Witt algebra

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In mathematics, a Witt algebra, named after Ernst Witt, is the Lie algebra of Killing vector fields defined on the Riemann sphere. Witt algebras occur in the study of conformal field theory.

Even though the symmetry group of a Riemann sphere is only PSL(2,C), the group of Möbius transformations, there exist Killing vector fields over the Riemann sphere which do not generate any conformal symmetries because of topological obstructions. In fact, any holomorphic function over the Riemann sphere corresponds to a conformal Killing vector field (other than the constant infinite function).

A basis for these Killing fields are given by the holomorphic functions L_n=-z^{n+1} \frac{\partial}{\partial z}, for n in \mathbb Z. This is nothing else than a Laurent expansion.

The Lie bracket of two Killing fields

[L_m,L_n]=\left( z^{m+1}(n+1)z^n-z^{n+1}(m+1)z^m \right)\frac{\partial}{\partial z}=(n-m)z^{m+n+1}\frac{\partial}{\partial z}=(m-n)L_{m+n}.

Consider the commutative algebra of functions with two arguments which are holomorphic in z in the first argument and antiholomorphic in z in the second argument. This algebra gives an algebraic description of the Riemann surface. From this, we can see there are actually two sets of generators.

L_n f(z,\bar z)=-z^{n+1}\frac{\partial}{\partial z}f(z,\bar z)

and

\bar L_n f(z,\bar z)=-\bar z^{n+1}\frac{\partial}{\partial \bar z}f(z, \bar z).
[\bar L_m,L_n]=0 and [\bar L_m,\bar L_n]=(m-n)\bar L_{m+n}.

This algebra can be centrally extended to give the Virasoro algebra.

When analyzing conformal field theory in spacetime as opposed to Euclidean space, z and \bar z now become the real lightcone coordinates and are truly independent.

[edit] References

  • , "Witt algebra" SpringerLink Encyclopaedia of Mathematics (2001)
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