en.wikipedia.org/wiki/Virasoro_algebra
1 correction found
It follows that the symmetry algebra of two-dimensional conformal field theory is the Virasoro algebra.
For a full 2D conformal field theory, the symmetry algebra is not a single Virasoro algebra but two commuting copies of it: a left-moving and a right-moving sector.
Full reasoning
This sentence drops one of the two chiral halves of the symmetry.
In a two-dimensional CFT, local conformal transformations split into holomorphic and antiholomorphic parts. After central extension, that gives two commuting Virasoro algebras, usually written as (\mathrm{Vir}\oplus \overline{\mathrm{Vir}}), not just one Virasoro algebra.
Authoritative summaries say this explicitly:
- The Wikipedia article on Two-dimensional conformal field theory states: "The symmetry algebra of a CFT is the product of two commuting copies of the Virasoro algebra: the left-moving or holomorphic algebra ... and the right-moving or antiholomorphic algebra ..."
- CFT Zoo likewise states: "2d conformal field theories transform under two copies of the Virasoro algebra" and notes that only chiral CFTs use a single copy.
So the sentence would be accurate only for a chiral/holomorphic sector of a 2D CFT, not for the full theory.
2 sources
- Two-dimensional conformal field theory - Wikipedia
The symmetry algebra of a CFT is the product of two commuting copies of the Virasoro algebra: the left-moving or holomorphic algebra, with generators L_n, and the right-moving or antiholomorphic algebra, with generators L̄_n.
- Virasoro Algebra [CFT Zoo]
2d conformal field theories transform under two copies of the Virasoro algebra, 𝒱 × V̄. Chiral Conformal Field Theory transform under a single copy of the Virasoro algebra.