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:

  1. 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...)

  2. 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.

  1. 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.
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    $\begingroup$ Actually, you did the right thing - questions don't get added to existing topics. Each individual question should be asked on its own. In fact, you should probably split up the three questions you have here and post each one separately - see meta.physics.stackexchange.com/q/13 $\endgroup$
    – David Z
    Apr 12 '11 at 22:33
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    $\begingroup$ "Since I don't know how to add another question to an already existing topic, I'm opening a new threat."---Hmmm, people generally don't respond well to threats :) $\endgroup$
    – Gordon
    May 15 '11 at 4:01

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.

  • $\begingroup$ Third equation from below - isn't it Kac-Moody algebra? $\endgroup$
    – mavzolej
    Nov 23 '16 at 1:20

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