Virasoro algebra

In mathematics, the Virasoro group is a central extension of the orientation-preserving diffeomorphism group of the circle. Its complexified Lie algebra, the Virasoro algebra, is spanned by elements

L_i for i in Z

and c with

L_n+L_-n

and c being real elements. Here the central element c is the central charge. The algebra satisfies

[c,L_n]=0

and

[L_m,L_n]=(n-m)L_m+n+c/12 (m³-m)δ_m,-n.

The factor of 1/12 is merely a matter of convention.

The Virasoro algebra generates both a centrally-extended orientation-preserving diffeomorphism group, and a centrally extended orientation-preserving homeomorphism group of the circle. The difference lies in the topology chosen.

The Witt algebra is the complexified Lie algebra of the diffeomorphism group of the circle. Therefore the Virasoro algebra is a central extension of the Witt algebra.

See also:

• Witt algebra
• Kac-Moody algebra
• WZW model
• conformal field theory