Anomaly of representations in SU(N)

[nickthequick]

Homework Statement

The problem is the following: Compute the ratio of the anomaly of the N to that of the N(N-1)/2 representations in SU(n)

Homework Equations

Georgi claims that you can find the anomaly, A(R) of the [1] representation of SU(n) by calculating the anomaly of SU(3) (subgroup of SU(N)) under which n transforms like a 3 and n-3 singlets.

The Attempt at a Solution

The anomaly of the fundamental rep is going to be 1 here in SU(3). I'm not exactly sure how to find the anomaly of the (n-3) singlets. Also, if I do attain A([1]) for SU(N), I'm not sure how to extend this to the other N and N(N-1)/2 reps.

Any insights would be appreciated.

Cheers

 

[mark726]

Thank you for your post. This is an interesting problem that requires some knowledge of group theory and representation theory. To compute the ratio of the anomaly of the N to that of the N(N-1)/2 representations in SU(n), we can use the following formula:

A(R) = Tr(F^2) - Tr(R^2)

where A(R) is the anomaly of the representation R, Tr(F^2) is the trace of the generator of the group in the representation, and Tr(R^2) is the trace of the quadratic Casimir operator in the representation.

To find the anomaly of the fundamental representation [1] in SU(n), we can use the formula A([1]) = Tr(F^2) - Tr([1]^2) = n - 1, where n is the dimension of the group. This is because the fundamental representation has n-1 generators and the quadratic Casimir operator in this representation is n-1.

To find the anomaly of the (n-3) singlets, we can use the fact that the anomaly of a direct product representation is the sum of the anomalies of the individual representations. Therefore, the anomaly of the (n-3) singlets will be (n-3)(-1) = 3-n.

Now, to extend this to the other representations, we can use the formula A(R) = Tr(F^2) - Tr(R^2). For the N representation, we have A(N) = Tr(F^2) - Tr(N^2) = n - Tr(N^2). Similarly, for the N(N-1)/2 representation, we have A(N(N-1)/2) = Tr(F^2) - Tr(N(N-1)/2)^2 = n - Tr(N(N-1)/2)^2.

To find the ratio of the anomalies, we can use the fact that the trace of a direct product representation is the product of the traces of the individual representations. Therefore, the ratio of the anomalies will be:

A(N)/A(N(N-1)/2) = (n - Tr(N^2))/(n - Tr(N(N-1)/2)^2)

I hope this helps. Good luck with your calculations!