[e, f] = h$ , and normalized form $B (h, h) = 2$ , $B (e, f) = B (f, e) = 1$ .

[F1]

Dual Coxeter number for the normalized invariant form: $h^{\lor}$ is half the adjoint Casimir scalar.

[F2]

Dual Coxeter number for the normalized invariant form: the adjoint Casimir acts as the scalar $γ_{B} I$ .

[F3]

Sugawara modes and their normal ordering: the normalized mode formula divides by $2 (k + h^{\lor})$ and is defined only when this scalar is nonzero.

Counterexample

1.1givenF1F2algebra

The $B$ -dual basis to $(h, e, f)$ is $(h / 2, f, e)$ , so $Ω_{B} = h^{2} / 2 + e f + f e$ . In the adjoint representation, $ad (Ω_{B}) e = \frac{1}{2} [h, [h, e]] + [e, [f, e]] + [f, [e, e]] = 2 e + 2 e + 0 = 4 e$ . By [F2] the operator is the scalar $γ_{B} I$ , so its eigenvalue on $e$ is $γ_{B} = 4$ ; [F1] then gives $h^{\lor} = γ_{B} / 2 = 2$ .

2.1F3step 1.1algebra∎

At $k = - 2$ , the denominator in [F3] is $2 (k + h^{\lor}) = 2 (- 2 + 2) = 0$ . Division by zero is undefined in $C$ , so the displayed normalized operator formula does not define $L_{m}$ at this level. This refutes the statement; it does not assert that every possible renormalized critical-level construction is impossible.

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Sources

David Ben-Zvi and Edward Frenkel, Geometric Realization of the Segal–Sugawara Construction, §3.4.1 (standard reference, not scraped)