The global conformal group of the Riemann sphere is the group of Möbius transformations P S L 2 ( C ) , which is finite-dimensional. On the other hand, infinitesimal conformal transformations form the infinite-dimensional Witt algebra : the conformal Killing equations in two dimensions, ∂ ∂ μ μ ξ ξ ν ν + ∂ ∂ ν ν ξ ξ μ μ = ∂ ∂ ⋅ ⋅ ξ ξ η η μ μ ν ν , reduce to just the Cauchy-Riemann equations, ∂ ∂ z ¯ ¯ ξ ξ ( z ) = 0 = ∂ ∂ z ξ ξ ( z ¯ ¯ ) , the infinity of modes of arbitrary analytic coordinate transformations ξ ξ ( z ) yield the infinity of Killing vector fields z n ∂ ∂ z .