The String Coffee Table

The exchange of $n$ indistinguishable particles in $R^d$ for $d>2$ is governed simply by the symmetric group $S_d$. This simple fact gives rise to bose and fermi statistics of point particles.

But actually, the group that is at work in the background is not quite the symmetric group itself, but really a special case of the motion group

for the case $S=R^d$ and $\Sigma$ the disjoint union of $n$ points in $S$.

In general, the motion group $\mathrm{Mo}(S,\Sigma )$ is, roughly, the group of operations which take a submanifold $\Sigma \subset S$, move it around in $S$ and return it (possibly up to diffeomorphism) to its prior position.

Physically, we can think of the elements of $\mathrm{Mo}(S,\Sigma )$ as the essentially different ways to move “$p$-branes” with $p=\dim (\Sigma )$ around in spacetime $S$.

It so happens that for $p=0$ and $d>2$ this motion group is simply $S_n$ (as long as the first fundamental group of $S$ vanishes, at least), which is responsible for the fact that we see bosons and fermions around us.

The special case $d=1$ is uninteresting, but $d=2$ is very interesting. Here the motion group for $n$ point particles is the braid group $B_n$. This contains $S_n$ as a subgroup, but is much larger. Hence indistinguishable particles in 3 dimensions may in principle have statistics different from bose or fermi statistics, that arises as a representation of the braid group. This is indeed what happens in the fractional quantum hall effect. See for instance the introduction of

N. Read & G. Moore
Fractional quantum Hall effect and nonabelian statistics
hep-th/9202001 .

It is natural to wonder how the motion group looks like for $p>0$. For $p=1$ it describes the ways in which $n$ (noninteracting!) strings can be “permuted”. Let’s always assume that $S={ℝ}^d$ for simplicity. Consider two closed strings sitting in ${ℝ}^d$. There are two essentially different ways to exchange their position.

For one, we can move them around each other, as we could if we contracted both to point particles.

But there is potentially one more possibility now. We can also exchange positions by threading one string through the other.

Whenever $d>3$ this does not make any difference. But precisely for $d=3$ it does. In this case the motion group is isomorphic to something called the loop braid group.

Of course, $3=2+1$ and we see that the parameters of the motion group, namely the dimension $p$ of the branes and the dimension $d$ of the ambient space, have both increased by one unit compared to the braid group case for point particles.

(Notice how it is important here that we do not allow these strings to either self-intersect or to intersect with any other string.)

This means that, potentially, there can be something like a fractional quantum hall effect for string-like objects (not necessarily the fundamental strings of string theory, but who knows) whenever the system lives in three spatial dimensions.

From the perspective of “formal high energy physics” one interesting realization of such systems is in terms of $BF$-theory.

$BF$-theory is a gauge theory of $n$-gerbes with connection over $n+3$-dimensional spacetime $S$ (often, and for larger $n$ always, assumed to be trivial in the existing literature). The gerbe connection provides us, locally, on patches $U_i\subset S$, with an $n+1$-form

together with an auxiliary 1-form

for some group $G$ acting on another group $H$ in a certain way (compare section 3.9 of Girelli&Pfeiffer ($\to$) for the local idea and this for the global idea).

On local patches the action for the theory reads

where $F_A$ is the field strength of $A$.

This action is extremized for vanishing field strengths

hence its physical configuration space is that of (certain) flat connections on (certain) flat $n$-gerbes. These flat connections are of course completely characterized by their holonomies over cycles in $S$.

This is interesting in the present context, because the “$p$-branes” we are considering can naturally be taken to be singular sources for the fields $B$ and/or $A$. Therefore we are interested in sourceless $BF$-theory on $S$ with $\Sigma$ (being the loci of our $p$-branes) cut out.

Now, given how the motion group $\mathrm{Mo}(S,\Sigma )$ provides nontrivial diffomorphisms of this space, we see that we necessarily get a representation of this group on the configuration space (of gerbe connections) of our $BF$-theory.

In the above paper by Baez, Wise and Crans this is spelled out in great detail for the familiar case with $n=0$. Here we have point particles coupled to a flat 0-gerbe $\simeq$ fiber bundle with connection. The holonomies of the connection around the cycles where the point particles have been cut out of $S$ characterize the gauge connection. Moving the point particles around provides an action of the braid group (in $d=2$) on this space of connections.

The goal is to generalize this well understood setup to 1-branes in $d=3$-dimensional space.

(The main motivation for the present authors is that for $\dim (S)=2$ $BF$-theory for the gauge group $\mathrm{SO}(2,1)$ on manifolds with $n$ holes cut out describes 3-dimensional quantum gravity coupled to point particle matter, with each hole representing one point particle. The hope is to somehow generalize this to 4-dimensional quantum gravity.)

One would expect that in this generalization to $\dim (S)=3$ a flat 2-form connection $B$ and its surface holonomy around non-contractible cycles obtained by cutting out strings from $S$ plays a major role. The technology for working this out is in principle available, but this is not attempted yet in the above paper.

What is done (in section 6) is an analysis of the action of the loop braid group just on the space of flat 1-form connections $A$.