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in Blumenhagen Book on conformal field theory, It is mentioned that the algebra of infinitesimal conformal transformation is different from the conformal algebra and on page 11, conformal algebra is defined by a redefinition of generators of infinitesimal conformal transformation. I have three question about this :

  1. How this is possible that by a redefinition of generators, one could obtain a sub-algebra of an algebra? in this case one obtain conformal algebra as a sub-algebra of algebra of generators of infinitesimal conformal transformations?

  2. Does this is related to «special conformal transformation» which is not globally defined?

  3. How are these related to topological properties of conformal group?

Any comment or reference would greatly be appreciated!

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Let $$\begin{align} \overline{\mathbb{R}^{p,q}}~~:=&~~\left\{y\in \mathbb{R}^{p+1,q+1}\backslash\{0\}\mid \eta^{p+1,q+1}(y,y)=0\right\}/\mathbb{R}^{\times} \cr ~~\subseteq &~~ \mathbb{P}_{p+q+1}(\mathbb{R})~~\equiv~~(\mathbb{R}^{p+1,q+1}\backslash\{0\})/\mathbb{R}^{\times}, \qquad \mathbb{R}^{\times}~~\equiv~~\mathbb{R}\backslash\{0\}, \end{align}\tag{1}$$ denote the conformal compactification of $\mathbb{R}^{p,q}$. Topologically, $$ \overline{\mathbb{R}^{p,q}}~\cong~(\mathbb{S}^p\times \mathbb{S}^q)/\mathbb{Z}_2 .\tag{2} $$ The embedding $\imath: \mathbb{R}^{p,q}\hookrightarrow \overline{\mathbb{R}^{p,q}}$ is given by $$\imath(x)~:=~\left(1-\eta^{p,q}(x,x): ~2x:~ 1+\eta^{p,q}(x,x)\right). \tag{3} $$ Let $n:=p+q$. [If $n=1$, then any transformation is automatically a conformal transformation, so let's assume $n\geq 2$.]

  1. On one hand, there is the (global) conformal group $$ {\rm Conf}(p,q)~\cong~O(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \}\tag{4}$$ consisting of the set globally defined conformal transformations on $\overline{\mathbb{R}^{p,q}}$. This is a $\frac{(n+1)(n+2)}{2}$ dimensional Lie group. The connected component that contains the identity element is $$ {\rm Conf}_0(p,q)~\cong~\left\{\begin{array}{ll} SO^+(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \} &\text{if both $p$ and $q$ are odd},\cr SO^+(p\!+\!1,q\!+\!1) &\text{if $p$ or $q$ are even}.\end{array}\right.\tag{5}$$ The global conformal group ${\rm Conf}(p,q)$ has 4 connected components if both $p$ and $q$ are odd, and 2 connected components if $p$ or $q$ are even. The (global) conformal algebra $$ {\rm conf}(p,q)~\cong~so(p\!+\!1,q\!+\!1)\tag{6}$$ is the corresponding $\frac{(n+1)(n+2)}{2}$ dimensional Lie algebra. Dimension-wise, the Lie algebra breaks down into $n$ translations, $\frac{n(n-1)}{2}$ rotations, $1$ dilatation, and $n$ special conformal transformations.

  2. On the other hand, there is the local conformal groupoid consisting of locally defined conformal transformations. The local conformal algebra consisting of generators of locally defined conformal transformations.

    • For $n\geq 3$, (the pseudo-Riemannian generalization of) Liouville's rigidity theorem states that all local conformal transformations can be extended to global conformal transformations, cf. e.g. this & this Phys.SE posts. Thus the local conformal algebra is only interesting for $n=2$.

    • For the 2D Euclidean plane $\mathbb{R}^{2}\cong \mathbb{C}$, when we identify $z=x+iy$ and $\bar{z}=x-iy$, then the local conformal transformations are local holomorphic and antiholomorphic maps. The corresponding local conformal algebra becomes two copies of the Witt algebra, which is an infinite-dimensional Lie algebra.

    • For the 2D Minkowski plane $\mathbb{R}^{1,1}$, there is a similar story if we replace the complex coordinates $z$ and $\bar{z}$ with light-cone coordinates $x^{\pm}\in \mathbb{R}$.

References:

  1. M. Schottenloher, Math Intro to CFT, Lecture Notes in Physics 759, 2008; Chapter 1 & 2.

  2. R. Blumenhagen and E. Plauschinn, Intro to CFT, Lecture Notes in Physics 779, 2009; Section 2.1.

  3. P. Ginsparg, Applied Conformal Field Theory, arXiv:hep-th/9108028; Chapter 1 & 2.

  4. J. Slovak, Natural Operator on Conformal manifolds, Habilitation thesis 1993; p.46. A PS file is available here from the author's homepage. (Hat tip: Vit Tucek.)

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