In 1+1 dimensions, 2D Minkowski space, a conformal transformation is given by two real functions (of one variable). After Wick rotating the time dimension, giving us 2 dimensional Euclidean space, conformal transformations are given by a single holomorphic function, which is just two real functions (of two variables) that satisfy the Cauchy Riemann equations.

Is there any way in which these two groups of transformations are in bijection due to the analytic continuation?


No, the conformal transformations of 2d Euclidean and 2d Minkowski space are not isomorphic.

The global conformal group of $\mathbb{R}^{2,0}$ is $\mathrm{O}(3,1)/\{\pm 1\}$, that of $\mathbb{R}^{1,1}$ is the infinite-dimensional group $\mathrm{Diff}(S^1) \times \mathrm{Diff}(S^1)$. Clearly an infinite-dimensional Lie group and a finite-dimensional Lie group are not isomorphic.

However, both the Euclidean and Minkowski infinitesimal conformal transformations in 2d contain the infinite-dimensional Witt algebra.

For more details, see chapter 2 of Schottenloher's book on CFT.


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