Traceless metric in string spectrum

[haushofer]

Hi,

I've got a (confusing) question about string theory.

Analyzing the massless modes of the closed string gives me three fields, which correspond to the fact that reps of the group SO(D) (or SO(D-1,1) ) can be breaken apart into three irreps:

*A symmetric traceless part
*An antisymmetric part
* A trace part

The first is our graviton. But, for instance, in General relativity the metric is not traceless with respect to the Minkowski metric! So what's going on?

I know that in a lightcone analysis of the linearized Einstein equations in D dimensions you can show that the physical degrees of freedom are in the traceless symmetric (D-2)x(D-2) part of the metric. Does this have to do with my question? Or am I mixing up things now?

Thanks in advance! :)

 

[haushofer]

Ofcourse, the trace in the SO(D-1,1) group comes from the invariant symbol of that group, but this is the same object you use in the linearized theory to take traces; in the dynamical theory, like GR, it wouldn't make sense to take this trace wrt the Minkowski metric of the dynamical metric.

 

[suprised]

*A symmetric traceless part
*An antisymmetric part
* A trace part

The first is our graviton. But, for instance, in General relativity the metric is not traceless with respect to the Minkowski metric! So what's going on?

The trace part is called the dilaton.

 

[haushofer]

The trace part is called the dilaton.

Yes, I know, but in the rep of the Lorentzgroup the graviton is traceless. So what does this mean in the gravity theory? What kind of trace are we talking about?

And a related question: how do i figure out the spin of the 2-form?

 

[haushofer]

Does it only make sense to talk about what spin the particles have corresponding to the graviton, the Kalb-Ramond form and the dilaton after compactification?

In 4 dimensions I know how to figure out spins of particles; just build up all irreps of SO(3,1) by using the fact that its Lie algebra is isomorphic to two SU(2)'s; the spin of a vector field, which is the irrep (1/2,1/2) then becomes for instance 1/2+1/2= 1.

But if people say that the G in the massless sector of the closed strings has spin 2, in flat spacetime, what do they exactly mean by that? "It has spin two after compactifying it such that we remain with Minkowski spacetime"?

 

[haushofer]

No-one? ;(

 

[haushofer]

For those who are interested: it is explained in Zwiebach's book, chapter 13.3, in lightcone quantization.

 

[dodelson]

The graviton isn't the metric itself; its the deviation of the metric from the background Minkowski metric in linearized gravity, i.e. g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}.

We can always choose a gauge in which h_{\mu\nu} is traceless, the so-called transverse traceless gauge.

Cheers,
Matt

 

[haushofer]

Thanks, I was indeed mixing up the metric and perturbations of the metric! I find this double role of the graviton very curious in string theory, and I'm still not sure what to think about it. So if anyone has some nice references, besides the usual GSW, Polchinski etc. which addresses this double role, I would be very interested!

 

[atyy]

There is a discussion in section 1.6 of http://www.ift.uam.es/paginaspersonales/angeluranga/lect2.pdf , which also references Polchinski section 3.7.

 

[haushofer]

Great, I'll take a look at it tomorrow morning, thanks!