- #1
StenEdeback
- 65
- 38
- 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?
Calabi-Yau is an orbifold.Demystifier said:As for compactification, there are also orbifolds.
Yes. But some orbifolds are not manifolds, and obviously I meant those orbifolds.samalkhaiat said:Calabi-Yau is an orbifold.
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.StenEdeback said:I wonder if Calabi Yau are the only players in that game,
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?samalkhaiat said:compactification requires Ricci-flat
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]Demystifier said: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?
An interesting book on my endless book list is:StenEdeback said:Thank you Demystifier and samalkhaiat! I will have a look at orbifolds.
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?samalkhaiat said:You can't avoid supersymmetry in superstrings. Where do the fermions come from?
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].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?
samalkhaiat said: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)##.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?
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
jk22 said:in the case of 4D metrics, the manifold shall depend explictly on time, else the null component of the metric vanishes
What exactly does it mean that it comes "for free"?samalkhaiat said:In superstring, the Kahler metric comes for free and we can show that it is flat.
Hmmm, unfortunately I don’t have the time to introduce the ABC of complex geometry in here.Demystifier said:What exactly does it mean that it comes "for free"?
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.
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.
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.
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.
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.