# Notes on Lattice Topological Field Theory in Three Dimensions, Part IV

Now I continue this series (part I, part II, part III) with a discussion of…

Differential Characters

We must consider what happens if the class $\omega$ in cohomology which we choose to represent our Chern-Simons action

$S=\frac{1}{n}[\langle \Omega(F),[B] \rangle-\langle \gamma^*\omega, [B] \rangle ]$

contains torsion. If this element has torsion of order $p$, then our action will have a $\mathbb{Z}_p$ phase. This is a further ambiguity which we want to remove, by specifying an action which detects the presence of torsion in extension to the chain $B$. Dijkgraaf and Witten choose to do this using differential characters. I note that it is not at all obvious to me that this is a unique approach – it seems to work, but that doesn’t mean there aren’t other possible ways of going about doing this.

Differential characters were original described by Cheeger and Simons (1985) – I found that paper a bit difficult to read, but there is a very nice description from an arXiv paper by Baer and Becker (link – chapter 5 in particular). The generalization proceeds by identifying elements in cohomology that, rather then obeying $\partial ^2 \omega=0$, satisfy $h \circ \partial \in \Omega^k$. Specifically, differential characters are elements of the abelian groups

$\hat{H}^k(M;\mathbb{Z}):=\{h\in Hom(C_{k-1}(M;\mathbb{Z}),U(1))| h \circ \partial \in \Omega^k(M)\}$

$h(\partial c)=\exp\left(2\pi i \int_c \alpha(h) \right)$.

Here $\alpha(h)\in \Omega^k(M)$ is called the curvature of $h$ (and is a closed form). Note there is some consistency to the nomenclature here – “characters are exponentials of classes” – which is true for i.e. Chern characters as well. To every $h$ there is a corresponding form $\beta$ which takes values in $\mathbb{R}$, defined by $h(c)=\exp(2\pi i \beta(c))$. We also have the map

$\mu^{\beta}:C_k(M;\mathbb{Z})\to \mathbb{Z},~\mu^\beta(c)=\int_c \alpha(h) - \beta (\partial c).$

Since the curvature is a closed form, it’s easy to show that $\mu^\beta$ is closed, and thus $[ \mu^{\beta}]\in H^k(M;\mathbb{Z})$.

Now, observe what happens when we evaluate this character on a chain $z\in C_{k-1}(M)$ which represents a torsion class in $H_{k-1}(M)$. By some similar arguments as we used before, we can choose $x\in C_k(M)$ such that $z=\partial x /N$ as real cocycles. Then we evaluate the character on the torsion element:

\begin{aligned} h(z)&=\exp(2\pi i \beta (z))=\exp \left(\frac{2\pi i}{N} \beta (\partial x)\right)\\&=\exp\left(\frac{2\pi i}{N}\delta \beta (x)\right)=\exp\left(\frac{2\pi i}{N}\left(\int_x \alpha(h) - \mu^\beta (x)\right)\right)\end{aligned}

To go from the first to the second line we just used the fact that the coboundary operator is the dual to the boundary operator, and the last step uses the definition of $\mu^\beta$ from above. Thus, rather then define the action $S$, we define the exponential of the topological action as the differential character $h$, which is evaluated over the extensions of $M$ to chains $B$.

This reduces to the appropriate notions – if $B$ is closed, we find the action given by $\langle \alpha,[B]\rangle$, which is the same thing as we get when $[B]$ is not a torsion element. When it is, we get the action $\int_B \alpha -\mu^\beta(B)$. By identifying $\alpha \leftrightarrow \Omega(F)$ (they are both curvatures) and $\mu^\beta\leftrightarrow \gamma^*\omega$ (they are both cocycles in $H^4(B;\mathbb{Z})$), we get back to our original topological action. In fact, a pair $(\Omega(F),\omega)$ uniquely determines the reduction of a form $\delta\beta=\Omega(F)-\omega$ if $H^3$ vanishes, which it does for the classifying space. I skip the proof of this, because it will take me too far afield.

One note – the Dijkgraaf and Witten paper looks somewhat different than the above discussion because they often exchange the roles of $\alpha$ (the differential form) and $h$ (the exponential of the integral of that form). Since they uniquely determine each other, there is no real issue here but it does make things slightly confusing to try and work out. I have tried to keep things as consistent as possible.

The Dijkgraaf and Witten paper now continues to discuss manifolds with boundary, CFT, and spin theories. None of these are relevant to my particular interest at this moment, so the next post will continue by fixing a finite gauge group and looking at what happens on the lattice.