Microscopic relation between loop space connections and 2-group holonomies

Posted by urs

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Currently I am discussing aspects of surface holonomy in terms of loop space connections with Jens Fjelstad. One aspect that keeps puzzling us is the following:

Introduction:

Recall that a curve in loop space corresponds to a, possibly degenerate, surface in target space and that taking the holonomy of a connection on loop space over this curve hence associates a group element to that surface which is addressed as surface holonomy. But in general the group element obtained this way is not unique.

Some of this non-uniqueness can be understood and removed easily: As I noted in hep-th/0407122 from various points of view one is lead to restrict attention to connections on loop space which are flat. This condition suffices to ensure that homotopy equivalent curves in loop space with the same endpoints assign the same surface holonomy.

Moreover, it turns out that no generality is lost by restriction to flat connections: A large and ‘natural’ set of connections $𝒜$ on loop space is of the form

(1)

𝒜 = \int_{0}^{2 π} d σ W_{A} (σ) B_{μ ν} (X (σ)) W_{A}^{- 1} (σ) X^{' μ} (σ) {dX}^{ν} (σ)

with $A$ locally a 1-form and $B$ locally a 2-form on target space $ℳ$ , $X : (0,2 π) \to ℳ$ the given loop and $W_{A}$ the holonomy of $A$ along that loop.

In the above paper I showed that for gauge transformations on loop space to preserve the general form of this connection we need to have

(2)

B = - F_{A},

where $F_{A}$ is the field strength of $A$ and that this does imply that $𝒜$ is flat. But in hep-th/0309173 it was discussed (using 2-group technology) that $B = - F_{A}$ is also the necessary condition for a well defined surface holonomy. Hence nothing is lost by restricting to flat connections on loop space of this form.

But in the loop space framework a connection $𝒜$ of the above form with $B = - F_{A}$ alone does not seem to ensure a well-defined surface holonomy. Apparently there must be further consistency conditions.

We have tried to discuss here these problems for the case of tori recently. But let’s concentrate on topological spheres for a moment, where things are much simpler:

For the sphere one can convince oneself that all closed curves in (unbased, oriented, parameterized) loop space which map (without overlap) to a given sphere in $ℳ$ are continuously deformable into each other and hence do associate unique surface holonomy. All these closed curves correspond to a foliation of the sphere into circles which share a common base point.

But there are also open curves in loop space which map to that given sphere. These must all start and end at a constant loop, too, but not at the same one.

It is easy to see that these open curves in loop space cannot possibly associate a unique surface holonomy: By making a gauge transformation on loop space the holonomy of the open curve in loop space can be given any value.

So the question that Jens Fjelstad and I are thinking about is:

Question:

Is there a fix to the above problem which makes the surface holonomies associated with open curves in loop space well defined, or, if not, do we have a good reason not to compute surface holonomy using open curves in loop space?

I currently tend to think that the latter is true. For the following reason:

Attempt of an answer:

Since the 2-group scheme does associate a unique surface holonomy in every case, it must be that the computation of surface holonomy using closed curves in loop space which correspond to spheres can be mapped to a computation along the lines of section 2.5 in hep-th/0309173, while apparently for open curves in loop space this is not possible.

If this were true it would explain from the 2-group perspective why we should not want to compute surface holonomy using open curves in loop space - it would just not correspond to an honest 2-group calculation.

Therefore one should try to translate the computation of the holonomy of the loop space connection $𝒜$ given above to the computations used in 2-group theory and see under which conditions the two can agree. This is what I am going to do here. I am calling this the microscopic relation between the two approaches because the idea is to discretize both computations and see how the adding up of the discrete contributions in both cases compare. It turns out that apparently indeed only for closed curves in loop space which run over loops that all share a common point does the computation of the holonomy of $𝒜$ amount locally (and hence globally) to that of the corresponding 2-group computation.

This works as follows:

The details:

First recall the laws of 2-group computations:

I) review of 2-group computations

Recall that the computation of surface holonomy using 2-group technology works as follows:

The surface whose holonomy is to be computed is covered with a graph (whose average mesh size vanishes as we take the continuum limit) whose edges are labeled by group elements $g \in G$ and whose faces are labeled by group elements $h \in H$ and by a sort of orientation defined by giving two vertices touching that surface.

The labeling has to satisfy some obvious consistency relations together with the not-so-obvious but extremely crucial one which says that

(3)

h = g_{2} \cdot g_{1}^{- 1},

where $g_{1}$ (called the source) is the label of the edge running from one of these two vertices to the other around the surface labeled by $h$ one way and $g_{2}$ (called the target) the label of the edge running the other way.

(For simplicity and without lack of generality I will asumme that $G = H$ and that $g$ acts on $H$ by the adjoint action.)

Recalling the lattice definition of $F_{A}$ it is clear that this relation is nothing but the lattice version of $B = - F_{A}$ .

For computing the surface holonomy one picks any two vertices of the graph, picks any edge running between them and then ‘walks over the surfaces’ starting at this edge and again ending at this edge, while using the two elementary rules of 2-group multiplication to multiply up the contributions from the various faces:

Horizontal multiplication of two surfaces $(g_{1}, h, g_{2})$ and $(g_{1}^{'}, h^{'}, g_{2}^{'})$ which share a common vertex produces the total surface with label

(4)

(g_{1} g_{1}^{'}, h (g_{1} h^{'} g_{1}^{- 1}), g_{2} g_{2}^{'}) .

Essentially everything is multiplid in the naively obvious way, with the only difference that $h^{'}$ is adjoined by $g_{1}$ before multiplication with $h$ . This can be understood simply as a parallel transport of $h^{'}$ back to the source vertex of $h$ , where the two may be compared. One already sees that this precisely the same mechanism that is at work in the definition of the loop space connection $𝒜$ above.

Vertical multiplication of two surfaces with labeling as above which share a common edge gives

(5)

(g_{1}, h^{'} h, g_{2}^{'}) .

So the intermediate edge contribution cancels and the surface holonomy is simply multiplied.

Now let us see how this can be recovered from loop space holonomies:

II) relation to loop space computations

The connection $𝒜$ above involves the integral of the 2-form $B$ around the full loop.

This suggests to consider a long chain of horizontal composition of small surfaces $(g_{1} (σ), h (σ), g_{2} (σ))$ with $σ = n ϵ$ an integer multiple of some small parameter which we’ll send to 0 in the continuum limit.

The horizontal product of all these surfaces gives, by rule 1) above

(6)

h (0) \cdot (g_{1} (0) h (ϵ) g_{1}^{- 1} (0)) \cdot (g_{1} (0) g_{1} (ϵ) h (2 ϵ) g_{1}^{- 1} (ϵ) g_{1}^{- 1} (0)) \dots

Due to the smallness of these surfaces we expand as usual

(7)

g_{1} (σ) \approx 1 + ϵ A_{σ} (σ)

(8)

h (σ) = 1 \approx ϵ^{2} B_{σ τ} .

Here $σ$ is supposed to be an index roughly parallel to that chain of surfaces, while $τ$ is orthogonal to it (but still lying in the surface, of course).

When this is inserted into the above expression for the long horizontal product one obtains

(9)

\dots \propto 1 + ϵ \int d σ W_{A} (σ) B_{σ τ} W_{A}^{- 1} (σ) + 𝒪 (ϵ^{2}) .

This is precisely the expression appearing in the loop space connection $𝒜$ above.

Not too surprising, but maybe interesting, the integral over $σ$ which enters the above definition of the loop space connection is hence nothing but the computation of the first order term in a continuum horizontal 2-group product.

More precisely, the result is the surface label of the full $ϵ$ -thin but macroscopically long strip formed by all the small surfaces that were horizontally multiplied.

Now comes, finally, the crucial point: When we compute the holonomy of the loop space connection we actually compute

(10)

(1 + ϵ \int d σ W_{A} (σ) B_{σ τ} W_{A}^{- 1} (σ)) (τ = 0) (1 + ϵ \int d σ W_{A} (σ) B_{σ τ} W_{A}^{- 1} (σ)) (τ = ϵ) \dots .

But this is now seen to be nothing but the vertical 2-group product of all these strips at different $τ$ . Not too surprising either - but the point is that according to the above mentioned rules of 2-groups this is only defined when these strips completely share one total edge, including the vertex where the strip begins and ends.

Anyone wondering what I really mean should please have a look at equation (2.13) in hep-th/0309173, which makes it quite clear.

The conclusion is that the computation of the holonomy of a loop space connection of the form given at the very beginning of this text corresponds to a 2-group surface holonomy computation if and only if the loops that are integrated over all share a common point, namely that common vertex of the lattice approximation!

Phrased differently, when $𝒜$ is integrated over a curve in loop space which contains loops that do not share a common point, then the result cannot correspond to anything computed using 2-group theory. But since the rules of 2-groups are precisely those which guarantee a consistent surface holonomy, it is no wonder that open curves in loop space, which cannot satisfy this condition, do not yield a consistent surface holonomy.

Discussion:

It seems to me that this is telling us that when computing surface holonomy using loop space technology we must for consistency reasons restrict to loops in loop space whose elements are loops that all share a common point. Otherwise the loop space holonomy is not computing anything that can reasonably be interpreted as surface holonomy in target space.

Of course we knew this before, using the very simple argument involving gauge transformations on loop space which I mentioned before, but I find it illuminating to see in ‘microscopic’ detail how this hangs together with local 2-group computations.

On the other hand, in the string theory context and the boundary state formalism which I originally derived the ideas about nonabelian loop space connections in, this raises further quetsions:

Everything seems fine as long as one is considering tree-level amplitudes for open strings. Here the boundary state formalism is telling us to compute sphere diagrams, and everything mentioned above worked nicely for spheres.

But it does not work nicely for higher genus surfaces. There we do not even have foliations into loops which all share a common basepoint.

Probably to resolve this one has to decompose the surface into patches which each seperately allow foliation with common basepoint. But the details of this are not clear to me at all at the moment.

Posted at August 18, 2004 3:50 PM UTC

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August 18, 2004