Thursday, October 03, 2013

String-string duality: irrational exuberance of sophistication

As is well-known, any consistent quantum description of forces including gravity has to be based on one of the equivalence approaches to string/M-theory. It is the supersymmetric vacua of string/M-theory (where SUSY is at most spontaneously broken) that are relevant for any stable world with fermions and without tachyons.

From K3, the simplest one after the torus.

Five 9+1-dimensional "string theories" in the old jargon – maximally decompactified vacua of string/M-theory in the new jargon – and one 10+1-dimensional theory (M-theory) are known. They are all connected to one network thanks to dualities, i.e. highly non-trivial equivalences that look impossible to a beginner and that a beginner wouldn't guess immediately but once he guesses them, they may be verified to hold absolutely exactly.

The six vacua are:
  1. M-theory: the 11-dimensional vacuum without strings but with M2-branes, M5-branes, and 11D SUGRA approximation at long distances. The remaining vacua are 10-dimensional vacua with strings.
  2. Type IIA: equivalent to M-theory on a short circle if the coupling is weak. The spacetime is left-right-symmetric while the string is chiral
  3. Type IIB: T-dual to type IIA (one one circle \(R\) is the same as the other on the circle of radius \(\alpha'/R\)). The spacetime is chiral while the stringy world sheet is not (the opposite case than type IIA). S-dual to each other; the \(\ZZ_2\) group is actually enhanced to an \(SL(2,\ZZ)\).
  4. Type I: type IIB with some orientifold 9-planes and spacetime-filling D9-branes and their mirrors (this relationship to IIB isn't a real duality). Gauge group \(SO(32)\) is the consistent choice that, due to Green's and Schwarz's proofs of consistency, sparked the 1984 First Superstring Revolution.
  5. Heterotic \(SO(32)\): it's connected to type I by S-duality (one with coupling \(g\) is the other with the coupling \(1/g\)). It's T-dual to the other heterotic string theory below if the radii and Wilson lines are properly matched.
  6. Heterotic \(E_8\times E_8\): aside from this T-duality, it's also equivalent to M-theory on a strip of 11D spacetime with two 10D end-of-the-world domain walls where the two \(E_8\) gauge supermultiplets are located.
Look at these theories and the dualities connecting them (not counting the IIB S-duality) that are listed above. Five dualities are enough connect these six theories into a web, a tree, and show that there's only one string/M-theory.

But this network of ideas is actually connected in many more ways. It's not just a tree. These theories are the vacuum part of a large configuration space that is pretty much infinite-dimensional – the relevant dimension of the moduli spaces (of the vacuum states) is "dynamical" and changes on various branches. And in this infinite-dimensional or dynamically-dimensional space, you may walk in many directions (infinite-dimensional spaces have many directions, indeed), and connect the theories and their compactifications in many more ways.

For almost two decades, I liked to draw the network of string/M-theoretical vacua connected by the five dualities as follows:

Please learn the Czech language if you want to know what's written in the caption or the whole 1997 article, thank you.

You see that the big string/M-theory cleverly looks like a giant letter "M". The letter "M" has five vertices but there are six theories because both maxima of the "M" graph represent pair of theories related by T-duality (the left peak are the two heterotic theories; the right peak are the two type II theories). That would bring us to seven but there are just six because the right lower end of the letter "M" contains no new theory, just a sign that type IIB is S-self-dual.

In some sense, those dualities get rather mundane if you get used to them. T-dualities look simple on the world sheet; S-dualities exist in quantum field theories, too; the type IIA-M and heterotic-M relationships remind us of some basic Kaluza-Klein methods to compactify one dimension on a circle (or a line interval).

String-string duality is harder, more shocking

But the string-string duality (or "heterotic-K3 duality") is much more stringy. It requires some advanced geometry to be understood. Unlike the basic dualities above, it demands that we reduce the number of uncompactified dimensions to 6 or 7. This duality is sort of crazy because it says that
Type IIA on the four-real-dimensional K3 manifold is equivalent to a heterotic string theory on a four-torus.
Why do I say it's crazy? Because the K3 manifold is very complicated – a section of an example of the K3 manifold is shown on the animated GIF at the top. It's the quartic surface in \(\mathbb{CP}^3\). You take four complex coordinates, make the 4-complex-dimensional space projective by an identification so that it's just a 3-complex-dimensional one, \(\mathbb{CP}^3\), and then you write an equation of the fourth order for the original four coordinates \(z_i\). That reduces the number of complex dimensions by another one so you're left with a 2-complex-dimensional or 4-real-dimensional manifold. It may have various shapes, some of these parameters may be encoded in the coefficients of the quartic polynomial.

This K3 surface has many holes (20 holes of an intermediate dimension, as we will discuss soon) i.e. a rather complicated topology – although it's a manifold that the algebraic geometers know since their 101 class. It also has a nonzero curvature – nonzero Riemann tensor – although it's Ricci-flat.

If one of the anti-string crackpots increased his or her intelligence by 40 points so that he or she would be able to listen to a high-brow talk at least for a minute so that he or she would hear the name K3, he or she would probably think that it's one of the contrived, arbitrary, "unscientific" structures that string theory is flooding us with. You know what these breathtaking imbeciles are saying all the time.

So if a K3 is one of the insufferable geometric complications that string theory is making up, it shouldn't be equivalent to something that has no such complications. However, the compactification of type II string theory on this curved, nontrivial manifold is exactly equivalent to the compactification of heterotic string theories on a simple manifold, the four-torus, that is defined simply by making four flat Cartesian coordinates periodic.

String theory not only dictates the critical dimension; it dictates the amount of "mathematical sophistication" that can nevertheless take on many forms. This sophistication may be hidden in the extra left-moving bosons or fermions that produce the \(SO(32)\) or \(E_8\times E_8\) spacetime gauge group in heterotic string theories; or it may be hidden in the curved and sexy shapes of a K3 manifold. A similar theory with the same degree of SUSY but without either of these two features would be inconsistent. A theory with both features at the same moment would be inconsistent, too.

I will hopefully (maybe) complete this blog entry tomorrow. Please don't think that I've been playing with my new tablet all the time. Instead, lots of work as well as reading of sources about the MSD cooperative savings bank which lost the license to do banking, as we were told today (the central bank's verdict is legally ineffectual so far). I have much more money over there than what I have ever earned from this blog so you may imagine I am a bit interested in all the details.


  1. K3 as displayed is chiral. It is not superposable upon its mirror image. You prove that photon vacuum mirror symmetry assumed for (fermionic) matter is broken: parity violations, chiral anomalies, symmetry breakings; Chern-Simons parity repair of Einstein-Hilbert action. Vacuum is trace chiral anisotropic toward matter. Noetherian vacuum isotropy enforcing angular momentum conservation leaks for matter as MOND's Milgrom acceleration re Tully-Fisher. No dark matter.

    One observation repairs 10^500 sterile vacua. Do it, then rederive empirical string theory. Opposite shoes chiral vacuum embed (mount a left foot) with different energies. They vacuum free fall along divergent minimum action trajectories - Equivalence Principle violation. Crystallography's opposite shoes are visually and chemically identical, single crystal test masses in enantiomorphic space groups. Two geometric Eötvös experiments, 5×10^(-14) difference/average sensitive. 0.113 nm^3 volume/alpha-quartz unit cell. 40 grams as 8 single crystal test masses contrast 6.68×10^22 opposite shoes (enantiomorphic unit cell pairs). You could be a hero.

  2. "one one circle R" should be "on one circle R"

  3. I'm going to take a wild guess that Xemist is a crackpot.


  5. This looks like spam that should not be here on this nice article ...

  6. Provocative observation: the views of Richard Feynman about ST resemble those of PW :-)

    It is interesting what RF believes should be the future research direction in particle physics: to explain the masses of the particles, why the particles have the masses they do. That is a very interesting question. Could we get the values of the masses from some group theoretic arguments?

  7. Never seen you here before, Al. Welcome. From me at least. :-)