Virasoro algebra

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In mathematics, the Virasoro group (named after the physicist Miguel Angel Virasoro) 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 $i\in\mathbf{Z}$

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

$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m,-n}$

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.

The Virasoro algebra is obeyed by the stress tensor in string theory, since it comprises the generators of the conformal group of the worldsheet, obeys the commutation relations of (two copies of the) Virasoro algebra. This is because the conformal group decomposes into separate diffeomorphisms of the forward and back lightcones. Diffeomorphism invariance of the worldsheet implies additionally that the stress tensor vanishes. This is known as the Virasoro constraint, and in the quantum theory, cannot be applied to all the states in the theory, but rather only on the physical states (confer Gupta-Bleuler quantization).