Block form of the generators of the fundamental representation of SU(2N)

[Esbilon]

Homework Statement

I am calculating the corrections to the beta functions of a quite general SU(N) gauge-yukawa theory coming from coupling an electro-weak (SU(2)xU(1)) sector similar to that of the Standard Model.

To do this, I need to calculate
$$S'^r_{ab}T^A_{bc}T^B_{ca}S'^r_{de}T^C_{ef}T^D_{fd},$$
where i use the Einstein summation convention, $\delta_{bc}$ is the Kronecker delta function, $T^0_{bc}=\frac{\delta_{bc}}{\sqrt{4N}}$, $T^A_{bc}$ is a generator of the fundamental representation of SU(2N), $S'^r$ is block diagonal with the r'th pauli matrix in each of the blocks, $A, B, C, D=0,1,...,4N^2-1$ and $a, b, c, d, e, f=1,2,...,2N$.

Homework Equations

Since, this is the fundamental representation of SU(2N), we have
$$T^A_{ac}T^B_{ca}=C_2(2N)=\frac12.$$
I will also use the Fierz Identity
$$T^A_{ab}T^A_{cd}=C_2(2N)\delta_{ad}\delta_{bc},$$
which for the case of SU(2) can also be written
$$\sigma^r_{\alpha\beta}\sigma^r_{\gamma\delta}=2\delta_{\alpha\delta}\delta_{\beta\gamma}-\delta_{\alpha\beta}\delta_{\gamma\delta},$$
where $\alpha,\beta,\gamma,\delta=1,2$.

The Attempt at a Solution

I decompose the block diagonal matrix
$$S'^r_{ab}=S'^r_{\alpha k,\beta l}=\sigma^r_{\alpha\beta}\delta_{kl},$$
where $k,l=1,2..,N$, and I then use this and the Fierz identity above to write
\begin{align}
S'^r_{ab}S'^r_{cd}=S'^r_{\alpha k,\beta l}S'^r_{\gamma m,\delta n}&=\sigma^r_{\alpha\beta}\delta_{kl}\sigma^r_{\gamma\delta}\delta_{mn}\\
&=2\delta_{\alpha\delta}\delta_{\beta\gamma}\delta_{kl}\delta_{mn}-\delta_{\alpha\beta}\delta_{\gamma\delta}\delta_{kl}\delta_{mn}\\
&=2\delta_{\alpha\delta}\delta_{\beta\gamma}\delta_{kl}\delta_{mn}-\delta_{ab}\delta_{cd},
\end{align}
where I have recomposed the indices. Now my problem is that the first quartet of deltafunctions cannot be recombined using the same indices, so I've been trying to find out if it is possible, for a general N, to decompose the generators of SU(2N) in a similar form, that is find any objects U, V, and W such that
$$T^A_{ab}=T^A_{\alpha k, \beta l}=U^{ABC}V^B_{\alpha\beta}W^C_{kl}.$$

I've been working with this notation way too much, so feel free to ask questions if I'm being unclear ;-).

 

[mighty2000]

I would suggest approaching this problem by first breaking it down into smaller, more manageable steps. It seems like the main issue here is finding a way to decompose the generators of SU(2N) in a similar form to the one given for SU(2). So, my suggestion would be to focus on this step first and see if you can come up with a general method for decomposing the generators for any value of N.

One approach could be to look at the structure of the generators for different values of N and see if you can find any patterns or relationships that hold true. Another approach could be to use mathematical tools such as group theory to help you break down the problem.

Once you have a general method for decomposing the generators, the rest of the calculation should follow more easily. Good luck!