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Stress-tensor modes satisfy the Virasoro relation

Statement

Let D be a complex vector space on which the holomorphic modes of stress-tensor data act as endomorphisms and leave D invariant. Assume the TT principal part and the nested radial contour-product prescription in the stress-data definition. Define Lm=Res⁡zzm+1T(z) for every m∈Z. Then, on D,

[Lm,Ln]=(m−n)Lm+n+c12(m3−m)δm+n,0I.

With the Virasoro convention specified in the supplier definition, the operators Lm and central action C↦cI satisfy its defining bracket. The conclusion is conditional on the declared OPE, common domain, and contour-product rule; it does not construct stress data or assert analytic convergence.

Facts & Assumptions

Given: The stress-tensor modes on a common invariant domain, the TT principal part with central scalar c, and the nested residue rule.

[F1]

Holomorphic stress-tensor data in a plane CFT: The mode realization is the formal coefficient array T(z)=∑k∈Ztkz−k−2 with tk∈End⁡C(D), the specified injective map to the weak dual, and scalar Laurent/smearing compatibility. Thus Lm=Res⁡zzm+1T(z)=tm is an endomorphism, not a residue of an arbitrary distribution. The TT principal part is c/(2(z−w)4)+2T(w)/(z−w)2+∂wT(w)/(z−w), and the contour prescription is [Lm,Ln]=Res⁡wwn+1Res⁡z=wzm+1T(z)T(w).

[F2]

Virasoro algebra in the Lm convention: The defining bracket is [Lm,Ln]=(m−n)Lm+n+(m3−m)δm+n,0C/12, with C central; central charge c means C acts as cI.

[A1]

All displayed mode products and nested residues are defined on the same invariant domain D; no boundary or contact term modifies the declared contour rule.

Proof

Given: Integers m,n, [F1]–[F2], and the common-domain assumption [A1].

1.1givenF1A1algebra

Use the compatible formal representative in [F1], so every coefficient and mode product lies in End⁡C(D). Substitute the three singular terms from [F1] into the nested residue. In the inner residue, expand zm+1=∑k≥0(m+1k)wm+1−k(z−w)k; only k=3,1,0 contribute to the fourth-, second-, and first-order poles respectively.

2.1step 1.1algebra

The second-order pole contributes 2(m+1)Res⁡wwm+n+1T(w)=2(m+1)Lm+n. The simple pole contributes Res⁡wwm+n+2∂wT(w)=−(m+n+2)Res⁡wwm+n+1T(w)=−(m+n+2)Lm+n, because Res⁡w∂w(wm+n+2T(w))=0 coefficientwise. Their sum is (m−n)Lm+n.

2.2step 1.1algebra

The fourth-order pole contributes c2(m+13)Res⁡wwm+n−1I. Since Res⁡wwq=1 exactly for q=−1 and is 0 otherwise, this is c2(m+13)δm+n,0=c12(m3−m)δm+n,0I. The polynomial identity holds for every integer m using (m+13)=(m+1)m(m−1)/6.

3.1F2step 2.1step 2.2algebra∎

Adding steps 2.1–2.2 gives the stated mode commutator. Comparing with [F2] identifies the central action as C↦cI and verifies precisely the Virasoro bracket in the fixed convention.

Depends on

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