“The spaces Calabi envisioned not only were complex, but also had a special property called Kähler geometry. Riemann surfaces automatically qualify as Kähler, so the real meaning of the term only becomes apparent for complex manifolds of two (complex) dimensions or higher. In a Kähler manifold, space looks Euclidean at a single point and stays close to being Euclidean because when you move away from that point, while deviating in specific ways."
In “The Shape of Inner Space - String Theory and the Geometry of the Universe's Hidden Dimensions” by Shing-Tung Yau, Steve Nadis
“So what exactly had I [Yau] accomplished [proving the Calabi-Yau Conjecture]? In proving the conjecture, I had validated my conviction that important mathematical problems could be solved by combining nonlinear partial differential equation with geometry. More specifically, I had proved that a Ricci-flat metric can be found for compact Kähler spaces with a vanishing first Chern class, even though I could not produce a precise formula for the metric itself. [..] Although that might not sound like much, the metric I proved to be ‘there’ turned out to be pretty magical. For as a consequence of the proof, I had confirmed the existence of many fantastic, multidimensional shapes (now called Calabi-Yau spaces) that satisfy the Einstein equation in the case where matter is absent. I had produced not just a solution to the Einstein equation, but also the largest class fo solutions to that equation that we know of.”
In “The Shape of Inner Space - String Theory and the Geometry of the Universe's Hidden Dimensions” by Shing-Tung Yau, Steve Nadis
“And it is within this circle of tiny radius that the fifth dimension of Kaluza-Klein theory is hidden. String Theory takes that idea several steps further, arguing in effect that when you look at the cross-section of this slender cylinder with an even more powerful microscope, you’ll see six dimensions lurking inside instead of just one. No matter where you are in four-dimensional spacetime, or where you are on the surface of this infinitely long cylinder, attached to each point is a tiny, six-dimensional space. And not matter where you stand in this infinite space. The compact six-dimensional space that’s hiding ‘next door’ is exactly the same.”
In “The Shape of Inner Space - String Theory and the Geometry of the Universe's Hidden Dimensions” by Shing-Tung Yau, Steve Nadis
Disclaimer: I don’t believe in String Theory being the TOE. Despite this, Yau’s book gives us a very thorough state-of-the-art compendium on String Theory (and M-Theory) and its connection with Calabi-Yau Spaces.
The current cosmology standard theory (Lambda CDM) is like a table with four legs; cosmic background radiation, inflation, dark matter and dark energy. The problem is, all four "legs" don't stand up to scrutiny. He is also right about cosmologist can't explain the one universe we have, so invest multi-verses and branes etc, which does little more than shift the problem elsewhere. For 200 years everyone thought that Newtonian gravity was correct, until Einstein proved it wasn't. Einstein's theory is also just a good approximation, and needs to be replaced.
Penrose's fashion-faith-fantasy strictures on "string theory" apply also to "black-hole theory", but the latter has its origins in his own topology of a surface-of-separation around the singularity. To some of us, including Einstein, this meant this class of solution is unphysical. Penrose would do physics a great service if he recognised this basic flaw in his early work and looked properly into the solution class with ultra-high gravitational fields replacing the matter-singularity (gravastars and collapsars).
Two battery chickens are chatting. The first says that he believes that humans have the power of life and death over them, and apparently infinite control over their well-being. He also believes that humans not only create amazing music, but care about the well-being of chickens, and indeed of every single chicken in their care. The second chicken tells him not to be so stupid, how can that be when they live in such awful conditions, wings and beaks clipped, force fed, only to be slaughtered and then eviscerated by hideous metal contraptions. I posit that there are only two possible ways out of this paradox. One is that humans either don't exist or don't have much control over the lives of battery chickens. The other will, I am sure, occur to you over time.
It's a question of probability. The odds of a randomly generated universe being capable of generating the stability and complex chemistry necessary for any conceivable kind of life is mind-bogglingly staggeringly small. Therefore our universe is mind-bogglingly staggeringly improbable. It's not "arrogance" to notice this and ask why.
Hawking's original explanation was that there are a near-infinite number of other universe that we can't see or detect in any way, and we just happen to live in one of the exceptionally rare ones that support life. But this is a highly speculative and unsatisfactory explanation precisely because we have no evidence that a multiverse exists. It's hard to imagine a more extreme violation of the principle of Occam's Razor.
How would a young Sheldon Cooper look alike answer this? I thought he'd try asking how a 5D AdS=4 CFT theory could exist in a 4D=3+1 D space as our world appears to be. Rather than fob him off with some slightly patronising "we can also build a 4D AdS theory!" the answer he'd be seeking was related to compactification of the extra 6=10-(3+1) dimensions so a 5D AdS theory can indeed fit into our (theoretically plausible) 10D universe, that appears 3+1 D to us, with some dimensions left over..
Perhaps Lee Smolin's Fecund Universe explains the String landscape. In his model, black holes create new universes, with differing constants of nature, so there is a "selection" effect, where "successful" universes are the ones that make the most black holes. Perhaps the String landscape of different ways to do Calabi-Yau manifolds is reflected in the different universes produced by black holes? I read somewhere these C-Y manifolds can be a way to do Brahms-Dicke theory which as my limited understanding is, is kind of a "generalization" of general relativity, where Newton's constant G is actually a variable. I always enjoy generalizations wherever possible so I like Brahms-Dicke, and the fact that it can be incorporated into C-Y manifolds is even cooler. So if the landscape problem is only real problem, than maybe this can be solved by the Fecund Universe model. However if it is true that the Large Hadron Collider ought to have found super-symmetries (not my area for sure) and they have not, that is more concerning than the landscape issue. Maybe string theory will turn out to be a piece of an even bigger picture which would explain this problem with not seeing super-symmetries. Having a computer science background, I know a bit about Set Theory, so I am naturally intrigued by Causal Set Theory though in fairness I don't know too much about it. Maybe Causal Set Theory can one day replicate in some way the predictions of string theory (or could be "boiled down" so to speak to a string model) but also solve the shortcomings re. Supersymmetry. Would love a video to explain more about Causal Set Theory or other possible candidates out there to solve these issues. There will be no resolutions to any issues until the questions are answered: why are Haag's and Leutweyler's theorems true in Relativity (where c < ∞) but not in non-Relativistic Theory (where c → ∞?) ... these are the two No Go Theorems that prevent a straightforward treatment of interactions (both classically and quantum theoretically) in Relativity and make necessary all the convolutions used in quantum field theory. The paradigms differ only in the value of c. Call them P(1/299792458² s²/m²) for Relativity and P(0) for non-Relativistic theory, the general case being P(1/c²).
What is the minimum value of 1/c² in which Haag and Leutwyler's Theorem hold in paradigm P(1/c²)? It can't be 0, because that would give (in principle) a way to measure a continuously varying quantity to infinite precision -- which is impossible. Just think of the cases of the paradigms P(1/c²) where 1/c² is so close to 0 that it is impossible to empirically distinguish it from 0 (i.e. the case where, non-relativistic paradigm holds for all practical purposes). One has a valid interaction picture there. That, of course, is expressed directly by Newton's Third Law ... and this is what's absent in Relativity (by virtue of the 2 above-mentioned No Go theorems).
Theory-Space must be continuous in its parameters. [In a way, this is a statement of the Correspondence Limit principle]. So, there should be some unifying framework Π(1/c²) that (a) upwardly extends P(1/c²) but (b) is free of the problems presented by Haag and Leutwyler for all values of 1/c² (thus enabling a more direct, straightforward treatment of quantum field theory), and (c) is continuous as 1/c² → 0. Many have tried to tackle this issue; Dirac, Wigner, Feynman, and have all failed.
Yau’s take on String Theory is a very enlightening read, if nothing else, it sheds light on the musings of Mathematicians and Physicists of a certain philosophical leaning, and it also tries to answer the old Physics' question: "Why will some cats share litter boxes while others won't?"
