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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?

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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 $L_n = -z^{n+1}\partial$.

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  • $\begingroup$ 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? $\endgroup$ Aug 2 '21 at 14:23
  • $\begingroup$ A generic Laurent series is not defined in at least 2 points. $\endgroup$
    – Qmechanic
    Aug 2 '21 at 15:00

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