Questions tagged [affine-lie-algebra]
An infinite dimensional Lie algebra. This tag is not to be confused with the [lie-algebra] tag.
28 questions
3
votes
1
answer
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views
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 $d$ ...
- 505
1
vote
0
answers
82
views
How to derive WZW model’s energy-momentum tensor? The result is of course the Sugawara construction
I want to know how to derive WZW’s energy-momentum tensor. We know WZW action is
$$
S_{WZW}[g]=\frac{k}{16\pi}\int d^2x Tr(\partial g^{-1}\partial g) - \frac{ik}{24\pi}
\int_B d^3y \epsilon_{abc} Tr(h^...
- 174
0
votes
1
answer
143
views
Central Charge Calculation of $SL_k(2,\mathbb{R})$ WZW Model
According to P. Francesco et al. conformal field theory book the central charge of the enveloping Virasoro algebra of the affine Lie algebra $\hat{g}_k$ corresponding with Lie algebra $g$ which ...
2
votes
0
answers
57
views
WZW primary fields / correlations in terms of current algebra?
Cross-posted from a Mathoverflow thread! Answer there for a bounty ;)
Given the
$\mathfrak{u}_N$ algebra
with generators $L^a$ and commutation relations
$ [L^a,L^b] = \sum_c f^{a,b}_{c} L^c $ ,
the ...
- 766
1
vote
1
answer
109
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Discretization of derivative of delta function and affine Kac-Moody algebra
In equation (4.16) of https://arxiv.org/abs/1506.06601, a discretization of the (classical) affine Kac-Moody algebra is presented:
$$
\frac{1}{\gamma}\left\{J_{m}^{1}, J^2_{n}\right\}=J_{m}^{1} J_{n}^{...
- 1,191
1
vote
0
answers
62
views
String theory coset-space theories in the text book by Becker, Becker and Schwarz (BBS)
I am reading the string theory and M-theory by Becker, Becker and Schwarz (BBS). And I came across a section in chapter 3 called coset-space theories (after equation 3.58)
At the beginning of this ...
- 462
3
votes
0
answers
101
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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 "...
- 1,327
2
votes
0
answers
64
views
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(\...
- 1,191
2
votes
0
answers
67
views
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 ...
- 1,191
3
votes
0
answers
181
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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.
...
- 1,191
3
votes
1
answer
246
views
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 $ ...
- 739
1
vote
0
answers
114
views
The mathematical structure of $\widehat{su(2)}_k$
Some of my colleagues work on CFT's and quantum groups and I hear them talk a lot about $\widehat{su(2)}_k$ algebras. According to them (and the general physics literature) these are what ...
- 1,315
1
vote
0
answers
283
views
How to derive Kac-Moody and Virasoro algebras from their descriptions as central extensions?
I am following the notes (https://arxiv.org/abs/hep-th/9904145) to learn conformal field theory, and want to know how to derive the contributions to the Virasoro and Kac-Moody algebras from the ...
- 766
1
vote
1
answer
325
views
Factor of $1/2$ in the Sugawara construction
I'm trying to reproduce the Sugawara construction calculation using this reference (page 14).
The normal-ordering of two local operators is defined as
$$ N(XY)(w)=\frac{1}{2\pi i} \oint_w \frac{dx}{...
- 16.3k
2
votes
0
answers
168
views
OPE Kac-Moody Currents
We have the following operators:
\begin{align}
J^a(z) = \frac{1}{2}\psi_s^{\dagger}(z)\sigma^a_{s s'}\psi_{s'}(z), \hspace{10 mm} \bar{J}^a(z) = \frac{1}{2}\psi_s^{\dagger}(\bar{z})\sigma^a_{s s'}\...
- 41
3
votes
2
answers
255
views
Kac-Moody algebra from WZW model via Poisson brackets
In 'Non-abelian Bosonization in Two Dimensions', Witten shows that the Poisson brackets of the currents that generate the $G\times G$ symmetry of the WZW model give rise to a Kac-Moody algebra upon ...
- 1,191
1
vote
2
answers
178
views
Half Witt algebra
I have the following Lie algebra which is generated by $\{L_n|n\geq 0\}.$ It satisfies the following commutation rule
$$ \Big[ L_i ,L_j \Big]=\frac18 \frac{(2i+2j-1)(2j-2i)}{(2j+1)(2i+1)}L_{i+j-1}-\...
- 113
8
votes
0
answers
347
views
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{...
- 420
4
votes
0
answers
162
views
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
2
votes
1
answer
164
views
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 $\...
- 479
2
votes
0
answers
207
views
Physical meaning of the WZW action and Lagrangian
What is the (super)-WZW term physical meaning? I mean, what is the physical importance of the Wess-Zumino-Witten action/Lagrangian in superstrings/M-theory and or field theory (not stringy if ...
- 6,727
1
vote
1
answer
361
views
Affine space for Minkowski space time
I'm studying Minkowski space time (M4) and they say it's a 4 dimensions real affine space.
M4 is an affine space so there is a non-empty set A, a 4 dimension real vector space V, and there is a ...
- 1,978
5
votes
1
answer
310
views
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 ...
- 145
1
vote
0
answers
458
views
Free field (Wakimoto) representation in 2d CFT
This question is more a request for explanations. I'm reading now the Di Francesco book in attempt to understand how the free field representations of 2d CFTs are constructed. The first steps in ...
- 3,013
6
votes
2
answers
744
views
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 ...
- 13.5k
4
votes
1
answer
298
views
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 ...
- 4,709
2
votes
1
answer
216
views
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 ...
- 4,581
13
votes
2
answers
2k
views
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 ...
- 1,243