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I have taken a course on group theory a few years ago. I have also used a little bit of group theory in a particle physics course.

However, now I have started with AdS/CFT and string theory and I am always confused when references to group theory are made.

Here is one of many examples: Say I study $\operatorname{AdS}_5 \times S^5$. Then someone mention that the isometry group of this is $\operatorname{SO}(4,2) \times \operatorname{SO}(6)$.

I could never have guessed the underlying isometry group, and I know so little group theory I don't even know why I need it...

In papers, authors typically mention the underlying group of the theory, but never does anything explicitly with that information (to my knowledge) which has allowed me to be ignorant about this for so long.

What resources are good to be able to obtain/understand the underlying group and what it does for the theory?

I know the question is very general but any guidance is very much appreciated.

Samuel Adrian Antz
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Nikolo J Bar
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    Possible duplicates: https://physics.stackexchange.com/q/6108/2451 and links therein. – Qmechanic Sep 19 '22 at 10:41
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    You can guess the underlying symmetry group in this case. SO(4,2) is the isometry group of AdS$_5$ and SO(6) is the isometry group of S$^5$. – Prahar Sep 19 '22 at 13:06
  • Hi @Prahar. Thanks. How would you determine the symmetry group of some other space? Is this something people remember or is there a way to quickly see it from a given space(time)? – Nikolo J Bar Sep 19 '22 at 16:21
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    You can of course derive it. You have to work out all the killing vectors of the spacetime and then check the structure of their algebras. However, through practice and experience, I now remember the symmetry groups of several relevant spacetimes such as Minkowski$_D$, AdS$_D$, CFT$_D$, S$^n$, etc. – Prahar Sep 19 '22 at 17:13
  • Thanks for this @prahar. Are you aware of any good resources that consider such examples explicitly? – Nikolo J Bar Sep 27 '22 at 10:37

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