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Can we view the Bianchi identity as a local coherence law?
Let me explain. On a vector bundle with connection and holonomy group $G$, each path on the base manifold gives an isomorphism between the fibers at the endpoints. The fibers are all isomorphic, but if the connection is not exact, there is no canonical choice of isomorphism between two fibers. If we now pick two homotopic paths $\gamma,\gamma'$ with the same endpoints, the “difference” (in a multiplicative sense) between the two holonomies is given by the parallel transport along the loop $\gamma'\circ\gamma^{-1}$. It’s an element in the holonomy group $G$. Let’s call it “difference of holonomies”. Now, we said that to compare two fibers (at different points on the base space) we have to choose not just the two fibers, but also a path. But to compare two paths (with the same endpoints), we don’t have to make any choice, right? The “difference of holonomies”, as I called it above, only depends on the paths, not on a 2-cell connecting them. (I’m talking of 1-bundles here.) Is this an example of coherence law?
Now, take two different 2-cells $\sigma,\sigma'$ connecting $\gamma,\gamma'$. We can “integrate” the curvature form on these 2-cells (possibly in a suitable non-Abelian sense), the result of the integral will be the same, and it will be equal to the “difference of holonomies” above.
Now there are two possibilities:
1) $\sigma,\sigma'$ are homotopic;
2) $\sigma,\sigma'$ are not homotopic.
If 1): then I can find a 3-cell connecting them. We can “integrate” the (exterior covariant) derivative of the curvature form on this 2-cell, that form is zero because of the Bianchi identity, and so the “integral” is the identity in $G$. So, the Bianchi identity is a local way to say that the difference of holonomies will not depend on the choice of the 2-cell, as long as it is between the right 1-cells.
If 2): the Bianchi identity is useless here, because there is no 3-cell along which we can do that “integral”. But the difference of holonomies still has to depend only on the two paths! So we can take the 2-cell $\sigma'\circ\sigma^{-1}$, and “integrate” the curvature form on it. The result should still be the identity in $G$.
Now as an example consider the tangent bundle to an orientable surface with Riemannian metric, with the Levi-Civita connection. This way the holonomy group is $SO(2)$, which is Abelian, and so all the “integrals” above are really (exponentials of) integrals of differential forms in the usual sense. In particular, curvature 2-form $R$ evaluated on the 2-cell $\sigma'\circ\sigma^{-1}$ is simply:
$\exp \Bigg( i\int_{\sigma'\circ\sigma^{-1}} R \Bigg)\;.$I have argued above that by “coherence”, this quantity should be the identity on $G=SO(2)$. This implies that:
$\int_{\sigma'\circ\sigma^{-1}} R$must be an integer multiple of $2\pi$.
Now, this is not the entire Gauss-Bonnet theorem, but part of it! Is this reasoning correct? Can it be generalized?
Sorry if what I ask is trivial (or wrong), I’m moving my first steps in the n-world. Please also tell me if anything is unclear.
Good question. So about 2/3rds of what you are saying may be captured like so:
given a principal connection $\nabla$ for a Lie group $G$, then the pair consisting of $\nabla$ and its curvature $F_\nabla$ jointly constitute the data of a “2-connection” for parallel transport over surfaces, with gauge 2-group the 2-group sometimes known as $\mathbf{E}G$, the thing coming from the crossed module of groups whose differential is simply the identity on $G$. The induced parallel transport over spheres, with values in $G$, that’s just the expression $\int_{\sigma' \circ \sigma^{-1}} R$ that you consider.
This is discussed in section 3.4 of “Connections on non-abelian gerbes and their holonomy” (arXiv:0808.1923). The last sentence of that section is precisely the observation that the flatness of the 3-form curvature of this 2-connection is equivalently the Bianchi identity of the 2-form curvature of the original connection.
One just needs to be careful with this when speaking fully homotopy-theoretically, because $\mathbf{E}G$ is weakly equivalent to the trivial 2-group. For some purposes this collapse is just what one wants to see, for the discussion of non-flat 1-connections one may want to add a constraint that disallow this collaps. Following this thought through to the end leads one to discover differential cohomology theory. This is the content of the section “Introduction – Geometry – Principal connections” in dcct. See section 1.2.7 in v2, which is section 1.2.6 in v1 .
This is very interesting. Is there any introductory text/paper/lecture on these concepts? What I’ve looked up so far is still a little too advanced for me, I don’t know where to start.
(For example, what is $EG$ exactly? Something like $G$ quotiented weakly by its action on itself? I guess it has its own nLab page, but what name do I look for?)
Also, is the thing with the multiples of $2\pi$ above relevant, or just a coincidence in low dimension?
Thank you!
Try groupal model for universal principal infinity-bundles. There’s also universal principal bundle and universal principal infinity-bundle. classifying space eventually gets round to describing models.
Yes, $\mathbf{E}G = G//G$ is the action groupoid of $G$ acting on itself by (left, say) multiplication. This action groupoid happens to retain group structure itself, which makes it a 2-group. I used to call it the inner automorphism 2-group.
And so what I meant to highlight is that if $\nabla$ is a $G$-connection, then the surface holonomy of its associated flat “curvature 2-transport” is such that it necessarily assigns elements in $G$ to spheres, too. That’s why you get $SO(2)$ in your example. But in fact the surface transport will take values in the center of $G$.
The more elementary discussion of this is in “Smooth Functors vs. Differential Forms” (arXiv:0802.0663). We could have (maybe should have) included the example of the “curvature transport” there, but for some reason it ended up in the other article that does the full gerbe story. (Well,in the beginning it was a single article, and then the journals made us split it into three separate pieces).
Nice. Thank you very much!
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