This is a fascinating story about the development of the mathematical concept of extra spatial dimensions known as Calabi-Yau spaces and its application in the string theory. The author speaks candidly, and describes his excitement at emerging new ideas in physics and mathematics, and how it progressed in string theory, and in the process changed his perspectives. Over the last 35 years this idea has shaped our thought on the nature of physical reality and involved an entire generation of theoretical physicists in research. This is partly autobiographical and hence makes it very interesting to read as he explains his odyssey. We get to read the contributions of leading physicists in this adventure; the growth of string theory as major force in theoretical physics. This is an outstanding book to read, but requires undergraduate level physics and strong interest in geometry.
A summary of this book is as follows: In string theory, the myriad of fundamental particle types is replaced by a single fundamental building block, a string. As the string moves through time it traces out a tube or a sheet (the two-dimensional string worldsheet), and different vibrational modes of the string represent the different particle types. The particles known in nature are bosons (integer spin) or fermions (half integer spin). By introducing supersymmetry to string theory both bosons and fermions could be accounted for, and with ten-dimensions, the mathematical requirements of string theory are completely satisfied. In addition, the anomalies and inconsistencies that plagued string theory are vanished. Until superstring theory came into existence, any predictions and calculations yielded nonsensical results, and were incompatible with quantum physics. The ten-dimensions consist of two sets four-dimensional spacetime we live in, and six-spatial dimensions in a hidden state in an invisible state because they are compactified to minute size. In this geometry, every point has a six-dimensional Calabi-Yau manifold in a compactified form, thus bringing physicists to the doorsteps of Calabi-Yau geometry.
Some physicists had originally hoped that there was only one Calabi-Yau manifold that would uniquely describe the hidden dimensions of string theory, but there are a large number of such manifolds each having a distinct topology. Within each topological class there are an infinitely large number of such Calabi-Yau manifolds. The Calabi-Yau space is further complicated by the fact that it has twisting multidimensional holes (about 500) running through the space. Another problem is; what makes the six-dimensions of space stable in a compactified form? It would be like constraining an inner tube with a steel belted radial tire. Just as the tire will hold back the tube as you pump air into it. All the moduli of the Calabi-Yau, both shape moduli and size moduli needs to be consistently stabilized. Otherwise the there is nothing to keep six hidden dimensions from unwinding and becomes infinitely large. It turns out that the D-branes of string physics can curb the tiny manifold's inclination to expand.
Some physicists have considered other types of spaces besides Calabi-Yau manifolds; they include non-Kahler compactification, and some non-geometric compactification postulates. In the beginning of the book , the author states : If Einstein's relativity is proof that geometry is gravity, string theorists hope to carry that notion a good deal further by proving that geometry, perhaps in the guise of Calabi-Yau manifolds is not only gravity but physics itself." In the latter part of the book the author takes a conciliatory mode by stating "Despite my affection for Calabi-Yau manifolds - a fondness that has not been diminished over the past thirty-some years - I am trying to maintain an open mind on the subject," ............."If it turns out that non-Kahler manifolds are ultimately of greater value to string theory than Calabi-Yau manifolds, I'm OK with that."
There are many success stories of mathematical reasoning; one such is the prediction of positrons by Paul Dirac. The biggest shortcomings of the Calabi-Yau space and the superstring theory and brane world is even though there is beauty and elegance in the setup but it still needs to make predictions which can be confirmed by the experiments. The results of LHC experiments so far have not resulted in satisfactory conclusions.
Reference: e Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions
A summary of this book is as follows: In string theory, the myriad of fundamental particle types is replaced by a single fundamental building block, a string. As the string moves through time it traces out a tube or a sheet (the two-dimensional string worldsheet), and different vibrational modes of the string represent the different particle types. The particles known in nature are bosons (integer spin) or fermions (half integer spin). By introducing supersymmetry to string theory both bosons and fermions could be accounted for, and with ten-dimensions, the mathematical requirements of string theory are completely satisfied. In addition, the anomalies and inconsistencies that plagued string theory are vanished. Until superstring theory came into existence, any predictions and calculations yielded nonsensical results, and were incompatible with quantum physics. The ten-dimensions consist of two sets four-dimensional spacetime we live in, and six-spatial dimensions in a hidden state in an invisible state because they are compactified to minute size. In this geometry, every point has a six-dimensional Calabi-Yau manifold in a compactified form, thus bringing physicists to the doorsteps of Calabi-Yau geometry.
Some physicists had originally hoped that there was only one Calabi-Yau manifold that would uniquely describe the hidden dimensions of string theory, but there are a large number of such manifolds each having a distinct topology. Within each topological class there are an infinitely large number of such Calabi-Yau manifolds. The Calabi-Yau space is further complicated by the fact that it has twisting multidimensional holes (about 500) running through the space. Another problem is; what makes the six-dimensions of space stable in a compactified form? It would be like constraining an inner tube with a steel belted radial tire. Just as the tire will hold back the tube as you pump air into it. All the moduli of the Calabi-Yau, both shape moduli and size moduli needs to be consistently stabilized. Otherwise the there is nothing to keep six hidden dimensions from unwinding and becomes infinitely large. It turns out that the D-branes of string physics can curb the tiny manifold's inclination to expand.
Some physicists have considered other types of spaces besides Calabi-Yau manifolds; they include non-Kahler compactification, and some non-geometric compactification postulates. In the beginning of the book , the author states : If Einstein's relativity is proof that geometry is gravity, string theorists hope to carry that notion a good deal further by proving that geometry, perhaps in the guise of Calabi-Yau manifolds is not only gravity but physics itself." In the latter part of the book the author takes a conciliatory mode by stating "Despite my affection for Calabi-Yau manifolds - a fondness that has not been diminished over the past thirty-some years - I am trying to maintain an open mind on the subject," ............."If it turns out that non-Kahler manifolds are ultimately of greater value to string theory than Calabi-Yau manifolds, I'm OK with that."
There are many success stories of mathematical reasoning; one such is the prediction of positrons by Paul Dirac. The biggest shortcomings of the Calabi-Yau space and the superstring theory and brane world is even though there is beauty and elegance in the setup but it still needs to make predictions which can be confirmed by the experiments. The results of LHC experiments so far have not resulted in satisfactory conclusions.
Reference: e Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions
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