Some questions on Conformal Field Theory, Current algebras and the Sugawara construction Since I don't know how to add another question to an already existing topic, 
I'm opening a new thread. However I'm referring to:
Beginners questions concerning Conformal Field Theory 
As noted, a few weeks ago I started reading about Conformal Field Theory. 
I'm actually from a more mathematical background, however I'm not very familiar 
with Quantum Field Theory. Though I'm quite familiar with Quantum Mechanics/Classical Mechanics. 
Now again some questions again turned up: 


*

*Think of a theory with an energy-momentum-tensor that is given 
on the plane. Let's assume the most general form 
\begin{equation}T(z)=\sum z^{-n-2} L_{n} \quad  \text{and} \quad
L_{n} = \frac{1}{2 \pi i} \oint dz z^{n+1} T(z).
\end{equation} 
Now some of my reference (such as David Tong in the reference question above) point
out that $L_{0}$ generates scalings/rotations and $L_{1},L_{-1}$ generate translations. 
So let's consider the example of a rotation. The generator of a rotation is $z \frac{\partial}{\partial z}$. Now in order to show that $L_{0}$ actualy generates 
this rotation one needs to show that $[L_{0},\phi]$ =  $z \frac{\partial}{\partial z} \phi$.
I've shown this for the example of the free boson, however I'm not 100% sure how to prove it in the general case. Can someone help me? (Maybe it's related to Operator Product Expansions...) 

*The second question goes a bit deeper into the theory. 
It concern Current Algebras. I've read some articles on the Sugawara construction 
and there Mr Sugawara proposes an Energy-Momentum-Tensor of the form
\begin{equation}
T(z)  = \gamma \sum_{a=1}^{dim g} : j^{a}(z) j^{a}(z): .
\end{equation}
However I don't really see 
how he comes up with it or why this seems to be a "natural choice" of an Energy-Momentum tensor. I've heard that it includes the Energy Momentum Tensor of the free boson 
(given by $T(z)=\partial_{z} \phi \partial_{z} \phi$) as a special case. 
For me this is not so obvious. Can someone please explain to me how he comes up that
in an easy way. I don't think it's necessary to show me all the calculations. 
Just the basic idea would be useful to get some intuition. 


*I'm having some troubles on understanding the intuition behind current algebras. 
(I haven't read about WZW Models yet). The Virasoro algebra appeared to me in a kind
of natural way in the example of the free boson. The generalization is then pretty much
straight forward. However I don't have that kind of intuition for current algebras. 
I've read that they provide some "additional symmetry structure" which reduces the number of possible correlation functions. But I don't know any details. 
I'd be more than happy if someone could comment on that.

 A: This question is pretty open ended.  The second part of this involves the $SL(2,{\mathbb R})$ subgroup of the Virasoro algebra.  So I thought that at the risk of giving answers which might not be relevant I thought I would try to connect this with Lie theory.  The Lie algebra g has a maximal set of commuting matrices that define the Cartan center $H^i$, $i~=~1,\dots,~ rank(g)$.  These operators act upon the remaining operators $E^\alpha$ as $[H^i,~E^\alpha]~=~\alpha^i E^\alpha$, where $\alpha^i$ are the roots of the algebra.  The Jacobi theorem 
$$
[[H^i,~E^\alpha],~E^\beta]~+~[[E^\beta,~H^i],~E^\alpha]~+~[[E^\alpha,~E^\beta],~H^i]~=~0
$$
permits us to compute
$$
 [E^\alpha,~E^\beta]~=~\matrix{ C(α,β)E^{α+β}~& :~\alpha~+~\beta~a~root \cr
2\alpha\cdot H/\alpha^2~&    ~: \alpha~+~\beta~=~0\cr
0~&     ~: otherwise}
$$
The structure constant $|C(\alpha,\beta)~=~\pm 1$ and in the second of these the contraction of $H^i$ with the root $\alpha^i$ is a trace of $H^i$ and is used in a normalization $E^\alpha E^{-\alpha}~= 2/\alpha^2$.  
The operators for the string modes obey a Virasoro algebra,
$$
{[L^{a_j},~L^{b_j}]~=~(a_j~-~b_j)L^{a_j + b_j}~+~c(a_j ,b_j ).} 
$$
The Virasoro generators are expanded according to the Laurent expansion
$$
L^{a_n}~=~\oint\frac{dz}{2\pi iz}z^{a_n+2}T(z),~T(z)~=~-\sum_{a_n=\infty}^\infty\frac{L^{a_n}}{z^{m+2}}.
$$
Commutators of the Virasoro generators $L^{-1},~L^0,~L^1$ produce the $SL(2,{\mathbb R})$ algebra
$$
[L^0,~L^{-1}]~=~L_{-1},~[L^0,~L^1]~=~-L^1,~[L^1,~L^{-1}]~=~2L^0.
$$
This is the same in form as the $SU(2)$ algebra for the angular momentum operators $L_\pm,~L_z$, but is noncompact.  
A general commutator of an element $T^a~=~T^a(z)$ in the vector space of a Lie algebra obeys $[T^a,~T^b]~=~iC^{ab}_cT^c$.  The inner product of these elements defines a positive element $\langle T^a,~T^b\rangle~=~h^{ab}$.  This serves as a metric in the vector space of the Lie algebra.  This defines a rule 
$$
\langle[T^a,~T^b],~T^c\rangle~+~\langle T^b,~[T^a,~T^c]\rangle~=~0.
$$
So the metric $h^{ab}$ defined in some representation, $r$, of matrix element $t^a_r$ then gives the Schur’s lemma result $tr(t^a_rt^b_r)~=~T_rh^{ab}$.  This further gives the definition of the Coxeter number cox(g)
$$
-\sum_{cd}C^{ac}_dC^{bd}_c~=~cox(g)(α_L)^2h^{ab}
$$
for $\alpha_L$ any long root.  
With some of these Lie algebraic basics down operator produce expansions (OPE) can be found.  The bosonic vertex operator for the heterotic string is of the form $j(z)\phi^i({\bar z})exp(ik\cot X)$, for $X$ the string world sheet.  A gauge bosonic vertex operator is similarly $j(z){\bar\partial}X^i({\bar z})exp(ik\cdot X)$.  The current is holomorphic in the complex $z$, and stress-energy constructed from currents in order to be conformal must also be holomorphic.. The most basic form of a an OPE is the $(1,0)$ holomorphic current is
$$
j^aj^b~\sim~ k^{ab}/z^2~+~i(c^{ab}_c/z)f^c(0).
$$
The algebraic content is found by taking the Laurent expansion of the current
$$
j^a(z)~=~\sum_{m=-\infty}^∞\frac{j^a_m}{z^{m+1}},
$$
where the current coefficients satisfy a Lie algebra
$$
[j^a_m,~j^b_n]~=~mk^{ab}\delta_{m,-n}~+~iC^{ab}_cj^c_{m+n},
$$
which is a Virasoro algebra.  The coefficients $k^{ab}~=~kh^{ab}$.  For $m = 0,\pm 1$ the Virasoro algebra obeys a closed algebra of commutators
$$
[j^a_0,~j^b_{\pm1}]~=~ic^{ab}_cj^c_{\pm 1},~ [j^a_1,~ j^b_{-1}]~=~2J_0,
$$
which is an $SU(2)$ algebra of the elements $2\alpha\cdot H/\alpha^2$, $E^\alphaα_0$, $ E^{-\alpha}_0$, or the elements $(2\alpha\cdot H~+~k)/\alpha^2$, $E^\alpha_1 E^{-\alpha}_{-1}$.  So we connect with the Lie algebraic construction above.  The Coxeter number above defines an OPE stress-energy
$$
T~=~[(k~+~cox(g))(\alpha_L)^2]^{-1}:jj(z):
$$
With :  : meaning a normalization.  With additional work the current algebra of the system constructs OPE expansions for relevant terms.  In this way a conformal consistent stress-energy can be constructed.  
