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The critical-level sl2 affine vertex algebra and its current OPE

Example

Take g=sl2(C) with the standard basis e,f,h and level k=−2. The universal affine vertex algebra V−2(sl2) exists. Its current fields satisfy Je(z)Jf(w)∼Jh(w)z−w−2id⁡(z−w)2. For this normalized form h∨=2, so the denominator in the displayed normalized Sugawara formula is zero. The example separates existence of the affine vertex algebra from that particular conformal-vector formula; it asserts no alternative critical-level stress tensor.

Facts & Assumptions

Given: The normalized sl2 basis, the affine current bracket, the universal affine vertex algebra theorem, and the Sugawara formula convention.

[F1]

Universal affine vertex algebras exist at every complex level: Vk(g) is a vertex algebra for every complex level k.

[F2]

The special linear Lie algebra sl_2: [h,e]=2e, [h,f]=−2f, and [e,f]=h.

[F3]

Residue two cocycle on a loop algebra: for normalized long-root-square-2 form, opposite root vectors with [e,f]=h=α∨ satisfy B(e,f)=1.

[F4]

Dual Coxeter number for the normalized invariant form: h∨=γB/2, where the Casimir ΩB for B acts in the adjoint representation by γBI.

[F5]

Sugawara modes and their normal ordering: the normalized Sugawara formula has denominator 2(k+h∨) and is defined only when k+h∨≠0.

[F6]

Affine current fields are local with the normalized current OPE: the singular OPE of currents is Jx(z)Jy(w)∼J[x,y](w)/(z−w)+kB(x,y)id⁡/(z−w)2.

Proof

1.1F1given

Since −2∈C, [F1] gives the universal affine vertex algebra at this level; its construction does not divide by k+h∨.

1.2F2F3F4algebra

By [F2] and [F3], invariance gives B(h,h)=B([e,f],h)=B(e,[f,h])=2. It also gives B(h,e)=B(h,f)=B(e,e)=B(f,f)=0: for example 2B(e,e)=B([h,e],e)=B(h,[e,e])=0, and 2B(h,e)=B([h,e],h)=−B(e,[h,h])=0; the f identities follow from [h,f]=−2f in the same way. Thus the B-dual basis to (h,e,f) is (h/2,f,e). The adjoint Casimir is therefore ΩB=12ad⁡(h)2+ad⁡(e)ad⁡(f)+ad⁡(f)ad⁡(e). On h its last two terms give 2h+2h=4h; on e its first two nonzero terms give 2e+2e=4e; on f its first and last nonzero terms give 2f+2f=4f. Hence γB=4 and [F4] gives h∨=2.

1.3F2F3F6algebra

Applying [F6] to x=e,y=f, with [e,f]=h, B(e,f)=1, and k=−2, gives the stated current OPE.

2.1F1F5step 1.1step 1.2algebra∎

At k=−2 and h∨=2, the denominator in [F5] is 2(−2+2)=0. Therefore this normalized Sugawara expression is undefined, even though step 1.1 supplies the affine vertex algebra. No Sugawara or Virasoro structure is inferred.

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