Unsere Welt zu verstehen:  Dualities

 Beitrag 0-171

The Dualities of M-Theory


Michio Kaku remembering die 2nd String Revolution:
First indications that duality might apply in string theory were found by K. Kikkawa and M. Yamasaki of Osaka Univ. in 1984. They showed that if you "curled up" (compactified) one of the extra dimensions into a circle with radius R, the theory was the same as if we curled up this dimension with radius 1/R.
This is now called T-duality: Torus of radius R is equivalent to Torus of radius 1/R.

    When applied to various superstrings, one could reduce 5 of the string theories down to 3. In 9 dimensions — with one dimension curled up — the Type IIA and IIB strings were identical, as were the E(8) × E(8) and O(32) strings.
    Unfortunately, T duality was perturbative.

The next breakthrough came when it was shown that there was a second class of dualities, called S-duality, which provided a duality between the perturbative and non-perturbative regions of string theory.

    The existence of S-duality in string theory was first proposed by the Indian physist Ashoke Sen in 1994. One year later Witten saw
    that type IIB string theory with the coupling constant g is equivalent — via S-duality — to the same string theory with the coupling constant 1/g.
    Similarly, type I string theory with the coupling g is equivalent to the SO(32) heterotic string theory with the coupling constant 1/g.
    Unlike the T-duality, however, S-duality has not been proven to even a physics level of rigor for any of the aforementioned cases. It remains — strictly speaking — a conjecture, although most string theorists believe in its validity. (Wikipedia)

Another duality, called U-duality, is even more powerful: The U-duality group of a given string theory is a group which comprises T- and S-duality and embeds them into a generally larger group with new symmetry generators.

    The main example is the system type IIA/IIB in d ≤ 8 on T10-d (T for Torus).
    These two 10-dimensional theories are different limits of a single space of compactified theories, which are called the moduli space of type II theory (meaning all compactifcations of type IIA and IIB).

Then Nathan Seiberg and Witten showed how another form of duality could solve for the non-perturbative region in four dimensional supersymmetric theories.
What finally convinced many physicists of the power of this technique was the work of Paul Townsend and Edward Witten: They caught everyone by surprise by showing a duality between 10-dim Type IIa strings and 11-dim supergravity! The non-perturbative region of Type IIa strings, which was previously a forbidden region, was revealed to be governed by 11-dim supergravity theory, with one dimension curled up.

    At this point, I remember, many physicists — myself included — were rubbing our eyes, not believing what we were seeing.
    I remember saying to myself, "But that’s impossible!"

All of a sudden, we realized that perhaps the real "home" of string theory was not 10 dimensions, but possibly 11, and that the theory wasn’t fundamentally a string theory at all! This revived tremendous interest in 11 dimensional theories and p-branes. Lurking in the 11th dimension was an entirely new theory which could reduce down to 11-dim supergravity as well as 10-dim string theory and p-brane theory.
Vafa added a strange twist to this when he introduced yet another mega-theory, this time a 12-dimensional geometric, non-perturbative version of stringtheory called F-theory (F for "father") which explains the self-duality of the IIB string. Though F-theory is formally a 12-dimensional theory, the only way to obtain an acceptable background is to compactify it on a two-torus. By doing so, one obtains type IIB superstring theory in 10 dimensions.
So, will the final theory have 10, 11, or 12 dimensions?
Schwarz feels that the final version of M-theory may not even have any fixed number of dimensions. He feels that the true theory may be independent of any dimensionality of space-time, and that 11 dimensions only emerge once we try to solve it.
Townsend seems to agree, saying "the whole notion of dimensionality is an approximate one that only emerges in some semiclassical context".
Witten believes
  • that the underlying degrees of freedom of M-theory are yet to be discovered,
  • that we are on the right track,
  • but will need a few more "revolutions" to finally solve the theory. So far (2013) there are no signs yet that the next one will occur very soon.

Source: Michio Kaku, see also Dieter Lüst, section 1.3.2

Kaku says "F-Theorie is rather strange: it has two time coordinates, not one, and actually violates 12 dimensional relativity".
This contradicts Wikipedia, where we read "F-theory has metric signature (11,1). It is not a two-time theory of physics".
Who is right?


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