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The diagonal SU(2) coset central charge

Physical scope

Conditional chiral central-charge calculation for the compact SU(2) affine coset SU(2)_k times SU(2)1 over diagonally embedded SU(2){k+1}; it makes no claim of a completed full local CFT or physical realization.

Example

Let $k \in Z_{> 0}$ . Take independent restricted affine current sectors for $s u (2)_{C} ≅ {s l}_{2} (C)$ at levels $k$ and $1$ , and embed the denominator algebra diagonally by $x \mapsto (x, x)$ . Assume the Sugawara modes and common stress-tensor data satisfy the exact restricted-module and embedding conditions of the GKO coset definition. The induced denominator level is $k + 1$ , and the chiral coset stress tensor is Virasoro with $c = 1 - \frac{6}{(k + 2) (k + 3)} .$ This calculation specifies only the central charge of the chiral coset data. It does not assert a full left-right state space, its OPE or crossing data, unitarity, modular invariance, or a physical realization.

Example references

Compact Wess–Zumino–Witten model and its left-right completion

GKO coset CFT with a specified affine embedding

Sugawara embeddings give a Virasoro coset stress tensor

Facts & Assumptions

Given: The compact WZW current convention, the GKO coset data, the diagonal ${s l}_{2}$ embedding, and $k \in Z_{> 0}$ .

[F1]

Compact Wess–Zumino–Witten model and its left-right completion: the basic invariant form has long roots of squared length $2$ , the current algebra is at level $k$ , and the Sugawara central charge on a restricted module is $k \dim g / (k + h^{\lor})$ .

[F2]

GKO coset CFT with a specified affine embedding: the denominator level is induced by the embedding, and $T_{G / H} = T_{G} - T_{H}$ is defined on the specified common restricted-module data.

[F3]

Casimir operator relative to an invariant form: for a basis and its dual with respect to $B$ , the adjoint Casimir is $C_{B} = \sum_{a} ad (x_{a}) ad (x^{a})$ .

[F4]

Dual Coxeter number for the normalized invariant form: if the adjoint Casimir for the normalized form acts as $γ_{B} I$ , then $h^{\lor} = γ_{B} / 2$ .

[F5]

Sugawara embeddings give a Virasoro coset stress tensor: for simple factors with levels $k_{i}$ , embedding indices $I_{i}$ , and induced level $ℓ = \sum_{i} I_{i} k_{i}$ , the difference of the numerator and denominator Sugawara modes is Virasoro with the stated difference of central charges, conditional on the exact restricted-module and noncritical-level hypotheses.

[A1]

The two numerator current factors have no cross commutators, and the diagonal map uses the identity embedding into each factor.

[A2]

On a basis $H, E, F$ of ${s l}_{2} (C)$ , the brackets are $[H, E] = 2 E$ , $[H, F] = - 2 F$ , and $[E, F] = H$ , and the basic form has $B (H, H) = 2$ , $B (E, F) = B (F, E) = 1$ , with all other basis pairings zero.

[A3]

The modules are restricted affine-current modules and all levels used here are noncritical; the selected compact model data meet the GKO definition's common-domain conditions.

Verification

Given: [F1]–[F3], [A1]–[A2], and the positive integer $k$ .

1.1F2F5A1givenalgebra

In each component the diagonal map is the identity on ${s l}_{2}$ , so $B_{i} (x, y) = B_{h} (x, y)$ and the two embedding indices are $I_{1} = I_{2} = 1$ . The level in [F5] is therefore $ℓ = 1 \cdot k + 1 \cdot 1 = k + 1$ .

1.2A2F3F4algebra

The basis in [A2] gives $\dim {s l}_{2} = 3$ . Its $B$ -dual basis is $(H / 2, F, E)$ , so [F3] gives $C_{B} = \frac{1}{2} ad (H)^{2} + ad (E) ad (F) + ad (F) ad (E)$ . On $H$ , the last two summands each give $2 H$ and the first is zero; on $E$ , the first gives $2 E$ , the second gives $2 E$ , and the third is zero; on $F$ , the first gives $2 F$ , the second is zero, and the third gives $2 F$ . Thus $C_{B} = 4 I$ on this basis, and [F4] gives $h^{\lor} = 2$ .

2.1F1step 1.2algebra

By [F1], the numerator factors have central charges $3 k / (k + 2)$ and $3 / (1 + 2) = 1$ , using $\dim {s l}_{2} = 3$ and $h^{\lor} = 2$ from step 1.2.

3.1A3F5step 1.1step 1.2step 2.1algebra

Apply [F5] with denominator level $k + 1$ from step 1.1 and $h^{\lor} = 2$ from step 1.2. The coset central charge is therefore $c = 3 k / (k + 2) + 1 - 3 (k + 1) / (k + 3)$ . The noncritical conditions hold because $k + 2$ , $1 + 2$ , and $k + 3$ are positive.

4.1step 3.1algebra

Put $D = (k + 2) (k + 3)$ . The numerator after bringing the expression in step 3.1 to denominator $D$ is $3 k (k + 3) + (k + 2) (k + 3) - 3 (k + 1) (k + 2) = k (k + 5) = D - 6$ . Thus $c = (D - 6) / D = 1 - 6 / ((k + 2) (k + 3))$ , as stated.

5.1givenstep 4.1algebra∎

Since $k \geq 1$ , $D \geq 12$ , so $0 < 6 / D \leq 1 / 2$ . Therefore $1 / 2 \leq c < 1$ , with $c = 1 / 2$ at $k = 1$ . These bounds are numerical consequences only; they assert no unitarity or minimal-model identification.

Remarks

The full coset module decomposition, selection rules, field identifications, left-right pairing, and local OPE data remain separate inputs. The cited central-charge agreement does not establish any of them.

The diagonal SU(2) coset central charge

Example

Example references

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources