Questions tagged [affine-lie-algebra]
An infinite dimensional Lie algebra. This tag is not to be confused with the [lie-algebra] tag.
27
questions
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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 ...
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votes
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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{...
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2
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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 ...
- 13.2k
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votes
1
answer
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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 ...
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1
answer
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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 ...
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4
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answers
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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 ...
user145190
3
votes
2
answers
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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 ...
- 1,171
3
votes
1
answer
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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 $ ...
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3
votes
0
answers
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Applications of Dynkin Diagrams in Physics [closed]
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 "...
- 1,213
3
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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.
...
- 1,171
2
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1
answer
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Kac-Moody algebra, proof of parameters calculation
I'm following the notes "Ginsparg - Applied Conformal Field Theory" (https://arxiv.org/abs/hep-th/9108028) and I'm stuck on a proof at page 140 about Kac-Moody algebras.
I would like to prove that $\...
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votes
1
answer
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Current operators for compactified CFTs
Intuitively I feel that if you compactified open bosonic strings on a product of $n$ circles such that each radius is fine-tuned to the self-dual point then the CFT of these $n$ world-sheet fields ...
- 4,561
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0
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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 ...
- 726
2
votes
0
answers
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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 (33):
\begin{equation}
\bigg[\frac{1}{2\pi}\bigg(\...
- 1,171
2
votes
0
answers
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Why does Witten not include higher order quantum corrections when quantizing Poisson brackets in the WZW model?
In 'Non-abelian Bosonization in Two Dimensions', Witten argues 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 ...
- 1,171