The String Coffee Table

Somebody wrote:

I have been trying to determine how to caluclate the surface holonomy for the $B$-field on a surface with a boundary. I would like to calculate it explicitely by triangulating the surface and then integrating over the triangulation. The relevant literature (by Kapustin hep-th/9909089) cites Gawedzki’s paper ‘Topological Actions in 2-d QFT’ where this holonomy is defined but without any motivation. It is very clear how to define the holonomy of a 1-form along a line and also for a 2-form on a closed surface (simply by requiring it to be gauge invariant as in Alvarez ‘Topological Quantization and Cohomology’) but I am not sure how to even motivate the definition of the holonomy in the case when the surface is not closed. As you and others are always discussing a similar problem but for non-abelian field 2-forms I presume the abelian case is already very well understood. Would you mind pointing me to a good reference?

First let me recap what that formula is that we are talking about, for the convenience of those following this.

An abelian gerbe with connection and curving is represented by the choice $\mathcal{U} \to M$ of a good covering of base space by open contractible sets $U_i \subset M$ together with a collection of 0-forms $\mathrm{ln} f_{ijk}$ on triple overlaps $U_{ijk} = U_i \cap U_j \cap U_k$, a collection of 1-forms $a_ij$ on double overlaps $U_{ij}$ and a collection of 2-forms $B_i$ on each $U_i$ itself.

Now, given any closed surface $\Sigma \subset M$ the ‘connection and curving’ of the gerbe associates to it a complex number

called its (surface-)holonomy.

There is an ‘intrinsic’ formula in terms of bundle gerbes for this $\mathrm{hol}$, as for instance described by Mackaay and Picken or as discussed in

A. Carey, S. Johnson & M. Murray
Holonomy on D-branes
hep-th/0204199.

In that reference it is also demonstrated how that ‘intrinsic’ formula is equivalent to a more local version, which is given by the following algorithm:

Choose a good covering $\mathcal{U}$ and a triangulation of $\Sigma$ subordinate to $\mathcal{U}$ (i.e. such that each face sits in a single, each edge in a double and each vertex in a triple overlap). Then multiply all the following elements of $U(1)$ together: For each vertex $x \in U_i$ take $\exp(i \mathrm{ln} f_{ijk})$, for each edge $\gamma \subset U_{ij}$ take $\exp(i \int_\gamma a_{ij})$ and for each face $\Sigma_i \subset U_{i}$ take $\exp(i \int_{\Sigma_i} B_i)$. (Obviously there is an issue of how to choose orientations which I am glossing over here.)

As far as I am aware this formula has indeed first been discussed by Alvarez in a formal physics context. It can sort of be guessed as a generalization of the respective formula for computing the holonomy of a curve in a bundle with connection, but as far as I am aware a really systematic motivation of this formula was unknown.

Now to answer the above question: what do we do if we want to compute the abelian surface holonomy of a surface $\Sigma$ which is not closed?

One case that is sort of obvious is the one where the open surface $\Sigma$ can be closed by gluing on a disk $D$ such that restricted to $D$ the gerbe is trivial. In that case one can compute the closed surface holonomy of $\Sigma \cup D$ and multiply the result by the inverse of the exponential of $B_D$ over $D$. This corresponds to using the above algorithm but leaving out te contribution from the patch $U_D \simeq D$.

And this is what is done for open strings on $D$-branes, as for instance reviewed on p. 8 of hep-th/0409200.

But one would certainly want to understand why this should be the right answer and how it works in the more general case.

I am not aware of any full discussion of this point in previous literature. But I do believe that what I have worked out together with John Baez does provide the answer, as described here (see also here).

Namely, the $\mathrm{hol}$ of a gerbe is really something that assigns 2-morphisms of 2-group 2-torsors to surface elements. Locally we can identitfy these 2-morphisms with group elements and hence obtain the more ‘naive’ notion of $\mathrm{hol}$ discussed above.

This also means that the $\mathrm{hol}$ of an open surface is really a 2-torsor 2-morphism from a 1-morphism associated to a singled out source component of the boundary of that surface to the 1-morphism associated to the remaining target component of the boundary of that surface.

When the surface is closed, source and target edges coincide; a local trivialization lets us associate a 2-group 1-morphism with it (which is always trivial in the case of abelian gerbes) and hence we are left with a 2-group 2-morphism associated to that surface, which in turn gives rise to an ordinary group element. By working out the details one finds that this group element is precisely given by the algorithm discussed above, which goes back to Alvarez.

How this works I have describe recently here, which is now section 12.4 in here. This gives a sensible motivation of the above algorithm, as is best seen by looking at the diagrams presented in these documents.

Now, suppose we have an open surface such that its boundary sits in a single contractible patch. Then locally identifying the 2-torsor 2-morphism associated to it by $\mathrm{hol}$ with a group element as just indicated yields the above prescription for how to compute the holonomy of an open surface.

Everything works quite analogously in the more general context wher in-boundary and out-boundary do not sit inside a single patch. Only subtlety here is that in order to obtain a group element from the 2-torsor 2-morphism associated to the surface we need to specify how we trivialized in the patch of the in-boundary as well as in the patch of the out-boundary.

Compare this to the situation of ordinary holonomy, which is completely analogous.

[…]

Oops, I gotta run now. Still no doplhins in sight, I am requested to do something about it. I’ll be glad to provide more details upon request.