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I was reading the book "A Fortunate Universe" by Geraint Lewis and Luke Barnes and something caught my attention:

At page 195 the authors say that universes with different symmetries could be modeled and they would have dramatic results like having different conservation laws.

I asked Mr. Lewis if he could give me more information about this and he told me this:

We can play with equations and add or subtract dimensions in our universe – like adding another time dimension to relativity. These other hypothetical universes could have different symmetries, so things like electric charge might not be conserved, while kinetic energy is. This would give very different physical outcomes. And yes, we could get very different physical laws with interactions that we just don’t see in our universe.

He mentioned that these universes with different symmetries and non-conserved quantities would be the result of "playing" with the dimensions of mathematical models of universes. This reminds me a lot to what happens in String Theory compactification or in M Theory.

And also, at another book by Kurt Sundermeyer called "Symmetries in Fundamental Physics", he says at page 466:

It is an astounding state of affairs that string theories in 10 dimensions vibrate with symmetries, but that after compactification we may get into worlds with no symmetries at all

I have always read that in string theory landscape the different vacua would have the same symmetries but they would be broken differently, or at least that is what I have understood until now. But I have never read anything about vacua with different fundamental symmetries in string theory.

Therefore, does String Theory or M Theory propose that these universes with very different fundamental symmetries may exist? Do any of these theories speculate that universes with different kinds of symmetries (different global symmetries, local symmetries, gauge symmetries...etc) may exist?

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I think that when talking about symmetries in string theory compactification what is most often referred to are supersymmetries. In this sense, what you try to do when compactifying is to get a 4 dimensional theory with N=1 supersymmetry and keeping that supersymmetry unbroken is already quite hard (you need to compactify on specific compact manifolds etc.) because the supersymmetries are very easily broken by compactification. As far as I know, most of the research going on stops at that stage, as in when a low-energy N=1 supergravity theory is obtained and it is considered Standard-model like if it is actually MSSM-like (minimal supersymmetric standard model). Now, when compactifying you have some freedom as to exactly which supersymmetries you are breaking but (as far as I know) it doesn't really matter as long as you end up with N=1 supersymmetry at the end (this N=1 criterion dictates what kind of fields you have).

I am not sure if this answers your question. When it comes to modiying the number of dimensions, you can compactify your string theory on a manifold of any dimension you want and end up with 8d, 6d etc theories. The 6d theories are of some use because they make the math a lot simpler but they cannot represent reality for simple reason that they are 6d. On the other hand, you can modify the dimensions of your original string theory by considering what is called non-critical string theories but then you have to be careful about anomalies that arise in the theory.

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