Saturday, August 25, 2007

Gerard 't Hooft, QCD, and bosonic string theory

Prof Gerard 't Hooft has a new interesting preprint about massive gravity, QCD, and bosonic string theory.

The Brout-Englert-Higgs mechanism for gravity that he talks about generalizes the usual symmetry breaking in gauge theories. However, you need four scalars, not just one, and their vevs are spacetime-dependent and equal to x,y,z,t, respectively. This induces additional masses and a cosmological constant.

't Hooft mentions that this setup could be relevant for cosmology but his real interest is elsewhere: in string-theoretical models of QCD.

He proposes bosonic string theory compactified on a 22-manifold to be a dual description of QCD. It is not really holographic but it could be useful anyway. The symmetry breaking mentioned above is supposed to kill the tachyons and massless scalars and make spin 2 glueballs massive.

Could the scalars with vevs x,y,z,t be actually embedded in string theory?

Well, I think that if one uses the correct rules of string theory, the answer is No. No scalars in string theory are just dumb fields added for fun. In fact, all fields in string theory play some function. They always have some interpretation - usually a geometric one - and their configuration space is moreover compact. What I talk about are geometric moduli of the compact manifolds, Wilson lines, embeddings of D-branes, and others.

In 't Hooft's setup, one needs a non-compact configuration space for the four scalars: that's the first contradiction. Moreover, if you want to preserve translational symmetry, the shift of these scalars by a constant must be a global symmetry. But there are no continuous global symmetries in quantum gravity or string theory. So it can't quite work, I think.

Gauge-theoretical beta-function from string theory

The calculation of the beta-function in non-Abelian gauge theories is a subtle thing. Gerard 't Hooft has been one of the first people who was able to do it. A key result is proportional to -11/3 and comes from a careful consideration of the gauge field and the friendly ghosts.

Is there a more conceptual way to get the number -11/3? Of course, there is once you know string theory. The dilaton around the D3-brane can be shown to be constant. That implies that the low-energy effective theory must have a vanishing beta-function. In fact, it must be conformal. That means that the gauge sector contributes as much as the 3 complex scalars, their 3 Dirac superpartners, plus one Dirac superpartner of the gauge field. We have 3 complex scalars and 8 Weyl fields. Each gives the same contribution. 3+8=11 and it must be canceled by -11 contributions of the gauge field. You may forget about the Faddeev-Popov ghosts. It is much safer that you will avoid mistakes in the calculation in this paragraph.

Beta-function from bosonic string theory

This calculation was arguably too complicated because it included scalars as well as fermions. Is there a simpler, purely bosonic calculation? Yes, there is. I've known this trick from 2000 when we talked about it with Josh Gray in Santa Cruz but later I learned that Gerard 't Hooft was aware of it, too.

Take D3-branes in 26-dimensional bosonic string theory. Again, there must exist an argument why the one-loop beta-function must be zero even though I can't make the argument too rigorous (the background AdS_5 x S^{21} is not an exact solution of bosonic string theory). The fields that contribute to the running in the effective field theory are the gauge field and 26-4=22 real scalars. They're equivalent to 11 complex scalars and again, the gauge field must contribute -11 units to the beta-function in order to make the sum vanish.

It seems that whenever there is a number in a field theory whose origin looks extremely strange, there exists an alternative and much more illuminating calculation of this number based on string theory.

And that's the memo.

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