Difference Between Algebra of Infinitesimal Conformal Transformations & Conformal Algebra

Question

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!

Answer 1

1. Let us first review the construction of the conformal compactification $\overline{\mathbb{R}^{p,q}}$ of $\mathbb{R}^{p,q}$. $$\begin{align} \overline{\mathbb{R}^{p,q}} ~:=~~&{\cal N}/\mathbb{R}^{\times} ~\subseteq ~~ \mathbb{P}_{p+q+1}(\mathbb{R}) ~\equiv~~ (\mathbb{R}^{p+1,q+1}\backslash\{0\})/\mathbb{R}^{\times}, \cr {\cal N}~~:=~~&\left\{y\in \mathbb{R}^{p+1,q+1} \backslash\{0\}\mid y\cdot y=0\right\},\cr y\cdot y^{\prime}~~:=~~& \sum_{M,N=0}^{p+q+1}y^M\eta^{p+1,q+1}_{MN} y^{\prime N},\cr \eta^{p+1,q+1}~~:=~~&\sum_{M,N=0}^{p+q+1}\eta^{p+1,q+1}_{MN} \mathrm{d}y^M\odot\mathrm{d}y^N,\cr \mathbb{R}^{\times}~~\equiv~~& \mathbb{R}\backslash\{0\}. \end{align}\tag{1}$$

2. Define a contraction $$\kappa:~~\mathbb{R}^{p+1,q+1}\backslash\{0\} \quad\longrightarrow\quad \mathbb{R}^{p+1,q+1}\backslash\{0\}\tag{2}$$ by $$\begin{align} \kappa(y_+, y_-)~:=~& \frac{2(y_+, y_-)}{\sqrt{(y_+,0)\cdot(y_+,0)}+\sqrt{-(0,y_-)\cdot(0,y_-)}} ,\cr &\qquad y_+~\in~\mathbb{R}^{p+1}, \qquad y_-~\in~\mathbb{R}^{q+1}, \cr \kappa({\cal N})~=~&\mathbb{S}^p\times \mathbb{S}^q .\end{align} \tag{3}$$

3. Topologically, the conformal compactification $$ \overline{\mathbb{R}^{p,q}}~\cong~ (\mathbb{S}^p\times \mathbb{S}^q)/\mathbb{Z}_2 .\tag{4} $$ The $\mathbb{Z}_2$-action in eq. (4) identifies points related via a simultaneous antipode-swap on the spatial and the temporal sphere. The unit spheres $\mathbb{S}^p\times \mathbb{S}^q$ are equipped with the metric induced from the ambient space $\mathbb{R}^{p+1,q+1}\backslash\{0\}$.

4. Let $n:=p+q$. [If $n=1$, then any transformation in $\mathbb{R}^{p,q}$ is automatically a conformal transformation, so let's assume $n\geq 2$. If $p=0$ or $q=0$ then the conformal compactification $\overline{\mathbb{R}^{p,q}}~\cong~\mathbb{S}^n$ is an $n$-sphere.]

5. The embedding $$\imath_{\pm}:~~\mathbb{R}^{p,q}\quad\hookrightarrow\quad \mathbb{R}^{p+1,q+1}\backslash\{0\}\tag{5}$$ is given by $$\begin{align}\imath_{\pm}(x)~:=~& \left(\pm 1\mp x\cdot x, ~2x,~ 1+x\cdot x\right)\cr ~&\in~{\cal N}~\subseteq~ \mathbb{R}^{p+1,q+1}\backslash\{0\}, \cr x\cdot x^{\prime}~~:=~~& \sum_{\mu,\nu=1}^{p+q}x^{\mu} \eta^{p,q}_{\mu\nu}x^{\prime\nu},\cr \eta^{p,q}~~:=~~&\sum_{\mu,\nu=1}^{p+q}\eta^{p,q}_{\mu\nu} \mathrm{d}x^{\mu}\odot\mathrm{d}x^{\nu} ~=~\imath_{\pm}^{\ast}(\eta^{p+1,q+1}) ,\cr \imath_{\pm}(x)\cdot \imath_{\pm}(x) ~=~&(\pm 1\mp x\cdot x)^2 +2x\cdot 2x- (1+x\cdot x)^2\cr ~=~&\ldots~=~0. \end{align} \tag{6} $$

6. The inversion $x\mapsto \frac{x}{x\cdot x}$ satisfies $$ i_{\pm}\left(\frac{x}{x\cdot x}\right)~=~\frac{i_{\mp}(x)}{x\cdot x}. \tag{7} $$

7. It is not hard to see that the composition $$\kappa\circ\imath_{\pm}: \mathbb{R}^{p,q}\to \mathbb{S}^p\times \mathbb{S}^q\tag{8}$$ is a conformal map.

Now let us address OP's question.

1. On one hand, there is the (global) conformal group $$ {\rm Conf}(p,q)~\cong~ O(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \}\tag{9}$$ 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. E.g.

   • An (indefinite) orthogonal matrix $\Lambda\in O(p,q)$ is implemented via the matrix $$ \begin{pmatrix} 1& \cr &\Lambda&\cr && 1 \end{pmatrix}~\in~ O(p\!+\!1,q\!+\!1).\tag{10}$$

   • An infinitesimal translation $x\mapsto x+\epsilon$ in $\mathbb{R}^{p,q}$ is implemented via the matrix $$ \begin{pmatrix} 0&\mp\epsilon_{\mu}&0\cr \pm\epsilon^{\mu}&0&\epsilon^{\mu}\cr 0&\epsilon_{\mu}& 0 \end{pmatrix}~\in~ o(p\!+\!1,q\!+\!1).\tag{11}$$

   • An infinitesimal dila(ta)tion $x\mapsto (1+\epsilon)x$ in $\mathbb{R}^{p,q}$ is implemented via the matrix $$ \begin{pmatrix} 0&0&\mp\epsilon\cr 0&0&0\cr \mp\epsilon&0&0 \end{pmatrix}~\in~ o(p\!+\!1,q\!+\!1).\tag{12}$$

   • A special conformal transformation is a composition of an inversion (7), a translation (11), and another inversion (7).

   The quotients in eqs. (4) & (9) are remnants from the projective space (1).

   The connected component that contains the identity element is $$ {\rm Conf}_0(p,q)~\cong~ \left\{\begin{array}{ll} SO^+(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \} &\text{if both $p$ and $q$ are odd},\cr SO^+(p\!+\!1,q\!+\!1) &\text{if $p$ or $q$ are even}.\end{array}\right.\tag{13}$$ The two cases in eq. (13) correspond to whether $-{\bf 1}\in SO^+(p\!+\!1,q\!+\!1)$ or not, respectively. 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 $$ {\rm conf}(p,q)~\cong~so(p\!+\!1,q\!+\!1)\tag{14}$$ 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 $$ {\rm LocConf}(p,q)~=~ \underbrace{{\rm LocConf}_+(p,q) }_{\text{orientation-preserving}} ~\cup~ \underbrace{{\rm LocConf}_-(p,q) }_{\text{orientation-reversing}} \tag{15}$$ consisting of locally defined conformal transformations. Let us denote the connected component that contains the identity element $${\rm LocConf}_0(p,q)~\subseteq~ {\rm LocConf}_+(p,q). \tag{16}$$ The local conformal algebroid $$ {\rm locconf}(p,q)~=~ {\rm LocConfKillVect}(\overline{\mathbb{R}^{p,q}})\tag{17}$$ consists of locally defined conformal Killing vector fields, i.e. generators of 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 transformations are only interesting for $n=2$.

   • For the 1+1D Minkowski plane we consider light-cone coordinates $x^{\pm}\in \mathbb{S}$, cf. e.g. this Phys.SE post. The locally defined orientation-preserving conformal transformations are products of 2 monotonically increasing (decreasing) diffeomorphisms on the circle $\mathbb{S}^1$ $$\begin{align} {\rm LocConf}_+(1,1)~=~& {\rm LocDiff}^+(\mathbb{S}^1)~\times~ {\rm LocDiff}^+(\mathbb{S}^1) \cr &~\cup~ {\rm LocDiff}^-(\mathbb{S}^1)~\times~ {\rm LocDiff}^-(\mathbb{S}^1),\cr {\rm LocConf}_0(1,1)~=~& {\rm LocDiff}^+(\mathbb{S}^1)~\times~ {\rm LocDiff}^+(\mathbb{S}^1).\end{align}\tag{18}$$ A orientation-reversing transformation is just a orientation-preserving transformation composed with the map $(x^+,x^-)\mapsto (x^-,x^+)$. The corresponding local conformal algebra $$ {\rm locconf}(1,1)~=~{\rm Vect}(\mathbb{S}^1)\oplus {\rm Vect}(\mathbb{S}^1) \tag{19}$$ becomes two copies of the real Witt algebra, which is an infinite-dimensional Lie algebra.

   • For the 2D Euclidean plane $\mathbb{R}^{2}\cong \mathbb{C}$, when we identify $z=x+iy$ and $\bar{z}=x-iy$, then the locally defined orientation-preserving (orientation-reversing) conformal transformations are non-constant holomorphic (anti-holomorphic) maps on the Riemann sphere $\mathbb{S}^2=\mathbb{P}^1(\mathbb{C})$ $$\begin{align} {\rm LocConf}_0(2,0) ~=~&{\rm LocConf}_+(2,0)\cr ~=~&{\rm LocHol}(\mathbb{S}^2), \cr\cr {\rm LocConf}_-(2,0)~=~&\overline{{\rm LocHol}(\mathbb{S}^2)} ,\end{align}\tag{20}$$ respectively. An anti-holomorphic map is just a holomorphic map composed with complex conjugation $z\mapsto\bar{z}$. The corresponding local conformal algebroid $$ {\rm locconf}(2,0) ~=~{\rm LocHolVect}(\mathbb{S}^2)\tag{21}$$ consists of generators of locally defined holomorphic (sans anti-holomorphic!) maps on $\mathbb{S}^2$. It contains a complex Witt algebra.

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.)

5. A.N. Schellekens, CFT lecture notes, 2016.