Questions tagged [affine-lie-algebra]

An infinite dimensional Lie algebra. This tag is not to be confused with the [lie-algebra] tag.

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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. ...
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Branching rules for coset construction: $\widehat{su}(4)_2 \rightarrow \frac{\widehat{su}(4)_2}{\widehat{su}(2)_4} \oplus \widehat{su}(2)_4$

I am trying to find the branching rules for the coset construction: $\widehat{su}(4)_2 \rightarrow \frac{\widehat{su}(4)_2}{\widehat{su}(2)_4} \oplus \widehat{su}(2)_4$, where the subscript indicates ...
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A specific coset model

When studying certain 4D $\mathcal{N} = 2$ SCFT, we are led to considering the following coset (having in mind $su(4) \oplus u(1) \subset so(8)$) $$ \frac{\widehat {so(8)}_{-2} \oplus \widehat{su(4)}...
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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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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 ...
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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 ...
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1answer
89 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}{...
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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'}\...
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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 ...
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157 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}-\...
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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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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 ...
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1answer
90 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 $\...
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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 ...
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1answer
133 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 ...
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1answer
180 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 ...
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238 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 ...
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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 ...
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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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1answer
181 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 ...
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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 ...