# A Does String Theory incorporate Kaluza Klein? How?

Tags:
1. May 4, 2017

### arivero

From time to time, I point to string theoretists that they should have considered more seriously to use Kaluza-Klein theory and they invariably answer me "we do", and move forward. So I am starting to thing that perhaps I am wrong and I have missed some developing of the theory the the XXIth century so that now they actually are well beyond the torus compactification of last century textbooks. So the question for this thread: do they use the KK ansatz in some way hidden in the modern notation, with D-Branes, fluxes and all that stuff?

To be sure, the Kaluza Klein Ansatz was fixed by Witten 1981 (you can find the article in page 30 of "the World in 11 dimensions" or other recopilations, but some of them seem not to be online). Basically it says the part of the metric between the compacted $\phi^k$ and the macroscopic $x^\alpha$ dimensions has the form:
$$g_{\mu i}=\sum_a A^a_\mu(x^\alpha) K^a_i(\phi^k)$$with $K^a_i$ the Killing vectors associated to the symmetries of the compact manifold.

Then $A^a_\mu$ emerge as gauge fields, and this is the thing one expects to see down in the low energy theory. Of course in string theory a lot more fields can happen, from the gauge fields already present in 10 or 11 dimensions. But these ones from the metric, or an explanation of how do they dissappear, are the ones I miss in string theory lectures... are they just hidden in the notation, somehow?

2. May 4, 2017

### arivero

A thing that I could be missing is that somehow the symmetry associated to Killing vectors in the Torus T6 is enhanced when stringers do the orbifold/orientifold/whateverfold operation, in ways similar for instance to quotienting a T2 by a discrete symmetry to map it into a sphere S2. If such sort of enhancements happen, I have never read of them, at least not explicitly.

3. May 5, 2017

### mitchell porter

Becker, Becker & Schwarz write: "In Calabi-Yau compactifications no massless vector fields are generated from the metric since [first Betti number] b1 = 0. A closely related fact is that Calabi-Yau three-folds have no continuous isometry groups."

I would like to dig deeper into this topic (though maybe not today...) for a variety of reasons. Some of the manifolds from the high point of Kaluza-Klein unification are still studied in string theory (though not in string phenomenology because of the chiral fermion problem); what has been learned? Also, it would be interesting to compare how the 4d gauge bosons are created in those vacua (from the stringy graviton), with how they are created in ordinary string phenomenology. They seem to be quite different; but is there some higher perspective (like generalized geometry) in which they are variations on the same theme?

4. May 5, 2017

### arivero

Yep, now I check it, also the textbook of Ibañez et al mentions this fact. Not sure how/if it extends to other kind of compactifications, nor if there is some twist to it.