The String Coffee Table

2-Holonomy LEGO

Posted by Urs Schreiber

Recently I had reported work on 2-connections in 2-bundles. There are several loose ends that still need to be tied up, though. For instance I claimed that one of the important points of hep-th/0412325 is the conception of global nonabelian surface holonomy. But the formula for computing this from the global nonabelian 2-connection has not made it into that, already long, paper. So I’ll present it here. In fact, not a formula, but a picture.

The definition and consistency proof of global nonabelian 2-holonomy is a rather enchanting exercise in ‘category LEGO’, also known as diagrammatic reasoning.

Instead of writing down formulas it proves to be way more elegant to draw diagrams and fit them together like LEGO blocks. In fact, this is not just more elegant. This ‘higher dimensional algebra’ reduces the definition and consistency of global 2-holonomy from something totally impenetrable in terms of formulas into an easy LEGO game. (If you don’t believe this and feel bored, try to rederive the following in terms of formulas.)

So to get a feeling for what I am talking about first consider global line holonomy in an ordinary fiber bundle.

This involves partitioning your manifold in open sets $U_i$ and proceeding as indicated in this picture:

Paths ${\gamma }_i$ get mapped to group elements $W({\gamma }_i)$. Note how the point $x$ is sort of ‘blown up’ under the holonomy map, being sent to a nontrivial group element $g_{\mathrm{ij}}(x)$.

Under gauge transformations the local holonomy transforms under what I shall call here a 1-dimensional onion move for reasons to become clear shortly:

Gauge invariance of the global holonomy is a consequence of cancelling of adjacent ‘onion slices’:

Here and in the following $\widetilde{(\cdot )}$ is the gauge transformed version of $(\cdot )$.

Note how it is sufficient for proving global gauge invariance to prove that every piece of holonomy transforms under gauge transformations only by being conjugated at its ‘boundary’ while the ‘internal’ conjugations mutually cancel.

That shall suffice as introduction. Now consider a principal 2-bundle. (For convenience, restrict attention to the special case where base 2-space has only identity morphisms and where the structure 2-group is strict.)

The usual identities between transition functions become 2-arrows in the 2-group:

From this one reads off the gauge transformation of the transition 2-morphism $f$ by looking at the 2-dimensional onion move

Now put a 2-connection on this 2-bundle. A gauge transformation on a piece of local 2-holonomy (which is a 2-morphism in a 2-group) is a ‘2-conjugation’:

In particular the transition from one patch $U_i$ to another one $U_j$ has been shown to be of this form:

But a local gauge transformation is completely analogous:

Note that 2-conjugation is a 2-similarity transformation and respects 2-compositon:

This means that, just as in the ordinary bundle holonomy discussed above, it is sufficient for proving the gauge invariance of some quantity to demonstrate that any piece of it under gauge transformations gets 2-conjugated only from the periphery.

We shall be particularly interested in the surface part of the 2-conjugation of 2-holonomy:

This is the onion skin $a_{\mathrm{ij}}$ in the 2-conjugation law for the transition law of local 2-holonomy drawn above. (I realize that the resolution of the graphics is a little too low for the labels to be easily readable. You can find a better readable pdf file here.)

By performing a 2-transition first in one gauge and then in the other, one can read off the gauge transformation law for the $a_{\mathrm{ij}}$ by combining all of the above diagrams iteratively into the following :

See why I call this ‘onion moves’?

By the magic of diagrammatic math we can simply read off the sub-onion move which describes the gauge transformation of $a_{\mathrm{ij}}$:

Ok, so much for the derivation of 2-gauge transformation in a 2-bundle with 2-connection. Now let’s define global 2-holonomy.

First triangulate the surface whose holonomy is to be computed. This can always be done in such a way that the resulting graph is trivalent and that every face comes to lie in a single $U_i$, every edge in a double overlap and every vertex in a triple overlap (just as in the prescription for 2-holonomy in an abelian gerbe, which we will recover as a special case of our non-abelian surface holonomy).

Certainly we have to assign local 2-holonomy (computable in terms of path space holonomy) to faces contained in a single $U_i$. Then, there is only one candidate 2-group element to be assigned to edges in double overlaps, namely that $a_{\mathrm{ij}}$. Furthermore there is only one candidate 2-group element to be assigned to vertices in triple overlaps, namely $f_{\mathrm{ijk}}$. Finally, there is only one way to glue all them together.

Hence category-LEGO logic tells us that the definition of global 2-holonomy must be as follows:

There is no other choice, hence this must be right.

As a first check, those who know the well-known definition of global surface holonomy in an abelian gerbe and who recall how we derived that the above symbols are related to the gerbe cocylce data will see at a glance that this diagram correctly reproduces the gerbe holonomy formula in the case where the 2-group involved is abelian.

But we should directly prove that the above is a well defined prescription for global 2-holonomy in the general (non-abelian) case. It’s easy using diagrams:

First consider computing global surface holonomy in the gauge $\widetilde{(\cdot )}$:

Now express all the $\widetilde{(\cdot )}$ objects here in terms of the original gauge, by using the onion moves derived above on each piece:

One sees that one can cancel 2-morphism $a$ against their reversed version in six places, which have been shaded in the diagram. Doing so yields:

Here we also turned some arrows around by ‘whiskering’ so that now 2-morphisms $p$ cancel against their reverses:

What is left is the 2-holonomy in the original gauge, 2-conjugated on the periphery. As discussed above, this is sufficient for proving full gauge invariance, since these 2-conjugations will cancel against their counterparts at the other vertices and edges.

(Of course in the end, when computing 2-holonomy of a closed surface, two unpaired edges will be left, the source and the target edge. Just like for line holonomy, they must be paired by a suitable trace operation.)

So this is how global nonabelian surface holonomy works in a principal 2-bundle with 2-connection. As a special case this reproduces the well-known formula for global abelian 2-holonomy as derived in the context of abelian gerbes.