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- Summary:
- Alternatives to Calabi Yau?

Are there alternatives to Calabi Yau spaces describing dimensions in superstring theory? If yes, what are they? If no, why?

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- Thread starter StenEdeback
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- #1

- 32

- 10

- Summary:
- Alternatives to Calabi Yau?

Are there alternatives to Calabi Yau spaces describing dimensions in superstring theory? If yes, what are they? If no, why?

- #2

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Moderator's note: Moved thread to the Beyond the Standard Model forum.

- #3

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Yes, D-brane worlds.

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Thank you, Demystifier!

Sten E

Sten E

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haushofer

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You mean by "describing dimensions" "compactifications to 4 spacetime dimensions?"

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Yes, that is a good way of putting it. I wonder if Calabi Yau are the only players in that game, or if there are other ways of handling all these extra dimensions.

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As for compactification, there are also orbifolds.

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samalkhaiat

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Calabi-YauAs for compactification, there are also orbifolds.

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Thank you Demystifier and samalkhaiat! I will have a look at orbifolds.

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Yes. But some orbifolds are not manifolds, and obviously I meant those orbifolds.Calabi-Yauisan orbifold.

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- #11

samalkhaiat

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Yes, they seem so. Broadly speaking, compactification requires Ricci-flat compact (complex) space. Ricci-flatness and compactness are what CY spaces have. Of course, finding the right one is another story.I wonder if Calabi Yau are the only players in that game,

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Why does compactification requires Ricci flatness? Does it depend on a requirement that some supersymmetry survives at low energies, or is there an argument that does not depend on supersymmetry?compactification requires Ricci-flat

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samalkhaiat

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Consider [itex]M^{D} = M^{d} \times K^{D-d}[/itex]. From Supergravity action you obtain the Einstein equation in [itex]M^{D}[/itex]: [tex]R^{(D)}_{AB} = 0.[/tex] This implies, [tex]R^{(d)}_{\mu\nu} = 0, \ \mbox{&} \ R^{(D-k)}_{mn} = 0.[/tex]Why does compactification requires Ricci flatness?

I would agree with that, if I was a phenomenologist. Mathematically, world-sheet supersymmetry means that there are Killing spinors, [itex]\nabla \epsilon = 0[/itex], on the target space of the critical dimension [itex]M^{10}[/itex]. Then, one can easily show that [itex]\nabla \epsilon = 0 \ \Rightarrow \ R^{(10)}_{AB} = 0[/itex]. So, if you consider the solution [itex]M^{10} = M^{4} \times K^{6}[/itex], then [itex]\mbox{Ric}(K) = 0[/itex].Does it depend on a requirement that some supersymmetry survives at low energies,

You can't avoid supersymmetry in superstrings. Where do the fermions come from?is there an argument that does not depend on supersymmetry?

- #14

MathematicalPhysicist

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An interesting book on my endless book list is:Thank you Demystifier and samalkhaiat! I will have a look at orbifolds.

https://www.amazon.com/dp/0521870046/?tag=pfamazon01-20

Seems like a good starting place, but other than that I don't know since I haven't even started reading this book, I have other books on my reading right now.

Cheers!

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Well, at least for academic purposes one can study compactification in 26-dimensional bosonic string theory. By replacing supergravity action with bosonic field action, your argument can be used to argue that we would need Calabi-Yau even then, am I right?You can't avoid supersymmetry in superstrings. Where do the fermions come from?

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samalkhaiat

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No. In order for the compact internal space [itex]K^{D-4}[/itex] to be a CY space, itWell, at least for academic purposes one can study compactification in 26-dimensional bosonic string theory. By replacing supergravity action with bosonic field action, your argument can be used to argue that we would need Calabi-Yau even then, am I right?

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Consider .

There is an alternative, but it is almost unknown and unexplored. It is usually considered ##M^{4}\times K^{D-4}## where both components of the product vary. However, there is an alternative when the space is fixed and the vector field is varied. For example, if you take an 8-dimensional space with a neutral metric and vary the vector field in it, you can get interesting things. First, if we consider the Lie algebra of linear vector fields annihilating the gradient of a quadratic metric interval, then note that this is the algebra of tangent vector fields of a 7-dimensional hypersphere of a space with a neutral metric, which is isomorphic to the Lie algebra of the matrix Dirac algebra. Second, if we take a doublet of Minkowski spaces with an inverse metric, then we obtain an 8-dimensional space with a neutral metric, therefore, a consistent local deformation of a linear covector field in the Minkowski space and its dual space induces a pseudo-Riemannian manifold. Third, if we compactify a space with a neutral metric, then the symmetries of the compactified isotropic cone will (as expected) correspond to the group ##SU(3)\times SU(2)\times U(1)##.Summary::Alternatives to Calabi Yau?

Are there alternatives to Calabi Yau spaces describing dimensions in superstring theory? If yes, what are they? If no, why?

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Where $$r^i:\mathbb{R}^4\rightarrow\mathbb{R},i=0,...4$$

The background metric is periodic, so are the vectors determining the vector r in 5D space, but with an incommensurable period ?

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Is there an approach where the metric of GR is obtained from a manifold but only in the average sense over a hidden parameter

Where are you getting this from? Personal speculation is off limits here.

in the case of 4D metrics, the manifold shall depend explictly on time, else the null component of the metric vanishes

What does this mean? It doesn't make sense to me.

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What exactly does it mean that it comes "for free"?In superstring, the Kahler metric comes forfreeand we can show that it is flat.

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samalkhaiat

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Hmmm, unfortunately I don’t have the time to introduce the ABC ofWhat exactly does it mean that it comes "for free"?

To understand what I meant, try to do the following exercises:

1) Given chiral superfields [itex]\Phi^{i}[/itex] and their conjugates [itex]\Phi^{+ i}[/itex], write down the most general (kinetic) Lagrangian for [itex]N = 1[/itex] chiral superfields in superspace. Work out the form of the Lagrangian in terms of the ordinary field components [itex]A^{i} = \Phi^{i}|_{\theta = 0}, \ \mbox{and} \ \chi^{i}_{\alpha} = D_{\alpha}\Phi^{i}|_{\theta = 0}[/itex].

2) Non-linear-sigma-model (

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