# B String theory, Calabi–Yau manifolds, complex dimensions

1. Jan 9, 2018

### Spinnor

So in string theory at each point of Minkowski spacetime we might have a 3 dimensional compact complex
Calabi–Yau manifold? We can have curved compact spaces without complex numbers I assume, what is
interesting or special about complex compact spaces?

Thanks!

2. Jan 10, 2018

### MathematicalPhysicist

I don't understand your question, I mean $\mathbb{C}$ is isomorphic to $\mathbb{R}^2$.
You can look at complex numbers as real vectors, nothing special about them.

I mean obviously we have an additional structure of addition and multiplication defined on real vectors given by complex addition and multiplication.

I.e from $(a+bi)(c+di) = (ac-bd)+i(ad+bc)$, so we have also multiplication on real vectors as: $(a,b)(c,d) = (ac-bd,ad+bc)$.

3. Jan 10, 2018

### Spinnor

Thank you for your reply. So I know the basics of complex numbers above. Let me try again.

Is it this additional structure above that differentiates say a 6 dimensional real compact space (again, I assume such a compact space exists) from a 3 complex dimension compact space? It seems that complex numbers have additional structure that nature may need that real numbers don't provide in regards to the proposed compact additional dimensions of string theory? Hoping for an "a ha" moment, a quantum jump in understanding.

Thanks!

4. Jan 10, 2018

### MathematicalPhysicist

You should wait for others that learned string theory, I haven't yet had the opportunity to read Zweibach's book, maybe next year.

BTW, are you reading technical texts or popular texts?

5. Jan 10, 2018

### MathematicalPhysicist

BTW don't restrict yourself only to complex numbers, we also have octonions,quartenions and whatnot

I.e, if the universe is infinite then every mathematical model has some manifestation in reality.

6. Jan 10, 2018

### Spinnor

Part of the problem is I have not poised my question better. I have had time to think about this and maybe can come up with better questions?

So I had a thought that is maybe a bit related to my question, real numbers are fine for general relativity, complex numbers help with Maxwell's equations but are not needed, complex numbers are necessary for quantum mechanics and quantum field theory, and it appears that complex compact spaces are required for string theory. So one might say, well yes, maybe you need complex compact spaces because at its heart nature is quantum mechanical and you need complex numbers to come from somewhere?

But back to maybe a better iteration of my question. So we are told that the number of distinct Calabi-yau 3 manifolds is larger then the big number 10^500 and if you look at a Google image search of Calabi-yau manifolds it is a wonderful assortment of shapes, for example.

So, do real 6 dimensional compact spaces come in the "crazy" forms above or does that require complex numbers?

If you had a dozen random real 6 dimensional compact spaces would they have similar features as a dozen random compact 3 complex dimensional Calabi-yau spaces but also differences?

I guess there is a simple reason for why nature needs complex and compact versus real and compact and if I did read about that reason I can not recall it.

Google takes me to lots of places but there does not seem to be much middle ground, it can only be made so simple it seems but fortunately many experts try and make it more simple. Like this little bit, https://www.maths.ox.ac.uk/about-us...d-mathematics-alphabet/c-calabi-yau-manifolds

Thanks!