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Affine Current Algebras, Virasoro, and Sugawara: Examples and Counterexamples

1 · Prerequisites

2 · Summary

These worked calculations test the normalization and parameter boundary in affine-current-algebras-virasoro-and-sugawara. The level-one central-charge example is conditional on a module satisfying the A-page theorem’s restricted-module hypotheses; it does not infer unitarity for arbitrary affine modules. The level-zero example verifies the trivial representation directly. The critical-level counterexample shows that the displayed normalized quotient is undefined when its denominator vanishes; it does not rule out other critical-level constructions.

Examples

The level-one sl2 Sugawara central charge computes $h^{\lor} = 2$ and $c = 1$ for normalized ${s l}_{2}$ at level one.
The level-zero trivial sl2 module verifies the one-dimensional trivial level-zero representation and its zero Sugawara modes.
The normalized Sugawara formula is undefined at critical level computes the critical level $k = - 2$ for normalized ${s l}_{2}$ and checks the zero denominator.

Each item gives its hypotheses and calculation. These examples clarify the A-page formulas; they are not theorem suppliers for broader module classes.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

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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The level-one sl2 Sugawara central charge

Example

Let ${s l}_{2}$ have basis $e, f, h$ with $[h, e] = 2 e$ , $[h, f] = - 2 f$ , $[e, f] = h$ , and use the invariant form normalized by $B (h, h) = 2$ , $B (e, f) = B (f, e) = 1$ . Then $\dim {s l}_{2} = 3$ and $h^{\lor} = 2$ . Substituting level $k = 1$ into the Sugawara formula gives central charge $c = 1$ . Consequently every level-one module satisfying the hypotheses of Sugawara modes satisfy Virasoro at every noncritical complex level carries the normalized Virasoro action with that central charge.

Facts & Assumptions

Given: The displayed ${s l}_{2}$ basis and form normalization, and a level-one module satisfying the theorem's restricted-module hypotheses.

[F1]

Sugawara modes and their normal ordering: the central-charge denominator is $k + h^{\lor}$ , with critical level $k = - h^{\lor}$ excluded.

[F2]

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

[F3]

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

[F4]

Sugawara modes satisfy Virasoro at every noncritical complex level: under the stated module hypotheses, $c = k \dim (g) / (k + h^{\lor})$ .

Verification

1.1F2F3algebra

The given brackets make ${s l}_{2}$ simple: $ad (h)$ has the three distinct eigenspaces $C e, C f, C h$ ; polynomial projections in $ad (h)$ show any nonzero ideal contains one of $e, f, h$ , and bracketing that vector with the other basis vectors yields all three. 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]–[F3], $2 h^{\lor} = 4$ , hence $h^{\lor} = 2$ .

2.1F1F4step 1.1algebra∎

The dimension is three. [F1] and [F4] therefore give $c = 1 \cdot 3 / (1 + 2) = 1$ . The denominator is nonzero, so the example lies away from the critical level $k = - 2$ . This is a conditional module example: it computes the central charge for every level-one module meeting the theorem hypotheses with no unitarity or integrability requirement.

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The level-zero trivial sl2 module

Example

The one-dimensional trivial representation of the loop central extension, with central level $k = 0$ , is irreducible, unitary, and positive-energy in degree zero. Every Sugawara mode is zero on it, and its Virasoro central charge is $0$ .

Facts & Assumptions

Given: The one-dimensional complex Hilbert space $H = C$ , with every loop-current mode and the central generator acting by zero, and the degree-zero grading.

[F1]

Positive-energy unitary module for an affine current algebra: a positive-energy unitary module has nonnegative energy grading, skew-adjoint compact modes, and a scalar central level.

[F2]

Sugawara modes and their normal ordering: each Sugawara mode is a scalar multiple of a sum of normally ordered products of two current modes.

[F3]

Sugawara modes satisfy Virasoro at every noncritical complex level: the central charge is $k \dim (g) / (k + h^{\lor})$ in this module regime.

[F4]

Dual Coxeter number for the normalized invariant form: $h^{\lor} > 0$ for the normalized compact simple algebra.

Verification

1.1givenF1algebra

The zero action respects the affine bracket because both the current bracket and its central term act as zero; $H_{0} = H$ is one-dimensional and every positive energy space is zero. The inner product is positive and the adjoint relation $0^{*} = - 0$ holds, so the stated representation meets [F1] and is irreducible.

2.1F2F3F4algebra∎

Every summand in [F2] is a product of zero operators, hence $L_{m} = 0$ for all $m$ . In [F3], $k = 0$ and $h^{\lor} > 0$ by [F4], so the denominator is nonzero and $c = 0$ . Thus the Virasoro relation is the zero identity on this example.

Affine Current Algebras, Virasoro, and Sugawara: Examples and Counterexamples

1 · Prerequisites

2 · Summary

Examples

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

The normalized Sugawara formula is undefined at critical level

Statement refuted

Facts & Assumptions

Counterexample

The level-one sl2 Sugawara central charge

Example

Facts & Assumptions

Verification

The level-zero trivial sl2 module

Example

Facts & Assumptions

Verification

Sources