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 :

• 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?

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

• How are these related to topological properties of conformal group?

Any comment or reference would greatly be appreciated!

This post imported from StackExchange Physics at 2015-05-27 20:59 (UTC), posted by SE-user QGravity

Comments

Where precisely do the authors make that statement?

This post imported from StackExchange Physics at 2015-05-27 20:59 (UTC), posted by SE-user joshphysics

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in page 11 after defining conformal group and conformal algebra.

This post imported from StackExchange Physics at 2015-05-27 20:59 (UTC), posted by SE-user QGravity

Answer 1

Let $\overline{\mathbb{R}^{p,q}}$ denote the conformal compactification of $\mathbb{R}^{p,q}$. 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 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 (global) conformal algebra is the corresponding $\frac{(n+1)(n+2)}{2}$ dimensional Lie algebra, which is isomorphic to $so(p+1,q+1)$. 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 theorem states that all local conformal transformations can be extended to global conformal transformations. 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. R. Blumenhagen and E. Plauschinn, Intro to CFT, Lecture Notes in Physics 779, 2009; Section 2.1.

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

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.)

This post imported from StackExchange Physics at 2015-05-27 20:59 (UTC), posted by SE-user Qmechanic

Comments

Thank you, but what has been stated in Blumenhagen is more general than just 2D! I want to know that is this a mathematical result that the generators of infinitesimal of some transformations is bigger than algebra which characterizes the lie algebra of group of that transformation?

This post imported from StackExchange Physics at 2015-05-27 20:59 (UTC), posted by SE-user QGravity

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I updated the answer.

This post imported from StackExchange Physics at 2015-05-27 20:59 (UTC), posted by SE-user Qmechanic

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Dear Qmechanic, thank you very much for your answer! I am looking for third part of my question!

This post imported from StackExchange Physics at 2015-05-27 20:59 (UTC), posted by SE-user QGravity