4
votes
Accepted
Strings on group manifolds and critical dimension
Yes. You've described a way for strings in 28 dimensions, 25 of which are flat, to be anomaly free. Of course neither of these numbers is 4 which is one of the many reasons why this is a toy model. ...
- 8,900
3
votes
Accepted
Conformal invariance in Toda field theories
The simplest example of conformal Toda theory is Liouville theory. Conformal invariance of Liouville theory is explained in the Wikipedia article. The idea is that you should compute the dimension of ...
- 2,792
2
votes
Kac-Moody algebra, proof of parameters calculation
Rather than explicitly subtracting the singular term when writing $T(z)$, you could write it as a normal-ordered product $T(z)\propto \sum_a (J^aJ^a)(z)$, and use Wick's theorem to compute the OPE of $...
- 2,792
2
votes
Factor of $1/2$ in the Sugawara construction
I'm not completely sure what OP is asking (v4) but here are some hopefully helpful comments:
There is an implicit radial ordering ${\cal R}$ assumed on the right-hand side of eq. (2.56) and on the ...
- 213k
2
votes
Kac-Moody algebra from WZW model via Poisson brackets
I think he may be using the distributional identity
$$
f(x)\delta'(x-a)=f(a)\delta'(x-a) -f'(a)\delta(x-a)
$$
which comes from differentiating
$$
f(x)\delta(x-a)=f(a)\delta(x-a)
$$
with respect to $...
- 56.5k
1
vote
Central Charge Calculation of $SL_k(2,\mathbb{R})$ WZW Model
Not an answer, but a long comment.
I've figured it out. According to AccidentalFourierTransform here What is wrong with this proof that $h^\vee$ the dual Coxeter number is always 1?, the definition of ...
- 2,803
1
vote
Discretization of derivative of delta function and affine Kac-Moody algebra
Let $\varphi$ be a smooth compactly supported function. Then :
\begin{align}
\int \varphi(x)\delta'(x)\text dx &= -\varphi'(0) \\
&=\lim_{\Delta\to 0} -\frac{1}{\Delta}(\varphi(\Delta)-\varphi(...
- 6,119
1
vote
Kac-Moody primary OPE
I haven't looked at this stuff for many years, so take my answer with caution.
Note that while $J^a = J^0, J^\pm$ are generators of an affine Lie Algebra, their zero modes are generators of the ...
- 5,451
1
vote
Half Witt algebra
For any $\lambda$, consider the commutation relations
$$
[L_i,L_j] = (i-j)(L_{i+j} + \lambda L_{i+j-1})
$$
These relations obey antisymmetry and Jacobi identities so we have a Lie algebra. To make it ...
- 2,792
1
vote
Accepted
Half Witt algebra
We can start with the obvious rescaling
$$
\tilde{L}_i = (2i + 1)L_i\,,
$$
which turns the algebra to
$$
[\tilde{L}_i,\tilde{L}_j] = (i-j)\left(\tilde{L}_{i+j}-\frac{1}{4}\tilde{L}_{i+j-1}\right)\,.
$$...
- 6,926
1
vote
Accepted
Affine space for Minkowski space time
The elements of A (the events in Minkowski space) exist indepently of your choice of coordinate system on A.
Similarly, one can define translations (the action by a vector) in a purely abstract ...
- 1,094
1
vote
What are the quantum dimensions of the primary fields for $SU(N)$ level-$k$ Kac-Moody current algebras?
The S matrix of affine Lie algebra can be calculated by the Kac-Peterson formula, which can also be found in Theorem 13.8 in Kac's book.
- 11
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