The Virasoro algebra is the complex Lie algebra with generators 
for 
and a central generator 
, satisfying
![[L_m,L_n]=(m-n)L_(m+n)+(m^3-m)/(12)delta_(m+n,0)C.](/images/equations/VirasoroAlgebra/NumberedEquation1.svg) |
The Kronecker delta gives the central term only when 
.
The quotient by 
is the Witt algebra of Laurent-polynomial vector
fields on the circle. In a vertex operator
algebra, the modes of the conformal vector represent the Virasoro algebra, with
acting by a scalar called the central
charge. These modes supply the conformal grading and translation operator.
