String Vacua: "Quantum Field Theory: A Modern Introduction International Student Edition" by Michio Kaku

“Quantum field theory has emerged as the most successful physical framework describing the subatomic world. Both its computational power and its conceptual scope are remarkable. Its predictions for the interactions between electrons and photons have proved to be correct to within one part in 108. Furthermore, it can adequately explain the interactions of three of the four known fundamental forces in the universe. The success of quantum field theory as a theory of subatomic forces is today embodied in what is called the Standard Model. In fact, at present, there is no known experimental deviation from the Standard Model (excluding gravity). This impressive list of successes, of course, has not been without its problems. In fact, it has taken several generations of the world's physicists working over many decades to iron out most of quantum field theory's seemingly intractable problems. Even today, there are still several subtle unresolved problems about the nature of quantum field theory.”

In “Quantum Field Theory - A Modern Introduction International Student Edition” by Michio Kaku

“One of the main problems in superstring research has been to find the true vacuum of the theory, either perturbatively or nonperturbatively. Therefore, intense research over the years has been spent trying to catalog the various possible four-dimensional compactified strings.

A few classes of these solutions include:

1. Calabi-Yau manifolds, which are highly nonlinear, nontrivial manifolds studied by mathematicians;

2. Orbifolds, which are certain manifolds which have fixed points on them (e.g., a cone is an orbifold);

3. Free fermion/free boson solutions.

Unfortunately, we now know millions upon millions of possible string vacua. In fact, it is conjectured that the complete set of all possible string vacua is the totality of possible conformal field theories (CFTs). Although there are an enormous number of possible four-dimensional string vacua, the surprising feature of string theory is that, with a few rather mild assumptions, one can come fairly close to describing the physical universe. Earlier, we saw that Kaluza-Klein theory was too restrictive to describe the physical universe. In particular, the Standard Model's gauge group and complex fermion representations could not be accommodated. However, the string model, because it is not based on Riemannian space, does not suffer from these problems.”

In “Quantum Field Theory - A Modern Introduction International Student Edition” by Michio Kaku

Is the universe fond of pi? Seriously, that is indeed strange. Since 3+1-space and 4-space are not necessarily identical in how things radiate, it seems sensible to ask if the extra pi applies in either cases or just one.

If there is a difference, then things get interesting further up. There are, at most, two time dimensions within the 11 defined dimensions in modern M-theory. If 3+1 acts differently to 4, then you should be able to make predictions and observations that distinguish 10+1, 9+2 and a pure 11. Much more importantly, though, given that we're looking at stuff that would be entirely visible from any 3-dimensional cross-section of a higher-dimensional radiating property, we're looking at stuff that proves whether these dimensions exist at all. In other words, experimental evidence that could test string theory, supergravity, holographic universe theory (since you run into limits on what you can compress in what way), etc (If leptons of any given type are indeed the visible protrusion of a higher-dimensional object, then provided the geometry of that object is fixed, there are other experiments you can perform, since you cannot rotate an n-dimensional object around n+1 axes. That's an aside, though.) At the same time, we're told that such theories are closer to philosophy and untestable.

Any thoughts on this Kaku?

At such short distances, space is not smooth. Because it is not smooth, the effective radius (the radius you get by creating a perfect sphere of equal surface area) is not the apparent radius (the radius you get from naive assumptions about spacetime). If you apply the effective radius, the inverse square law is a perfect fit. This means that at Planck lengths, you have to consider the fact that spacetime is nothing more than geodesics. Now, obviously flattened spacetime is not going to be identical to curved spacetime. But That Does Not Matter. It doesn't matter because if you know the geometry, you know the space/area/line that the force is occupying. If you double the size of the container, you halve what is in any given part of it. This is so incredibly elementary. Anyone who knows about Compton wavelengths knows about geometry and knows that for every convex hull there is a circle of equal circumference. Why should I have to explain this? Kaku, step forward!

Calabi-Yau Spaces 2: "The Shape of a Life: One Mathematician's Search for the Universe's Hidden Geometry" by Shing-Tung Yau, Steve Nadis

“My proof, I told them [Andrew Strominger and Edward Witten], was motivated by physics, specifically the notion that even in a vacuum, a space with no matter, gravity could still exist. I felt certain that this must be important for physics, though I was not sure of the exact ramification.“

In “The Shape of a Life - One Mathematician's Search for the Universe's Hidden Geometry” by Shing-Tung Yau, Steve Nadis

“String theory further postulated that we inhabit a ten-dimensional universe consisting of the three familiar (and infinitely large) spatial dimensions, one dimension of time, and six  additional miniature dimensions that are wound up into a tight coil and thereby hidden from view.  The question that Candelas and Strominger, among others, were grappling with concerned the geometry of the six shrunken, or ‘compactified’ dimensions. What, exactly, is the shape into which these extra dimensions are confined? Strominger knew they needed a manifold, or space, with well-defined properties, including a special kind of symmetry called ‘supersymmetry,’ which turns out to be an intrinsic feature of the manifolds, of the variety called Kähler, whose existence I had proved. Supersymmetry is also a requisite feature of many versions of string theory, which is why it’s sometimes called ‘superstring theory’ instead.”

In “The Shape of a Life - One Mathematician's Search for the Universe's Hidden Geometry” by Shing-Tung Yau, Steve Nadis

I wear a giant panda suit outside a Panda Burger giving out promotional leaflets. As this job is a bit easy and I can do it without too much conscious effort... the only thing I have to watch out for is farting as it is unpleasant trapped in that panda suit... anyhow I digress ... this gives me a LOT of time to think about serious issues such as time and the merits of thinking about Physics in a Panda suit in scorching hot Lisbon in the Summer... 

Mark Twain said that scientific facts give rise to speculations, which of course are tested if possible. For the most part, math is not about "numbers" but largely about properties of, and relationships among highly abstract objects. Indeed, mathematics as a profession is a risk and self-sacrifice. One has to devote time and effort to one's field before one gets to appreciate it and produce results worth of publication. But there is always a risk that, even if one gains an understanding - which in itself is rare and precious - it will not be followed by original results, stalling one's academic career. This stalling of career due to the lack of originality is normally a direct result of being risk averse and not pushing yourself hard enough. Mathematics is an essentially creative activity: you are bound to achieve something if you are genuinely interested...Tricky thing defining maths. Even if the definition is true, it never looks very interesting. Certainly not as interesting as mathematics itself. It's certainly made a wee bit of progress from counting. Over the last few thousand years... There was that Archimedes and that other Euclid guy. And that Al Khwarizmi dude. Some Newton bloke. Euler, Gauss, a whole truckload of Bernoullis, Fourier, Cauchy, Poincare, Riemann, Noether, Cantor, Goedel, Brouwer... feel like I've forgotten a few hundred really big names but I just can't put my fingers on them...Reducing maths to numbers is kind of like saying all cooking is really just a matter of making 2 minute noodles.

My querky moment while learning mathematics was during a moment of boredom when I took the differences of successive calculated polynomial values and continued taking differences of the results. It turns out this is the basis of the difference engine that Babbage designed, and how mathematical tables were created before the advent of electronic calculators and computers. Probably unsurprisingly I took up Engineering which makes use of a myriad of mathematical techniques and valid short cuts, many of which are never taught to scientists and mathematicians in my experience.

There's something sublime, mystical and ineffable about such problems. You'd think maths would be easy, just counting, but hidden within those ostensibly basic concepts are such convolutions and crenelations and complications. It's amazing that 1+1 can get to such things like Fermat's Last Theorem and imaginary numbers or that Calabi-Yau Manifolds can be applied to Physics, namely String Theory and General Relativity. Let alone whatever these things are on about.

I just wish Yau had written a more math-oriented biography. We don't really get math insights on how he got to prove some of the things important to Physics, namely the Calabi-Yau conjecture. It's all very vague... If you want that to dig deeper into the math part of some of these topics, you should read “The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions” by the same authors.

Coda: No-one uses Calabi-Yau in a sentence (apart from Woody Allen in a New Yorker piece). It inspired me...

I wish my house was a Calabi-Yau Space,

a place where I could tell fiction from fact

I'd invite politicians to sit in the middle

Then I'd focus the heat so it's hot as a griddle

I'd make then elucidate policies at length

And keeping them talking to sap all their strength

And right at the end I would shout and declare

"Your lies and deceit are now totally clear

My house has deciphered your thoughts and your words

And showed them as nothing but bright polished turds

I'm leaving you now and I'll never come back

This part of my house is now fading to black....

NB: It was kind of interesting to read about Yau’s take on the feud between Yau and Chern and also his attempt at explaining what happened with the Poincaré Conjecture (he was accused of “stealing” Perelman’s discovery by having some of his students develop a more rigorous proof of Perelman’s demonstration).