# Holonomy and Differential Characters

This question is going to be rather vague, but I'm just trying to see if there are obvious connections between these two concepts.

So the holonomy of a vector bundle with Lie group $G$ is $$h(A)=\mathcal{P}\exp\left(\int_\gamma A\right)$$ where $\mathcal{P}$ is the path-ordering symbol and the integral over the connection $A$ is taken over a curve $\gamma$. These elements form the holonomy group, which relates to the curvature of the connection via Ambrose-Singer.

A differential character is an element $$h\in Hom(C_{k-1}(M;\mathbb{Z}),U(1)),\quad h\circ \partial \in\Omega^k(M)$$ defined on a chain $c\in C_{k}(M;\mathbb{Z})$ to be $$h(\partial c)=\exp \left(\int_c \omega(h)\right)$$ where $\omega(h)$ is an element in $\Omega^k(M)$ (Called the curvature of $h$). Differential characters form a group $\hat{H}^*(M,\mathbb{Q}/\mathbb{Z})$, which are related to homology groups and are key objects in topological quantum field theory.

So my question is essentially how these two things are related to each other. For instance, one might think that the differential characters evaluated on points would be equal to the holonomies of a $U(1)$ bundle. Thus, can we think of differential characters as something like "higher-order holonomies"? At least of $U(1)$ bundles?

What if we generalize in the other direction, change the image of the exponentials to be a general Lie group $G$? Would this be a generalization of holonomies to a higher $k$-skeleton?

Does anyone know if what I propose is natural, totally wrong, or very complicated?

## 1 Answer

A differential character in degree 1 with coefficients in U(1) is precisely (an isomorphism class of) a principal U(1)-bundle with connection. (The group in your description should be R/Z=U(1), not Q/Z.)

With a little bit more effort one can recover the corresponding categories and not just isomorphism classes.

If you evaluate a differential character on a circle C, then you recover the holonomy of the corresponding principal U(1)-bundle around C.

Similarly, in degree n one recovers U(1)-bundle (n-1)-gerbes with connection. For an arbitrary abelian Lie group A we get A-bundle (n-1)-gerbes with connection.

All of this can be found on nLab, see, for example, http://ncatlab.org/nlab/show/Cheeger-Simons+differential+character.

For a general noncommutative Lie group G the construction can only work in degree 1, and once the definitions are set up properly (holonomy around a circle no longer makes sense as an element of G) one recovers principal G-bundles with connections, see, for example, Theorem 5.4 in http://arxiv.org/abs/0705.0452.