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Jul
17
awarded  Enlightened
Jul
17
awarded  Nice Answer
Jul
3
comment Expressing an adjoint representation Wilson line in terms of the fundamental representation
Sorry, $E^{ij}$ is the matrix with all elements zero except a unit at the place $i,j$
Jul
3
answered Expressing an adjoint representation Wilson line in terms of the fundamental representation
Jun
5
comment The $U(1)$ charge of a representation
@user41746, cont.Now, Slansky insists to identify the highest weight of the $SU(5) $ representation with the highest weight of the quark representation after the symmetry breaking. For that he needs to act on the weights with a “projection matrix” given in equation 6.7. After the action of the projection matrix the fourth component of the weights is lost, but nevertheless, the dual U(1) vector can be obtained by insisting that it has a constant scalar product with each subrepresentation weights. As you can see all the exercises lead to the same result for the hypercharges.
Jun
5
comment The $U(1)$ charge of a representation
@user41746, Sorry I noticed that I used $a_1, a_2$ for the $SU(3)$ labels and $a_4$ for the $SU(2)$ and in your example you used $a_3, a_4$ for $SU(3)$ labels and $a_1$ for SU(2). Thus you should use the second row in $G$ for your example. You will get the same result with the normalization factor $\frac{1}{3}$.
Jun
5
comment The $U(1)$ charge of a representation
@user41746 $SU(5)$ is $A_4$, so the metric tensor should be $4\times 4$ matrix, please see the first matrix in table 7 page 82. Its third row for n=4 is just the given vector. I can’t see this exercise on page 84 (This page belongs to the appendix and it contains tables 10 and 11a. However, on page 16 equation (3.3), Slansky obtained the same hypercharges as in the question.
Jun
5
revised The $U(1)$ charge of a representation
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Jun
5
answered The $U(1)$ charge of a representation
Jun
3
revised Coadjoint orbits in physics
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Jun
2
answered Coadjoint orbits in physics
May
29
revised Triality and charge
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May
29
answered Triality and charge
May
28
revised Why is the chiral symmetry $SU(2)_A$ not anomalous?
added 22 characters in body
May
27
answered Why is the chiral symmetry $SU(2)_A$ not anomalous?
May
22
comment The derivation of the Belinfante-Rosenfeld tensor
@ramanujan_dirac: The energy momentum tensor constitutes of the source term in Einstein equations: $R_{\mu\nu}-g_{\mu\nu}R = \frac{8\pi G}{c^4}T_{\mu\nu}$, and both the Ricci curvature and the metric tensor are symmetric.
May
4
answered The properity of $\mathbb{R}^4$ that has infinitely many differential structures is related to Yang-Mills field?
May
1
comment The curvature of the space of commuting hermitian matrices
@vesofilev: Sorry for not understanding you. The following reference arxiv.org/abs/0801.2913 contains a parameterization of the matrix $U$ for the flag manifold $\frac{SU(3)}{U(1)\times U(1)}$ on page 9, the third paragraph. (The authors call it the dressing matrix).
Apr
30
comment The curvature of the space of commuting hermitian matrices
@vesofilev: The $SU(2)$ case is a special case. Take for example the space of one commuting matrix in $SU(3)$, the spaces obtained from the adjoint action on $diag(0,0,1)$ and $diag(0,1,1)$ are disconnected.
Apr
30
comment The curvature of the space of commuting hermitian matrices
The space of $p$ commuting $n\times n$ matrices is not connected, since a unitary transformation does not change the rank of a matrix. Each connected component corresponds to a specific choice of the ranks of the commuting matrices. The connected components are generalized flag manifolds. Since they are homogeneous spaces, they are of constant curvature and there are many ways to calculate their curvatures.The following lecture notes may be of help to you uregina.ca/~mareal/flag-coh.pdf