Questions tagged [affine-lie-algebra]

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

Filter by
Sorted by
Tagged with
1
vote
1answer
58 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}{...
2
votes
0answers
39 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'}\...
0
votes
0answers
35 views

Unitary representations of a Kac-Moody algebra

I am reading one of the original papers about the GKO construction. They say that a representation of the (untwisted) Kac-moody algebra $\hat{g}$ is one where the generators are such that $T^{a\dagger}...
3
votes
2answers
90 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
vote
2answers
139 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}-\...
4
votes
0answers
160 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{...
3
votes
0answers
87 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 ...
2
votes
1answer
73 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 $\...
1
vote
0answers
67 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 ...
1
vote
1answer
70 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 ...
5
votes
1answer
110 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 ...
1
vote
0answers
170 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 ...
4
votes
1answer
154 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 ...
2
votes
1answer
176 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 ...
10
votes
2answers
1k 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 ...