Know more about String Theory applied to Calabi-Yau spaces
Comprehending what is Calabi-Yau space and its relation to String Theory
In the domains of both physics and mathematics, there’s a lot of interest in Calabi-Yau spaces. Those who wonder if they are somehow related to Calabi Yau by Tesseract, which is a song, should know the Calabi-Yau spaces in this post are different. Some may even find Calabi-Yau pronunciation a bit difficult but this doesn’t impact the fact that they are extremely important to String Theory.
String Theory applied to Calabi-Yau spaces
Calabi-Yau spaces (also called Calabi-Yau manifolds) play an important role in String Theory, where one model puts forward the universe’s geometry to consist of a 10-dimensional space, which is in the form of MxV, where M stands for a 4-dimensional manifold (time-space) while V refers to a 6-dimensional dense Calabi-Yau space. Those who wonder what is Calabi-Yau space may now have an answer.
Though Calabi-Yau spaces find major application in the domain of theoretical physics, especially when one considers String Theory applied to Calabi-Yau spaces, they are equally interesting even from a completely mathematical point of view.
It should be noted that while String Theory model does result in the required ten dimensions, the result of Super String Theory, or M-Theory postulated by Professor Witten at a Astrophysics seminar in 1995 resulted in a required eleven degrees of freedom to make the supersymmetry particles come in line for M-Theory. After saying this, there is an implied resultant concept due to the above that results in the maximum of 100 to a hundred power possible distinct and separate Universes of one which is our own.
Now there was an equation was given to Dr. Einstein by a Russian Mathematician named Kaluza and it rocked Dr. Einstein’s world. Initially the great Physicist scoffed at its simplicity. But when the good Dr. realized that Kaluza’s theory postulated the combination of Einsteinium space matrix of ten dimensions along with Maxwell’s field matrix of four dimensions resulting in a fifth dimension for Gravity, he immediately submitted the paper for publication. The equation is quite simple to state for it is but addition of valid variables, namely: 1+M(4)+E(10)=15. The parameters M(4) and E(10) are placed here to
to explain the variables ie: M(4) is the necessary dimensions required according to Maxwells field theory of the propagation of light through space while E(10) is the necessary dimensions specified by Dr. Einstein’s General Theory of Relativity. This parameter is of course determined by Dr. Einstein’s metric Tensor. Leaving the one which is the additional dimension necessary to explain the fifth dimension for gravity. It is of course added to the 4 dimensions of our space time continuum. Note please that the equations solution in this particular instance is 15 components which are readily inclusive of Riemann’s fifth dimensional gravity field as well as the 10 for Einstein’s and the four components of Maxwell’s field. By the way this equation also works for the compacted 6-dimensional Calabi-Yau spaces as well. The ending component space equals 11 dimensions specified by string theory ie: 6 for the Calabi-Yau manifold + 3 spacial dimensions of our reality + one for time + 1 for gravity equals the 11 dimensions required by string theory.
The curious case of Calabi-Yau manifolds
A strange observation was made by physicists when they were building examples of Calabi-Yau manifolds. They found that Calabi-Yau manifolds naturally seemed to crop up in pairs in which each Calabi-Yau can be linked to its “mirror.” What was more interesting is that with a minor change of string theory, it was possible to exchange a Calabi-Yau manifold with its partner (or mirror) sans any change in physics at all. This, in turn, encouraged physicists to suggest several correspondences between the geometry of Calabi-Yau manifolds and their “mirrors.” However, this proposition seemed illogical to mathematicians.
Additionally, within the compactification nature of Calabi-Yau manifolds there exist another hypothesis that states that the series of holes within the manifold and the resultant frequencies of the superstrings give rise to the very particles which make up out Universe.
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Dr Braun, Andreas. “C is for Calabi-Yau Manifolds.” University of Oxford Mathematical Institute, Updated June 7, 2018. https://www.maths.ox.ac.uk/about-us/life-oxford-mathematics/oxford-mathematics-alphabet/c-calabi-yau-manifolds