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

Continuing my notes from the Dijkgraaf and Witten paper (part I, part II)…

Topological Actions

This will be an extended discussion of the introduction in part I, with the basic goal to understand how to classify these theories by elements in the 3rd cohomology group. Before starting I should say I began a short discussion on MathStax about this – the focus being on the generalization of such actions into the language of higher stacky cohomology. The answer there is very well-written but deviates farther from Chern-Simons then I am interested in at the moment.

When the bundle $E$ over the 3-manifold $M$ is trivial, we can easily extend it to a 4-manifold $B$ which restricts to $M$ on its boundaries. If $E$ is not trivial (not globally $G\times M$), than this is not possible – since we are interested in a generic Chern-Simons form, this is exactly the case we want. First, rather than consider extensions to manifolds Dijkgraaf and Witten consider extensions to a “smooth, singular 4-chain” which is probably just an element $\sum_i n_i \sigma_i\in C_4(B)$ where $\sigma_i:\Delta^4 \to B$ – the “smooth” modifier is probably meant to suggest the map $\sigma_i$ is smooth. In order to have a bundle over this extension, the classifying map must also have an extension $\tilde{\gamma} :B \to BG$ with boundary $\gamma(M)$. This translates into a condition on homology – namely, $\tilde{\gamma }(B)\in C_4(BG)$ is an extension of $\gamma(M)\in C_3(BG)$ if $\partial \gamma(B)=\gamma (M)$. But then $\partial \gamma(M)=0$ and $[\gamma(M)]=0\in H_3(BG,\mathbb{Z})$. So the existence of an extension comes from the vanishing in the 3rd homology group.

This homology group consists entirely of torsion – this is because $H_{odd}(BG,\mathbb{R})=0$ for a Lie group or finite group (and in fact, all $H_*$ vanishes for finite groups). In the previous post we learned that the kernel of the map $\rho:H_k(M,\mathbb{Z})\to H_k(M,\mathbb{R})$ is the torsion elements, so by the first isomorphism theorem $H_{odd}(BG,\mathbb{Z})/Tor=0$. Torsion elements are cyclic of order $p$, so this implies that $p\cdot [\gamma(M)]=0$ (writing the groups multiplicatively). In some way there are “$p$ copies of $E$ in the extension $\tilde{E}$.” Thus we have a natural ambiguity in how we extend the bundle $E$ over $B$ – namely, we can only do so up to an integer $p$.

At this point, it is becoming obvious to me that a lot of the ambiguity coming from this construction has to do with on one hand needing to use differential forms (with coefficients in $\mathbb{R}$) to define the action but using chains (with coefficients in $\mathbb{Z}$) to determine when the construction makes sense. Perhaps this is a partial answer to my inevitable confusion about why torsion is important in homology – but I digress…

The differential form we care about is $\Omega=\frac{k}{8\pi^2}Tr(F\wedge F)\in H^4(B,\mathbb{R})$ for curvature $F$. This form can be described by a integer-valued cochain $\omega$ such that $\rho([\omega])=\Omega$, but only up to a torsion element. The idea is to use this ambiguity to resolve the ambiguity in the action from above. Thus, define the action

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

(here I am using the pairing $\langle \alpha, [M]\rangle \to \int_{M}\alpha$). Notice this does not depend on which cocycle $\omega$ we chose to represent the class $[\omega]$, since if we shift it $\omega \to \omega + \delta\epsilon$ where $\epsilon\in C^3(BG,\mathbb{Z})$ the action changes by

$\Delta S=-\frac{1}{n}\langle \gamma^*\delta\epsilon,[B]\rangle=-\frac{1}{n}\langle \gamma^* \epsilon,\partial B\rangle=-\langle \gamma^*\epsilon,M\rangle \in \mathbb{Z}$

(It’s amazing how much mathematics Stoke’s theorem can get you through…). We have used $n M=\partial B$ (the boundary of $B$ consists of $n$ copies of $M$), and the last equality comes from $\epsilon$ having integer coefficients. Since the action changes by an integer, it remains a gauge invariant.

The action above is the topological action for a bundle of order $n$. Dijkgraaf and Witten perform a few more technical consistency checks on it, but we will move on to more interesting things, since my focus is going to be on finite groups and lattice theories. The next short post will be about differential characters, which allow you to define such actions in slightly more generality.