# GUTs: Understanding SU(5), SO(10) & E6

• Jim Kata
In summary, the conversation discusses the use of a diagonal representation of su(5) in the context of GUTs. It is mentioned that there are 24 generators in su(5), but not all of them are used to avoid having 24 gauge bosons. Instead, a vacuum expectation value for a Higgs field in the adjoint representation breaks su(5) into su(3) x su(2) x u(1). The conversation also mentions the possibility of using the Cartan subalgebra and asks for good online resources on the topic.
Jim Kata
I don't have access to Georgi's and Wilcheck's work, if somebody has some good links to an overview of the su(5), so(10), and e6 GUTS that would be awesome.

Anyways, there's something I really don't get, let's take su(5) for example, su(5) has the root system $$A_4$$ now there is 4(4+2) = 24 generators of the lie algebra. Now I know they didn't use all 24 of the generators because it would mean that there were 24 gauge bosons, not including the Higgs. So I heard they used a diagonal representation of su(5). Something like

[su(3)0]
[0 su(2)]

Does that entail just using the Cartan subalgebra or what? What did they do?

All 24 gauge bosons are there, but SU(5) is spontaneously broken to SU(3)xSU(2)xU(1) by a vacuum expectation value for a Higgs fields in the adjoint representation.

Srednicki's QFT book (available free on his website) has a good basic discussion of the SU(5) model.

Jim Kata said:
I don't have access to Georgi's and Wilcheck's work, if somebody has some good links to an overview of the su(5), so(10), and e6 GUTS that would be awesome.

Anyways, there's something I really don't get, let's take su(5) for example, su(5) has the root system $$A_4$$ now there is 4(4+2) = 24 generators of the lie algebra. Now I know they didn't use all 24 of the generators because it would mean that there were 24 gauge bosons, not including the Higgs. So I heard they used a diagonal representation of su(5). Something like

[su(3)0]
[0 su(2)]

Does that entail just using the Cartan subalgebra or what? What did they do?

I'm not at all a specialist of GUT, but I think only generators from SU3 x SU2 x U1 are diagonal and commute. Other generators are non-diagonal and entail the breaking of SU5 into SU3 x SU2 x U1.
If you have good www ressources, I'm also interested.

## 1. What is the significance of SU(5), SO(10), and E6 in physics?

These are three of the most well-studied and influential Lie groups in particle physics. They are used to describe the symmetries and interactions between fundamental particles and are important in understanding the underlying structure of the universe.

## 2. How do these Lie groups relate to each other?

SU(5) is a subgroup of SO(10), and SO(10) is a subgroup of E6. This means that SU(5) is a smaller, more specific version of SO(10), and SO(10) is a smaller, more specific version of E6.

## 3. What do the numbers in SU(5) and SO(10) represent?

The numbers in these Lie groups represent the number of dimensions in the mathematical space used to describe the particles and their interactions. SU(5) has 5 dimensions, SO(10) has 10 dimensions, and E6 has 6 dimensions.

## 4. How are these Lie groups used in Grand Unified Theories (GUTs)?

GUTs attempt to unify the three fundamental forces (electromagnetic, weak, and strong) into a single force at high energies. SU(5), SO(10), and E6 are all potential candidates for this unification, with E6 being the most widely studied and successful so far.

## 5. Are there any experimental evidence supporting the use of these Lie groups in GUTs?

While there is currently no direct experimental evidence for the unification of forces using these Lie groups, there have been attempts to test the predictions of GUTs. For example, the proton decay, which is predicted by many GUT models, has been searched for but not yet observed. However, the mathematical elegance and success of these theories in explaining other phenomena make them a promising avenue for further research and exploration.

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