Kahler Manifolds and Supersymmetric NLSM: Understanding the Connection

In summary: Fourth, an 8-dimensional space with a neutral metric is the only one for which 2-dimensional Weyl transformations are defined. That is, you can write a Lagrangian in the form of a scalar product of two 4-dimensional metrics, and then the Weyl transformations are defined independently for these two metrics. However, if we are considering a scalar product in a 6-dimensional space with a neutral metric, then the Weyl transformations are defined only for this metric, and the 4-dimensional metric is defined by the Minkowski space or some other way. For example, if you take the 4-dimensional Minkowski space with an inverse metric and a 4-dimensional space with a quadratic metric, you
  • #1
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TL;DR 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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  • #2
Moderator's note: Moved thread to the Beyond the Standard Model forum.
 
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  • #3
Yes, D-brane worlds.
 
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  • #4
Thank you, Demystifier!
Sten E
 
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  • #5
You mean by "describing dimensions" "compactifications to 4 spacetime dimensions?"
 
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  • #6
Thank you haushofer!
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.
 
  • #7
As for compactification, there are also orbifolds.
 
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  • #8
Demystifier said:
As for compactification, there are also orbifolds.
Calabi-Yau is an orbifold.
 
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  • #9
Thank you Demystifier and samalkhaiat! I will have a look at orbifolds.
 
  • #10
samalkhaiat said:
Calabi-Yau is an orbifold.
Yes. But some orbifolds are not manifolds, and obviously I meant those orbifolds.
 
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  • #11
StenEdeback said:
I wonder if Calabi Yau are the only players in that game,
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.
 
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  • #12
samalkhaiat said:
compactification requires Ricci-flat
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?
 
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  • #13
Demystifier said:
Why does compactification requires Ricci flatness?
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]
Does it depend on a requirement that some supersymmetry survives at low energies,
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].
is there an argument that does not depend on supersymmetry?
You can't avoid supersymmetry in superstrings. Where do the fermions come from?
 
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  • #14
StenEdeback said:
Thank you Demystifier and samalkhaiat! I will have a look at orbifolds.
An interesting book on my endless book list is:
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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  • #15
samalkhaiat said:
You can't avoid supersymmetry in superstrings. Where do the fermions come from?
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?
 
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  • #16
Demystifier said:
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?
No. In order for the compact internal space [itex]K^{D-4}[/itex] to be a CY space, it must admit a flat Kahler metric. In superstring, the Kahler metric comes for free and we can show that it is flat. This is not so in bosonic string theory. The pure Plyakov action of bosonic string is very boring. However, interesting things happen if you include the NLSM torsion potential (known as the Kalb-Ramond field [itex]H_{AB}(X) = H_{[AB]}(X)[/itex]) in the action and assume that the metric [itex]G_{AB}(X)[/itex] and the torsion potential [itex]H_{AB}(X)[/itex] are independent of [itex]X^{0}[/itex].
 
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  • #17
samalkhaiat said:
Consider .
StenEdeback said:
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?
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)##.
 
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  • #18
Is there an approach where the metric of GR is obtained from a manifold but only in the average sense over a hidden parameter $$\lambda$$, like $$g^{avg background}_{44}=\int sgn(\cos(\lambda))\rho(\lambda)d\lambda=0$$ and giving equations to obtain the desired GR metric $$g_{\mu\nu}=\int\langle\frac{\partial r^i}{\partial x^\mu}|g^{background}|\frac{\partial r^i}{\partial x^\nu}\rangle \rho(\lambda)d\lambda,\mu,\nu=0,...3$$

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 ?
 
  • #19
Addendum : in the case of 4D metrics, the manifold shall depend explictly on time, else the null component of the metric vanishes. Does this mean that the other components depend on t too, hence the metric could not be static, implying that it is impossible to find an embedding in a higher space giving the Schwarzschild metric for example ?0
 
  • #20
jk22 said:
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.

jk22 said:
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.
 
  • #21
samalkhaiat said:
In superstring, the Kahler metric comes for free and we can show that it is flat.
What exactly does it mean that it comes "for free"?
 
  • #22
Demystifier said:
What exactly does it mean that it comes "for free"?
Hmmm, unfortunately I don’t have the time to introduce the ABC of complex geometry in here.
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 (NLSM): Show that a [itex]4D[/itex] NLSM has the [itex]N = 1[/itex] supersymmetric extension if and only if the NLSM target manifold [itex]\mathcal{\Sigma}[/itex] is Kahler.
 
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1. What are Kahler manifolds and supersymmetric NLSM?

Kahler manifolds are a type of Riemannian manifold that satisfies certain geometric conditions. They are important in differential geometry and complex analysis. Supersymmetric NLSM (non-linear sigma models) are quantum field theories that describe the dynamics of a particle moving on a target space. They are characterized by supersymmetry, a symmetry that relates bosons and fermions.

2. What is the connection between Kahler manifolds and supersymmetric NLSM?

The connection between Kahler manifolds and supersymmetric NLSM lies in the fact that the target space of the NLSM is often a Kahler manifold. This allows for the use of geometric and analytical tools from differential geometry to study the dynamics of the NLSM.

3. How does understanding this connection help in theoretical physics?

Understanding the connection between Kahler manifolds and supersymmetric NLSM provides insights into the behavior and properties of these quantum field theories. It also allows for the development of new techniques for solving and analyzing these theories, which can have applications in various areas of theoretical physics such as string theory and condensed matter physics.

4. What are some current research topics related to Kahler manifolds and supersymmetric NLSM?

Some current research topics include the study of supersymmetric NLSM on non-Kahler manifolds, the role of Kahler manifolds in string theory and mirror symmetry, and the application of these concepts to other areas of mathematics such as algebraic geometry and topology.

5. What are some potential real-world applications of this research?

While the direct real-world applications of this research are not yet fully understood, the development of new techniques and understanding of these quantum field theories could have implications in fields such as quantum computing, materials science, and high-energy physics. Additionally, the study of Kahler manifolds and supersymmetric NLSM has connections to other areas of mathematics and could lead to advancements in those fields as well.

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