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Gauge theory in 12 dimensions?

The highest dimensionality in which a non-gravitational supersymmetric effective field theory is understood is 10 - or 9+1 - dimensions. And the theory is nothing else than the 9+1-dimensional Yang-Mills theory.

It is non-renormalizable but it may emerge either as the effective description of non-gravitational parts of the 10-dimensional N=1 supersymmetric string vacua or it can be compactified to 3+1 dimensions or lower to produce extremely important (not only for Matrix theory and AdS/CFT) renormalizable supersymmetric gauge theories.

The reason why 9+1 dimensions is the maximum dimensionality for supersymmetry is rather simple.

Particles may be invariant at most under 1/2 of the supersymmetry. Because the remaining broken supercharges act as ladder operators within the supermultiplet and because we want the spin to be at most 1 (3/2 is already too much because the only consistent spin-3/2 massless fields are gravitinos), there can be at most 4 raising and 4 lowering generators "broken" by a particle - to get us from -1 to +1 by the 1/2-steps that supercharges offer.

However, that means that there can be at most 8 supercharges broken by a particle that preserves 1/2 of supercharges - and at most 16 supercharges in total. The maximum dimensionality where spinor representations with 16 real components exist is 9+1 dimensions. The corresponding theory has a spin-1/2 field with 16 real components (a Majorana-Weyl spinor) all of which turn out to be physical - which is inevitable for the Dirac spinors and their reductions. These 16 fields may be Fourier-transformed into 8 creation and 8 annihilation operators for fermions in the momentum space. That's great because the 8 fermionic polarizations match the 10-2 physical bosonic polarizations of a gauge field in 10 dimensions.

Two times and higher dimensions

That's fine but can't we go slightly above 10 dimensions? Isn't there a more fundamental reality in a dimensionality that slightly exceeds 10? After all, we've seen examples that this must be true to some extent. M-theory provided us with a more fundamental, 11-dimensional platform to discuss type IIA and heterotic E8 x E8 string vacua in 10 dimensions. And F-theory brought us a geometric, 12-dimensional perspective on type IIB vacua with the variable dilaton-axion field.

Itzhak Bars has been studying and promoting theories with two time coordinates for decades. And today, he and his collaborator reveal their construction of a 12-dimensional supersymmetric gauge theory:

Super Yang-Mills theory in 10+2 dimensions, the 2T-physics source for N=4 SYM and M(atrix) theory
As you see, the signature is 10+2. Such a theory has two times. It's clear that the conventional physical interpretation of dynamical theories is impossible if there are two time coordinates. After all, if there were at least two large time coordinates, there would exist a two-dimensional "Euclidean" plane with purely time-like directions.

In such a plane, you could draw a smooth circle. That would inevitably be a closed time-like circle (CTC), something that prevents you to organize physics in terms of initial and final conditions because it allows you to kill your grandfather before he had his first intimate contact with your grandmother, thus making your personal existence logically inconsistent. ;-)

To make things simpler, there would be no universal "after" and "before": the future light cones and the past light cones would be connected through the second time coordinate.

I think that Itzhak Bars is aware of this issue. His two-time constructions have always been less "physical" in this sense: he always meant the higher-dimensional two-time starting point to be a more symmetric "master theory" that sheds more light on the unity between the lower-dimensional theories and that makes many of their symmetries and properties manifest.

In my opinion, this is a beautiful template for a very serious and potentially important research. Except that I think that in the examples that Bars has studied, such a program - or working hypothesis - hasn't been successful yet.

In the new paper, the authors construct a 12-dimensional theory that may be reduced to the 9+1-dimensional gauge theory and others. But the actual reduction is not the familiar dimensional reduction. After all, they don't find any Kaluza-Klein modes etc. That means that even the counting of the physical degrees of freedom in the higher-dimensional, 10+2-dimensional theory has to be altered.

If you study their Lagrangian (1.1) for a while, you will see that they actually believe that there could be a gravitational theory in 12 large dimensions. That's an even stronger statement but let us follow their newest paper and focus on the non-gravitational truncation.

This theory has a gauge field, the "F_{mu nu}", with the familiar "F mu nu F mu nu term". It's multiplied by a power of the "dilaton" Omega (taken from the gravitational sector that we try to decouple now) and by other things. All the terms are also multiplied by a volume factor that depends on W, another scalar field, and by some undetermined extra factor K etc.

The overall normalization of the action looks truly strange with so many scalar factors - a hint that the 12-dimensional geometry is not "real".

But what is the key novelty is the terms for the fermions. Normally, in 9+1 dimensions or lower, you have a kinetic term and the coupling to the gauge field that combine into a gauge-invariant combination. However, in their case, they make this gauge-invariant term more complicated, namely
lambda Vslash D lambda.
The covariant derivative is also multiplied by Vslash which is the vector V contracted with the gamma matrices. And the vector V is the gradient of the scalar W (over two). So if you were imagining the scalar field W to be constant so far, well, then you see that the fermion terms would vanish.

In general, you don't want this term to identically vanish. That means that the non-constancy of W has to be crucial for the theory to be physically non-trivial. I feel that the variations of the fields in the direction along the gradient of W (and one more?) are kind of (identically) unphysical which makes the whole construction less meaningful and important.

However, there is some similarity with the Bagger-Lambert-Gustavsson (BLG) theory in 3 dimensions that sparked the "membrane minirevolution" but that was more or less found out to be a new fancy description of the 2+1-dimensional theory. It made the SO(8) symmetry of its conformal limit - which is so important for the Matrix theory of type IIB string theory - manifest - but otherwise you could a posteriori dismiss the whole BLG theory as a new notation for old physics. However, the BLG theory did lead to a class of new theories - the ABJM theories - that were directly inspired by the BLG construction.

If something like that were possible in the Bars-Kuo construction, it would already deserve your attention. However, I am not sure whether it's the case. So far, the extra two dimensions you may find in the 10+2-dimensional theory look very artificial to me. They just don't seem "real" in any way. They don't seem to "explain" any known property of the gauge theories that was previously unexplained (or unexplained in a manifest way).

We may see later.

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reader Brian G Valentine said...

I have long believed that Bars is correct on this, and for the early universe at any rate his construction provides a means to explain how events can apparently "precede" others (as viewed backward in E-1 time) and events are not casually related by "cause and effect" in a world of orthogonal time coordinates.

There is an interesting analogy of this 12-dimensional gauge theory and H Weyl's construction of the electromagnetic field from the unspecified six components of the contracted Riemann curvature tensor of empty spacetime

reader Alejandro Rivero said...

I wait for the invited post of prof. Bars, I hope he will find time to write it.

12 dimensions is also an intriguing topic, to me, in general Kaluza Klein theory. As it is well known, 11 dimensions are enough to produce SU(3)xSU(2)xU(1), and besides they naturally split to 7+4. But Salam et al. noticed that the charges are not right. One could speculate that the cause is the lack of the non-gauge U(1) of Weinberg model, the B-L charge, and that then one needs a jump to 12 dimensions but in some peculiar way to avoid gauging this group.
Also, in 12 dimensions one can produce the symmetry SU(3)xSU(2)xSU(2), and again one could hope for the extra dimension to be peculiar enough to force the breaking of SU(2)xSU(2) down to SU(2)xU(1). A indefinitely small extra dimension should be satisfactory, because then the scale of the breaking would be Planck scale.

reader dimension10 said...

This is not wrong. EM is the curvature of the U(1) bundle, Weak force is the curvature of the SU(2) bundle, Strong force is the "curvature" of the SU(3) bundle, gravity is the curvature of (3,1) dimensional spacetime. That's why Kaluza - Klein theory is 5-dimensional; (Gravity + EM; 4 + 1), ....

reader dimension10 said...

Supergravity is the expansion of KK theory, but it is the low - energy, classical limit of the more fundamental theory, string theory.