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Strings, loops, knots, and gauge fields.

The natural question is: Which gauge theory has Wilson loops (or more generally: network states) that behave

**exactly**like the fundamental strings of string theory. Surprisingly, an answer has been proposed already quite a while ago in the 90s by Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya: A review of their work is

hep-th/9908038 IIB Matrix Model .

I am trying to discuss this idea in the newsgoup sci.physics.research. But maybe here in this forum it will be interesting, too. This is what I originally wrote:

<quote from s.p.r (here is the full thread) >

"John Baez" <baez@galaxy.ucr.edu> schrieb I am Newsbeitrag

news:bnl7rt$d81$1@glue.ucr.edu...

> In article <Pine.LNX.4.31.0309270749590.32151-100000@feynman.harvard.edu>,

> Lubos Motl <motl@feynman.harvard.edu> wrote:

> >Once an infinite number of incorrect terms is removed and

> >the theory [LQG] is made equivalent to string theory, then - of course -

one

> >will obtain a consistent theory of gravity.

>

> Yeah, yeah. Even if you're right and we're all mixed up

> and need to change the model drastically and when we do it

> turns into something like string theory, we'll still have

> something interesting, namely a manifestly background-free

> formulation of string theory as a state sum model. I would

> not mind this at all.

I find it really interesting that the following looks like a way to "change

the model" and indeed turn it into string theory:

So let's suppose we want to formulate our fundamental theory in terms of

functional states on a space of gauge connections A_mu taking values in some

Lie algebra. We want to be really background free. In LQG one does away with

background _fields_ on spacetime, but one still does need a manifold to set

up the theory. Let's do away with the assumption of a

(topological/differentiable) manifold, too.

Without a manifold it makes no longer sense to have the A_mu(x) be functions

of coordinates. Therefore let's assume, being very naive, that the

connection is _independent_ of any coordinates.

As we learn from LQG, functions (states) on the space of connections

are spanned by generalized Wilson lines ("networks", graphs - I'll avoid the

word "spin" for the moment). In ordinary LQG these are embedded into a

manifold. But since we have just done away with this manifold we now have to

evaluate our connections on abstract networks that are not embedded into any

a priori structure. The natural way to do that is to equip the network that

comes with a given state (function on the space of connections) with a

D-tuple valued (piecewise defined) 1-form k and compute the holonomy of

Sum_mu k^mu A_mu

along the edges of the network, intertwining at the vertices as desired and

finally tracing over the result.

In other words, in this manifold-independent formulation of "Ashtekar

geometry" a network state is given not just by a coloring of edges {e} by

representations {r} (and coloring of vertices by intertwiners {i}) but

instead by a coloring by representations _and_ D-tuple valued 1-forms k.

(The information that was previously contained in the coordinates of a given

edge has now moved into the extra piece of data k.) So a basis of states

should now be of the form

{ psi_{e,r,i,k} }

where each element psi_{e,r,i,k} is associated with an abstract

combinatorial graph e colored by r,i, and k.

This basis spans the kinematical Hilbert space. Now we need dynamics.

Personally I feel that in LQG kinematics is very beautiful but that as soon

as the ordinary dynamics enters the game things become rather awkward. A

fundamental theory is not supposed to look awkward, so let's slighly modify

the ordinary LQG dynamics. Instead of using the action of B^F theory on the

space of connections A we'd rather use the simplest action quadratic in the

curvature:

S = Tr F^2 = Tr [A,A]^2 .

In other words, the slight modification of LQG that I am proposing here is a

theory whose configurations are given by _constant_ gauge connections A and

whose observables (correlation functions) are

<psi_{e1,r1,i1,k1} psi_{e2,r2,i2,k2}...psi_{en,rn,in,kn}>

=

int DA psi_{e1,r1,i1,k1}(A) ...psi_{en,rn,in,kn}(A) exp(-S(A))

with S and psi_{...} as defined above. The two major modifacations as

compared to ordinary LQG are the absence of the manifold background and the

switching from an action linear in the curvature to the simplest one

quadratic in the curvature.

The point of all this is the following: In 1996 the authors N. Ishibashi, H.

Kawai, Y. Kitazawa and A. Tsuchiya have proved for us (see hep-th/9908038

and references given there) that

if we identify Wilson lines (network edges) in the above theory with

fundamental strings

and if we use U(N>>>1) as the gauge group then the above action for the

connection induces on these Wilson line, which are now also regarded as

functionals (states) on the configuration space of the fundamental string,

the equations of motion of string field theory!

Voila.

I spent some time at the "Strings meet loops" symposium trying to find LQG

people who would find this as interesting as I do. That's because my

impression is that maybe the true value of this IKKT model is not properly

appreciated in the string community, which might have to do with its radical

background independence. Lubos mentioned that timelike T-duality which does

away with time (!) looks suspicious. But maybe it is just what we need, I

wonder. In any case, when looked at it from the proper perspective (as I

have tried to demonstrate above) the IKKT model looks much more like LQG

than like string theory. Of course, looked at it from another perspective it

completely looks like string theory. Great.

Luckily, I found several very friendly and open minded LQGists who did find

this interesting. I am looking forward to hearing what they come up with

when studying this in detail.

<end quote>