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Re: Meaning of dilaton field




On Tue, 1 Apr 2003 21:14:06 +0000 (UTC), Urs Schreiber <Urs.Schreiber@uni-essen.de> wrote:

> There is also a nice relation of the dilaton to membranes and
> 11-dimensional physics:
> 
> It turns out that one can think of the 10-dimensional spacetime that
> strings live in as 11-dimensional spacetime with one dimension
> compactified
> on a circle. The component of the 11 dim metric that measures the
> circumference of this circle is, form the 10 dim persepctive, the
> dilaton field, a phenomenon familiar from any KK theory. Furthermore,
> strings are really membranes wrapped around this 11th dimension. Hence
> the dilaton measures the "thickness" of strings along the 11th
> dimension, so to say. 
> 
> I'd guess (but I don't really know) that the relation between the size
> of the 11th dimension and the string coupling can therefore be
> understood from the fact that the action of membranes is essentially
> proportional to their worldvolume, so that membranes wrapped on a large
> S^1 (having a larger volume) couple more strongly than membranes on a
> small S^1. (Hm, is this in accordance with the sign of this effect?
> Dunno.)

No. This argument does NOT tell you anything about the coupling of the
string! What you derived is the tension. In 11D There is no
dimensionless coupling there is only the Planck length and membranes
have unit tension. If you wrap them on the circle ("double dimensional
reduction") you get strings whose mass is given by the length of the
string times the circumference of the circle (in Planck units). So
this tells you what alpha' is in terms of R_11 and l_P. As there are
no interacting strings (the string just sits there) it does not tell
you anything about the coupling.

You get a handle on the coupling by considering D0-branes in type IIA
string theory. As for any D-branes the mass (or tension resp.) is
given by the appropriate power of alpha' to get dimensions right
divided by the string coupling (this follows from considering the
string diagram in which a closes string is exchanged between two
D-branes: Using your nice description work out the power of g that
comes with it and then realize that the closed string contains a
graviton, so you are computing graviton exchange. But the coupling of
gravitons is given by the mass, so this amplitude should scale with
the mass of the D-brane sqared (one for emition one for absorbtion)). 

Then you have to use that D0-branes are KK-modes of gravity from the
11D perspective and use your standard formula for masses of KK-modes,
namely that they are inverse proportional to the radius. Putting
things together you find that the radius is proportional to the string
coupling.

Robert

-- 
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Robert C. Helling     Department of Applied Mathematics and Theoretical Physics
                      University of Cambridge
print "Just another   Phone: +44/1223/766870
    stupid .sig\n";   http://www.aei-potsdam.mpg.de/~helling