In Conformal field theory, physicist says, the conformal group in two dimensions is infinite dimensional, so the associated with the infinity of generators and infinity conserved charges provided.
Is the statement "the conformal group in two dimensions is infinite dimensional" is true in mathematics?
I tried to find the proof in CFT textbook, but I couldn't find it.
If you know some textbook or papers which provide the detail proof please let me know. Thanks.
As mentioned in Peter Franek's comment, it depends on the signature.
If you are considering $ \mathbb{R}^{1,1} $, then the conformal group mathematically turns out to be $ Conf(\mathbb{R}^{1,1}) \cong Diff_+(\mathbb{R})\times Diff_+(\mathbb{R}) $. Here $ Diff_+(\mathbb{R})$ means the group of all orientation preserving diffeomorphisms of $\mathbb{R}$. However, usually working with compact manifold is preferred so people often mean $ Conf(\mathbb{S}^{1,1}) \cong Diff_+(\mathbb{S})\times Diff_+(\mathbb{S}) $ when they say "the conformal group of the Minkowski plane". Both of these are indeed infinite dimensional.
If you are considering the Euclidean plane $ \mathbb{R^2}\cong \mathbb{C} $, then the conformal group is $ SO^+(3,1)$- the identity component (the maximal connected neighbourhood containing the identity) of the group $ SO(3,1)=\{ M\in M_4(\mathbb{R})\mid M^T g M= g,\, \det M = 1 \}$, $\,g= diag(1,1,1,-1)$, i.e. special indefinite orthogonal group preserving the symmetric bilinear form of signature (3,1). In this case it is NOT infinite dimensional.
However, physicists usually calculate locally and then make global and hence not necessarily true statements. Both of these groups have Lie algebras which contain the Witt algebra- the complex vector space with basis $\{L_n\}_{n\in\mathbb{Z}}$, where $ L_n:= -z^{n+1}\frac{d}{dz} $ and Lie bracket $ [L_m, L_n]=(m-n)L_{m+n}$. After quantization the Witt algebra is replaced by its essentially unique central extension- the Virasoro algebra with Lie bracket $ [L_m, L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m,-n}$. As you can see, both the Witt and Virasoro algebras are indeed infinite dimensional. Therefore, the statement that "in 2D the algebra of infinitessimal conformal transformations is infinite dimensional" is true and this is usually what matters to physicists.
In general it is quite tricky and you can find more about this in M. Schottenloher's book "A Mathematical Introduction to CFT" which is also availabe here. These questions are mostly addressed in Chapters 1 and 2.
EDIT:
There is also the restricted conformal group $ SO^+(2,2)/\{ \pm 1\} \subset Conf(\mathbb{S}^{1,1})$ which is generated by translations, Lorentz transformations, dilatations and special conformal transformations. It is the Minkowski plane analogue of $SO^+(3,1)$ and is usually the one physicists have in mind when they say "the conformal group of the Minkowski plane".