An infinite dimensional Lie algebra. This tag is not to be confused with the [lie-algebra] tag.
An infinite dimensional Lie algebra. This tag is not to be confused with the lie-algebra tag.
Questions tagged [affine-lie-algebra]
30 questions
13
votes
2 answers
Geometric/Visual Interpretation of Virasoro Algebra
I've been trying to gain some intuition about Virasoro Algebras, but have failed so far.
The Mathematical Definition seems to be clear (as found in http://en.wikipedia.org/wiki/Virasoro_algebra). I just can't seem to gain some intuition about it. As…
Michael
- 1,241
8
votes
0 answers
Free Field Realization of Current Algebras and its Hilbert space
I have some conceptual confusion regarding the interplay between current algebras, their free field representation and the Hilbert space generated from it.
Let's sketch a simple example, $\mathfrak{so}(n)_1$. As is readily seen, taking $n$ free…
thi
- 420
6
votes
2 answers
What are the quantum dimensions of the primary fields for $SU(N)$ level-$k$ Kac-Moody current algebras?
The CFT of the $\mathrm{SU}(N)$ level $k$ Kac-Moody current algebra has many Kac-Moody primary fields. I wonder if any one has calculated the quantum dimensions of those Kac-Moody primary fields.
I know that, for $\mathrm{SU}(2)$ level $k$ Kac-Moody…
Xiao-Gang Wen
- 13,725
5
votes
1 answer
Conformal invariance in Toda field theories
A standard Toda field theory action will be of the shape:
$$ S_{\text{TFT}} = \int d^2 x~ \Bigg( \frac{1}{2} \langle\partial_\mu \phi, \partial^\mu \phi \rangle - \frac{m^2}{\beta^2} \sum_{i=1}^r n_i e^{\beta \langle\alpha_i, \phi\rangle}…
Abelaer
- 145
4
votes
1 answer
Emergence of $SU(2)\times SU(2)$ at the self-dual point in bosonic string theory
I want to understand the derivation of the equations 8.3.11 in Polchinski Vol 1.
I can understand that at the self-dual point the Kaluza-Klein momentum index $n$, the winding number $w$, and the left and the right oscillator number $N$ and…
user6818
- 4,749
4
votes
0 answers
Geometry of Affine Kac-Moody Algebras
One can reconstruct the unitary irreducible representations of compact Lie groups very beautifully in geometric quantization, using the Kähler structure of various $G/H$ spaces.
Can one perform a similar construction for affine Lie algebras? In…
user145190
3
votes
1 answer
Mode expansion of vertex operator in $c=1$ CFT at rational radius?
In the context of the compactified free boson CFT ($c = 1$), we know that at special radii, additional vertex operators become dimension one and hence can be interpreted as conserved currents.
In other well-known cases (e.g., the stress-energy…
baba26
- 702
3
votes
1 answer
Strings on group manifolds and critical dimension
In their work, Witten and Gepner in "Strings on group manifolds" have shown that the central charge of the theory is
\begin{equation}
c=\frac{kD}{c^{\vee}+k}+d=26,
\end{equation}
where $d$ is the dimension of flat spacetime, $D$ is the dimension of…
Arian
- 545
3
votes
0 answers
WZW primary fields / correlations in terms of current algebra?
Cross-posted from a Mathoverflow thread! Answer there for a bounty ;)
Given the
$\mathfrak{u}_N$ algebra
with generators $L^a$ and commutation relations
$ [L^a,L^b] = \sum_c f^{a,b}_{c} L^c $ ,
the WZW currents of $U(N)_k$
$$ J(z) = \sum_{n \in…
Joe
- 796
3
votes
0 answers
Applications of Dynkin Diagrams in Physics
I've been studying Dynkin Diagrams for a while, but I can't grasp what are the applications in physics.
Can anyone help me understand where can we use Dynkin Diagrams in particle physics to "solve problems"?
RKerr
- 1,367
3
votes
0 answers
WZW model - affine Kac-Moody current algebra from Quantum Group exchange algebra
In `Hidden Quantum Groups Inside Kac-Moody Algebra', by Alekseev, Faddeev, and Semenov-Tian-Shansky, a relationship between quantum groups and affine Kac-Moody algebras is shown for the WZW model.
Towards this end, the authors show that a certain…
Mtheorist
- 1,211
3
votes
1 answer
Kac-Moody primary OPE
I am reading a paper and on page 13-14 (PDF page 15-16), they say that,
The fermionic generators [$G^\pm$ and $\tilde{G}^\pm$] are Virasoro
and affine Kac-Moody primaries with weights $h= 3/2 $ and $j=1/2$.
I understand the first statement to…
user2062542
- 739
3
votes
2 answers
Kac-Moody algebra from WZW model via Poisson brackets
In 'Non-abelian Bosonization in Two Dimensions', Witten shows that the Poisson brackets of the currents that generate the $G\times G$ symmetry of the WZW model give rise to a Kac-Moody algebra upon canonical quantization.
The Poisson brackets are…
Mtheorist
- 1,211
2
votes
0 answers
Introductory resources on Kac–Moody algebras for CFT
I'm self-studying CFT and I'm trying to understand more about Kac–Moody algebras and their role in CFT. While Blumenhagen and Plauschinn's (B&P for now) book has a pleasant discussion, it seems very disconnected from what I read elsewhere…
Níck Aguiar Alves
- 24,192
2
votes
0 answers
Symmetry generating commutator in Witten's treatment of WZW model
In 'Non-abelian Bosonization in Two Dimensions', Witten writes down the commutation relations between currents and fields of the WZW model in equation…
Mtheorist
- 1,211