Why do we consider the Witt algebra to be the symmetry algebra of a classical conformal field theory?

In standard physics textbooks, it is usually stated that the Witt algebra is the symmetry algebra of classical conformal field theories in two dimensions.

Following M. Schottenloher, A Mathematical Introduction to Conformal Field Theory and this Phys.SE post, we note that a more precise form of the preceding statement is: In Euclidean spacetime, the Lie algebroid of locally defined conformal Killing vector fields, or equivalently, the Lie algebroid of locally defined holomorphic vector fields in the Riemann sphere contains a complex Witt algebra.

Why do we use the complex Witt algebra to describe classical symmetries of a $${\rm CFT}_2$$? Why not $${\rm LocConfVec}(\mathbb{S}^2)$$ or any other Lie subalgebra contained in the Lie algebroid?

1 Answer

Well, this is likely because in physics textbooks on CFT in 2+0D (especially in string theory) we are rarely studying the conformal compactification = the Riemann-sphere $$\mathbb{S}^2$$ per se, but typically a double-punctured Riemann-sphere $$\mathbb{S}^2\backslash \{0,\infty\}\cong \mathbb{S}\times \mathbb{R}=$$ a cylinder, where the 2 punctures $$z=0$$ and $$z=\infty$$ are temporal infinities (= distant past & future).

A locally defined holomorphic vector field on $$\mathbb{S}^2\backslash \{0,\infty\}$$ is then expanded as a (possibly formal) Laurent series $$\sum_{n\in\mathbb{Z}} a_nz^n \partial ,\qquad a_n~\in~\mathbb{C}.$$ This leads to the complex Witt algebra $$L_n = -z^{n+1}\partial$$.

• How is the fact that we are working on $\mathbb{S}^2 - \{0,\infty\}$ related to being able to write the formal Laurent series? If instead we were working on the Riemman sphere $\mathbb{S}^2$ with no punctures, what would be the difference? Aug 2 '21 at 14:23
• A generic Laurent series is not defined in at least 2 points. Aug 2 '21 at 15:00