# 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!

7. Jan 26, 2018

### mitchell porter

Calabi-Yau manifolds have a neat combination of properties but they are not the whole story. They are complex manifolds (meaning they can be covered in complex-valued coordinate patches, with transition functions that are complex-differentiable), they are Ricci-flat (volume is conserved by tidal forces), they allow a constant spinor field (and thus a low-energy supersymmetry).

But once you start looking at compactifications with fluxes or that are metastable (i.e. the geometry is changing), any or all of those properties can be lost. Also, if you start with M-theory, in 11 dimensions, you compactify on a 7-manifold, so it can't be complex, though it can have the other properties.

Calabi-Yaus are just where string phenomenology got started. The math was more tractable and it took a while to realize other possibilities... There is a theory that the "landscape" of d=4 string vacua is numerically dominated by flux compactifications of F-theory on a particular CY4, but we don't know if that has anything to do with the real world.

8. Feb 12, 2018

### bluecap

Spinnor. Since you google a lot. Maybe you know the answer to the following related:

Do you believe there is no known mechanism in string theory (and none in the any foreseeable future) that would dynamically prefer Calabi-Yau compactifications over other compactifications". Can't anyone think of any exception? Couldn't Calabi-Yau spaces be like alphabets where they could create all sorts of possibilities?

Also why is the interest in CY-compactifications said to be "entirely driven by the prejudice that nature should feature one unbroken supersymmetry at low (here: weak breaking scale) energy?"

Lastly. For years I kept seeing the thread "the wrong turn of string theory".. I thought it was talking about wrong turn of string theory in general. I just found out a while ago that that thread was about the unusual possibility there are supersymmetric relations among the already known particles. I missed this for 5 whole years. So my messages there are indeed off topic. Sorry to the participants

9. Feb 12, 2018

### Spinnor

From my questions you should have gathered I am NOT an expert in string theory, but that said I think it is the only game in town right now.

My uneducated bet now (might change Friday) is that we don't need supersymmetry. Maybe you know if string theory can still work without it?

People make wrong turns all the time, eventually they usually get where they need to be.

Question for you bluecap, in Kaluza-Klein theory can I think of the electric field of an electron as some type of curvature of 5D spacetime caused by a point electron?

Presumably the curvature gets smaller the further we are away from the electron?

Presumably the curvature can be associated with energy?

If we move past the electron the energy of the curvature of the electric field looks like momentum?

Can we associate the magnetic field with that momentum?

Electric field <--> curvature
Magnetic field<--> flow of curvature?

Thanks!

10. Feb 12, 2018

### bluecap

My question is. If there is no supersymmetry.. could there still be Calabi-Yau compactifications? Are the two related? Can you have Calabi-Yau compactification without any supersymmetry (whether in the weak scale or among the known particles in some hidden symmetry?) Anyone?

I read a lot of Brian Greene a few years back and Woit's, Smolin's, Randall's and other popular books and Idiot's Guide to Superstrings. I'll review them again this week as well as the more advanced ones... as I really need to scrutinize this whole e8xe8 thing (do you know the second set of e8 are shadow matter).

11. Feb 12, 2018

### Spinnor

12. Feb 12, 2018

### VIctor Medvil

Super-symmetry is necessary to explain Anti-particles in all forms without it you could not have the "Negative" Valued states or Opposite Vector direction in my experience. For Instance, you would just have a Charge of +1 or 0 and not -1 charge in your model without it on the electric dimension if it where applied to that, or spins in the opposite direction or whatever dimension you are referring to that has symmetry or Super-symmetry. .

13. Feb 12, 2018

### bluecap

Thanks. Please share about the connection between supersymmetry and Calabi-Yau compactifications. If there is no supersymmetry.. could there still be Calabi-Yau compactifications? Are the two related? Can you have Calabi-Yau compactification without any supersymmetry (whether in the weak scale or among the known particles in some hidden symmetry?)?

14. Feb 12, 2018

### VIctor Medvil

This is a link you may find interesting about Mirror Symmetry and them. Unfortunately, it is pay walled. =)

https://arxiv.org/abs/hep-th/0211102

Here is the version that is not pay walled took me a moment to find.

https://arxiv.org/pdf/hep-th/0211102.pdf

Last edited: Feb 12, 2018
15. Feb 12, 2018

### bluecap

nice links but just a quick question (because it would take me a week to read it thoroughly).. if LHC won't detect any form of supersymmetry in any scale.. can there still Calabi-Yau manifolds at planck scale? just answer yes or no and a bit of explanations.. lol.. tnx..

16. Feb 12, 2018

### VIctor Medvil

We do..... ever heard of Antimatter? that is basically it on the Electromagnetic Force dimension and SNF Dimension, but yes they would just be differently shaped Calabi-Yau manifolds that do not include it with slightly different math equations that govern them.

17. Feb 13, 2018

### Spinnor

Darn it!

18. Feb 13, 2018

### Staff: Mentor

That is totally wrong. The electric dimension - there is no such thing, In your experience - have you even studied Quantum Field Theory?

Normal Quantum Field Theory explains all those things.

The basis for electrical charge is U(1) symmetry, and other symmetries explain a large amount of things in the standard model:
http://uw.physics.wisc.edu/~himpsel/449group.pdf

That's the dazzling beauty part of the standard model. The kludge is the number of constants that have to be put in by hand. That's what theories beyond the standard model hope to explain.

Thanks
Bill

19. Feb 14, 2018

### PAllen

... by the anthropic principle, since the constants per the landscape are taken to be more like mass ratios of planets in the solar system, rather than anything fundamental.

20. Feb 14, 2018

### VIctor Medvil

Yes, there is do you know the definition of a dimension in QFT/QM?

Do you see that wave-function over ortho group O(x) that is a dimension being Change over r being a orthogonal change over r being another wonderful effect of how wave-functions exist over space. being Phi over R.

The Light wave certainly does about the strange thing do you see the vibration field of E and M over Wave function.

Very basic Trig explains it , but yes there are many Special Orthogonal groups. that do spinors on wave fuctions over a dimension, QFT/QM say it is fine to do this.

which Feynman also changes over near light particle change by canceling of something.

Maxwell's equation say it will change with one of my favorite operators, a masterful work of Laplace' his operator.

which gives you the curl of any Charge dimension or any thing over (I,J,K) by Maxwell being a change in charge over a volume basically.

Whatever, it is, it is space filling.being something on or over it.​

Last edited: Feb 14, 2018
21. Feb 14, 2018

### Staff: Mentor

Sure - that's one thing string theory suggested - I think it was Susskind that was a vocal proponent it was what string theory was trying to tell us.

Thanks
Bill

22. Feb 14, 2018

### Staff: Mentor

In standard QM its 3D Euclidian - in QFT its 4D Minkowskian.

That's utter gibberish like the rest of your post. As a mentor I can assure you if you keep it up action will be taken. To avoid that I STRONGLY suggest you learn some actual physics.

Their is nothing at all strange about light waves. Its simply an elementary solution to Maxwell's equations in free space which in the Lorentz gauge is ∂u∂uAv =0 - its a very basic exercise in partial differential equations to show the simplest solutions are waves. Try it.

Rest of misconceptions mercifully snipped.

At the moment just a friendly warning - we generally only accept statements that can be backed up by peer reviewed papers or reputable textbooks. What you have been posting is, to be blunt, incoherent wild speculation. If you keep it up I can assure you stronger action will be taken.

Thanks
Bill

23. Feb 14, 2018

### Staff: Mentor

Since EM has been mentioned in this thread lets see just whats really going on at a slightly non-rigorous level, but somewhat above B level. Have a look at good old F=ma. We will generalize this a bit to get something I call J and have it equal to k(d^2/dt^2)A where k is some constant and like J will not say anything yet about A. Now we will assume something about J - it is a 4 vector so you get Ju = k (d^2/dt^2) Au. This does not conform to the requirement of relativity - it should have the same form in all frames - the means (d^2/dt^2) should be ∂v∂v (of course the v should be raised in the second ∂v but I dot want to go to the trouble of Latex). The equation then is Ju = k ∂v∂v Au. We will assume Ju whatever it is is conserved ie obeys the equation of continuity ∂uJu = 0. This means ∂v∂v∂μAu = 0. To make this always true we will take ∂μAu = 0. The is called the Lorentz Gauge condition. Au is called the vector potential and Ju is called the 4 current. These are Maxwell's equations in the so called Lorentz guage - given special relativity they are not hard to come up with.

K simply determines units so we will take it as 1 for simplicity.

The equations then becomes ∂u∂u Av - ∂u∂vAv = Jv (1) since ∂vAv = 0.

Take this equation and we notice something. If we substitute A'v = Av + ∂vΛ, where Λ is any function of the xi, for Av then we get exactly the same equation in A'v. In fact if we take ∂v∂vΛ= -∂vAv we have a differential equation in Λ, solve it for Λ, and you then get the Lorentz Gauge condition. Take (1) as our equation and we see it has this very interesting gauge symmetry. Its an example of whats called U(1) symmetry. I am not 100% who exactly showed this symmetry is in fact the rock bottom essence of EM - it may be Weyl or Schwinger or someone else - but physicists now know its really the essence of Maxwell's Equations.

What I wrote is in some sense a derivation, but its validity depends on what went into it. It really is just fooling around with equations and seeing what you get - Dirac did this sort of thing a lot. Mostly it leads nowhere - but sometimes its reveals something quite interesting like here.

Now how do we get the usual form of Maxwell's equations.

Simply define Fuv = ∂uAv - ∂vAu and you get the normal electric and magnetic fields:
https://quantummechanics.ucsd.edu/ph130a/130_notes/node451.html

Forget the C; we will, for simplicity work in units C=1.

We have ∂v∂u Fuv = 0 as can be easily seen. And ∂uFuv = Jv so, as we assumed ∂vJv =0 ie the 4 current is conserved. This is how the field equations are arrived at but how do charged particles move? For this we use Lagrangian Field Theory. Lagrangian's take the form of Lp + Li + Lf. Here Lf is the Lagrangian of the field. Li is the Lagrangian of interaction between the field and charged particle and Lp is the Lagrangian of a particle. We have the equation of the EM field - that's the equation when you have no charge - just the field and it gives ∂uFuv = 0. To get Jv we have the interaction term which must be -JvAv. You add that to the particle Lagrangian and you get how a charged particle moves ie the Lorentz Force law. The gory detail can be found here:
http://www.phys.ufl.edu/~pjh/teaching/phy4605/notes/chargelagrangiannotes.pdf

This is a beginner thread, and what I wrote is at least I level. Don't be worried if its above your level - that's not why I posted it. It was simply to show there is no mystery in Electromagnetism - it is so well understood you can almost pull it out of nothing.

What string theory deals with are not issues like why we have negative charge - that isn't really a worry - string theory deals with much deeper issues - such as quantum gravity or like was mentioned before - suggesting that we have a massive number of spaces string theory can be compacted into. We live in a world that simply has one - we have to have one - the exact one we got isn't important as far as explanation is concercerned.

Thanks
Bill

Last edited: Feb 15, 2018
24. Feb 15, 2018

### andrien

I am not sure what you are searching for but if you look at Holonomy group generated by covariant constant spinor, it gives you SU(N) (N=3 for the case you are considering) rather than SO(N). So the notion of Complex space rather than Real space need to be used. An even dimensional real manifold does not necessarily translates to Half the real dimension Complex manifold, in fact there would be a proper subset of Real manifolds which will translate to Complex Manifolds.

25. Feb 15, 2018

### bluecap

Bill and others knowledgeable in superstring...

I googled a lot about "superstrings without supersymmetry".. some come out but it is either not indepth or very complex.. even search within PF didn't produce a lot on this important viewpoints. So I picked up the dusty old book Not Even Wrong by Peter Woit and he gave some interesting mathematical history but some details were not given (my questions below). First to set the tone, Woit wrote that:

"Another reason for being interested in supersymmetry was the hope that it might help with the problem of constructing a quantum field theory for gravity. One of the main principles of general relativity is what is called 'general coordinate invariance', which means that the theory doesn't depend on how one changes the coordinates one uses to label points in space and time. In some sense, general coordinate invariance is a local gauge symmetry corresponding to the global symmetry of space and time translations. One hope for supersymmetry was that one could some make a local symmetry out of it. This would be a gauge theory and might give a new version of general relativity, hopefully one whose quantum field theory would be less problematics."

I'd like to know:
1. Superstring theory without supersymmetry is supposed to be called bosonic string theory.. can't you create pure string theory without supersymmetry and supergravity, why?
2. If a theory can produce bosons and fermions without strings and supersymmetry.. would it be viable? what BSM programmes do this (advanced version of LQG)?