What is the best PhD choice for gauge-gravity duality research? I am a mathematician but want to do a PhD in mathematical physics focusing on gauge-gravity duality and more generally quantum gravity, what is the most appropriate PhD choice to do research in this field, is it an astrophysics (cosmology) PhD or a particle physics one?
I am looking for a program that I can take a lot of relevant abstract mathematics courses like topological quantum field theory, functorial quantum field theory, algebraic geometry and topology, while at the same time I can get a good understanding of physics concepts about quantum gravity.
At least in the US, your first question should be whether to do a mathematics or a physics PhD.  It's possible to work as a physicist with a mathematics PhD and vice versa (and I know people who do it), but it's hard enough to find an academic job (which is the only kind of job that will pay you for theoretical research of the kind you seem to be interested in) as it is, and having background and teaching experience in the &quot;wrong&quot; field makes it much harder.
It's important to understand that physics and mathematics are different fields with different ideas of what it means for a statement to be true.  In mathematics, a statement is true if and only if someone has given a valid mathematical proof for it.  In physics - well I'm a mathematician who is ignorant of physics, so I think it's best to let physicists speak for themselves - but it's not what I just wrote for mathematics.  Their definition (in the mathematical sense of definition) of truth is different.  Which definition of truth do you want to work with?
Also important - would you rather teach introductory physics or calculus?
In either case, you want to be at a university where the mathematics and physics departments have a good relationship with each other.  (For this purpose unlike many others, having no relationship does not count as having a good relationship.)