The String Coffee Table

So the goal is to derive from the action

equations of motion for the expectation values

of observables $F(A,\Psi )$ and to show that for $F$ a Wilson line of $A$ around an abstract loop (in the beginning there is no spacetime in which this loop is embedded in this model, this spacetime arises as a derived concept!) these equations of motion describe propagation of relativistic strings as well as their splitting and joining interactions.

I’ll closely follow the papers by Kawai, Tsuchiya et al. but my goal shall be to focus on the cruciual steps that illuminate how strings and their propagation and interaction arises from calculation with matrices only.

To do so, my first stel shall be to ignore the fermionic contribution and concentrate on the bosonic part of the model, i.e. to consider the action

In order to get anything like a Wilson loop from this action consider the abstract circle ${\mathrm{S}}^1$ and any function

on this circle. To any such function $k$ we may associate an observable $w(k)$ on the space of large constant matrices $A_{\mu }$ by writing

As for now the $k^{\mu }(\sigma )$ have no physical interpretation and are just auxiliary functions that label all kinds of observables in the model. But we will see that $k$ acquires the meaning of the momentum density on a string’s worldsheet - momentum, that is, in a space which is nothing but the Fourier dual of the space of $k$ themselves.

In order to make things a little simpler lets consider a regularized version of these Wilson loop obervables. Introduce a large integer

set

and approximate

The point is that this will allow us to conveniently identify matrix multiplication with differentiation with respect to $k$ by means of the formula

This way the matrices $A_{\mu }$ become related to the ‘spacetime’ associated with the $k^{\mu }$.

Next we need to choose some sort of Schwinger-Dyson equation for the Matrix Model, re-express it but replacing matrix multiplication by differentiation and read off the sought-after equation of motion for the Wilson loops.

It turns out that a useful Schwinger-Dyson equation to consider is

Here we write

where $t^{\alpha }$ are the generators of U(N) normalized so as to satisfy

Evaluating the above expression using the product rule gives us three different kinds of terms, depending on whether thr derivative acts on the action, the first Wilson loop or the remaining Wilson loops:

I have called these terms $(f)$, $(s)$ and $(j)$ because they will be seen to describe the free propagation, the splitting and the joining of the strings desribed by the various $k^i$.

This works as follows:

The $(f)$ term yields

By replacing matrices by derivatives these can be taken out of the integral and become differential operators on the space of multi-loops. In fact, recalling that in the end $k^{\mu }$ will be identified with a momentum density on the string define the operator

on the space of (expectation values of) Wilson loops. The the above can be rewritten as

It’s nice how the big old matrix term condenses to this concise form, but this is not quite the free equation of motion for the string that we expected to see.

No problem. The reason is that the splitting term $(s)$ contains a contributions which doesn’t describe any splitting at all and has hence to be included in the free piece $(f)$:

One finds

It is fun to see how the partial derivative with respect to $A_{\mu }$ splits the string associated with $k^1$ in half and how the contraction of the $t^{\alpha }$ glues the remaining open ends so as to form two new closed strings. However in one of these processes a piece of string of vanishing length is split off and produces not another string but a kinematical term proportional to $k\cdot k$. Taking this term together with previously found $(f)$ gives

Now, with a little tweaking of $N,ϵ$ and $g$, taking careful limits, etc., this begins to look like the Hamiltonian constraint of the free superstring! (The limits are subtle and I think Aoki et al. dont have them under rigorous control, either. But they give a lot of arguments for why the advertised string field theory should be obtained in the correctly taken limit. Furthermore, my prefactors might contain errors. They are not precisely what I see in the literature, which, on the other hand, does not seem to be fully consistent, either. I need to check that.)

Good, finally one can convince oneself that the last term, $(j)$, really does describe the joining of two strings. One gets:

Here now the partial derivative with respect to $A_{\mu }$ cuts open one of the strings and the contraction over $t^{\alpha }$ glues the ends with the open ends of the first string.

The claim is that the three terms together give proper closed string field theory in an appropriate limit $ϵ\to 0$,. $N\to \mathrm{\infty }$. I very much enjoy the train of thoughts that leads to this result.