The String Coffee Table

(The LaTeXified version of the following text appears as section 3.4 in this pdf.)

Some flat connections on loop space were investigated in the context of integrable systems in [18]. The authors of that paper used the same general form (3.15) of the connection on loop space that dropped out from deformation theory in our approach. They then demanded that the $A$-curvature vanishes, $F_A = 0$, and checked that for this special case an $A$-covariantly constant $B$, as well as a an $A$-closed $B$ which furthermore takes values in an abelian ideal, is sufficient to flatten the connection $\oint_A (B)$. In the notation used here, this can bee seen as follows:

The curvature of the general connection (3.15) is

For vanishing $A$-field strength this reduces to

It is easy to see under which conditions both terms on the right hand side vanish by themselves (while it seems hardly conceivable that there are conditions under which these two terms cancel mutually without each vanishing by themselves), namely the first term vanishes when $d_A B = 0$, while the second term vanishes when the components of the 1-form $\oint_A (B)$ mutually commute.

This is for instance the case when $B$ takes values in an abelian ideal or if $B$ is $A$-covariantly constant and all components of $B$ at a given point commute. These are the two conditions discussed in section 3.2 of [18].

The motivation for setting $F_A = 0$ in [18] was to ensure that surface holonomies induced by integrating the loop space connection over loops in loop space is unique, i.e. independent of the parameterization of the surface, which is encoded in the choice of loop space loop.

For the local considerations that were presented here, however, the requirement for $F_A$ to vanish and for $B$ to be in an abelian ideal is too strong a condition, since it is gauge-equivalent to $A = 0$, which leaves us just with the abelian $B$.

For these reasons we now want to discuss conditions under which the flatness of the connection on loop space alone is a sufficient condition for well defined surface holonomies.

Denote by $\mathcal{L}\mathcal{L}(\mathcal{M})$ the space of parameterized loops in loop space. The holonomy of the loop space connection $\oint_A (B)$ around these loops in loop space is a map

where $G$ is the gauge group. This computes the surface holonomy of the (possibly degenerate) unbounded surface in target space associated with a given loop-loop in $\mathcal{L}\mathcal{L}(\mathcal{M})$.

In general, there are many points in $\mathcal{L}\mathcal{L}(\mathcal{M})$ that map in a bijective way to the same surface in $\mathcal{M}$ and that are related by reparameterization. Only if the function $H$ takes the same value on all these points does the loop space connection $\oint_A (B)$ induce a well-defined surface holonomy in target space.

In the case considered here, where, locally at least, $\mathcal{M} = \R^D$, all loops in $\mathcal{L}(\mathcal{M})$ are contractible. This means that when $\oint_A (B)$ is flat, $H$ maps all of $\mathcal{L}\mathcal{L}(M)$ to the identity element in $G$. In this case the surface holonomy is therefore trivially well defined, since all closed surfaces represented by points on $\mathcal{L}\mathcal{L}(\mathcal{M})$ are assigned the same surface holonomy, $H = 1$.

This is the case that has been found here to arise from boundary state deformations. It implies that unbounded closed string worldsheets do not see the nonabelian background. But worldsheets having a boundary that come from cutting open those surfaces corresponding to points in $\mathcal{L}\mathcal{L}(\mathcal{M})$ do. Is their surface holonomy also well defined?

For the special flat connection with $B = -F_A$ it is. This follows from the fact that this is just the trivial connection, which assigns the unit to everything, but gauge transformed with $\exp\left(\mathbf{W}\right) = \mathrm{P}\exp\left(\int_0^{2\pi}d\sigma\, A_\mu X^{\prime\mu}(\sigma)\right)$. Under a gauge transformation on loop space the surface holonomies of bounded surfaces coming from open curves in $\mathcal{L}(\mathcal{M})$ are simply multiplied from the left by the above gauge transformation function evaluated at one boundary $B_1$ and from the right by its inverse evaluated at the other boundary $B_2$:

But $\exp(\mathbf{W})$ is reparameterization invariant on the loop, as long as the preferred point $\sigma = 0$ is unaffected, at which the Wilson loop is open (untraced). Therefore the surface holonomy induced by our flat connection on open curves in $\mathcal{L}(M)$ depends (only) on the position of the preferred point $\sigma = 0$.

In the boundary state formalism, the point $\sigma = 0$ has to be identified with the insertion of the open string state which propagates on half the closed string worldsheet. So this dependence appears to make sense.

Next consider the case where $\pi_2(\mathcal{M})$ is nontrivial, so that not all loops in $\mathcal{L}(\mathcal{M})$ are contractible. Does a flat connection on loop space still induce a well defined surface holonomy for closed surfaces?

A sufficient condition for this to be true is that any two points in $\mathcal{L}\mathcal{L}(\mathcal{M})$ which map to the same given surface can be connected by continuously deforming the corresponding loop on $\mathcal{L}(\mathcal{M})$. This is not true for nondegenerate toroidal surfaces in $\mathcal{M}$. But it should be the case for spherical surfaces, for which the loop on $\mathcal{L}(\mathcal{M})$ begins and starts at an infinitesimal loop. All slicings of the sphere should be continuously deformable into each other, corresponding to a deformation of a loop on $\mathcal{L}(\mathcal{M})$, under which the holonomy of a flat connection is invariant.

Therefore even with nontrivial $\pi_2(\mathcal{M})$ a flat connection on loop space should induce a well defined surface holonomy on topologically spherical surfaces in $\mathcal{M}$. This would mean that the condition $F_A = 0$ in [18] can be relaxed and replaced by the more general requirement for a flat connection on loop space.

Whether the same statement remains true for toroidal surfaces is not obvious. But actually for open string amplitudes at tree level in the boundary state formalism spherical surfaces are all that is needed.