Virasoro Algebra

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Main Question or Discussion Point

Hi there,

It is said that the Virasoro algebra/constraint is the most important algebra is string theory.

(i) Why is that?

(ii) How is it related to the world-sheet conformal symmetry?

(iii) How do I see that the Virasoro constraints are the vanishing of the stress tensor, which can be solved classically in the light-cone gauge?

Thanks
 

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(1) The Virasoro Algebra is the quantum version of the conformal algebra in two dimensions. If you have conformal symmetry and you're in 2 dimensions, as is the case on the world sheet of the strings, then the representation theory of the fields living on the world sheet comes down to representation theory of the Virasoro algebra. Other bigger symmetries might be present, such as supersymmetry, but in the end the Virasoro algebra is always there.

2) It's the quantum version of the conformal algebra. Conformal symmetry in two dimensions is special as it can be enlarged to the infinite dimensional conformal algebra. The presence of this algebra imposes, in some sense, an infinite number of constraints on the correlation functions. Wether you work with or without supersymmetry or some other type of larger symmetry class, it will always contain the Virasoro algebra.

3) The stress energy tensor doesn't vanish, but its trace does. Tracelessness of the stress energy tensor at the quantum level always implies conformal symmetry (in 2D).

Having conformal symmetry in 2D is a very big deal, mostly because there are an infinite number of constraints present in all the correlators. Instead of the usual perturbative approach in QFT you can resort to exploiting these infinite number of symmetries to completely 'solve' the theory, regardless of the coupling constant (the coupling constant doesn't run).
 

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