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

I will continue this series (part I, part II, part III, part IV) by moving directly to considering topological field theories which have a finite gauge group. These are groups with a finite number of elements – that is, there is a bijection between the elements of $G$ and a subset of the natural numbers (integers greater than zero). Thus, $\mathbb{Z}$, $U(1)$, or $SU(2)$ are not finite groups, whereas $\mathbb{Z}_2$ and the symmetric groups $S_n$ are. I will be in particular interested in the symmetric groups. Do not confused “finite group” with “finitely generated” – $\mathbb{Z}$ is generated by a single element (“1” under addition), but is not a finite group.

Axioms of Quantum Field Theory

There are several different versions of these axioms – I will present those due to Atiyah, because they are more clearly presented than how Witten has done it. Strictly speaking, I won’t be relying too much on these axioms for these notes, but anyone interested in TQFT should know them. These axioms follow the structure of category theory.

Definition (Atiyah): A topological quantum field theory in dimension $d$ over a ring $\Lambda$ is a functor $\Phi: \mathcal{M}_d\to \mathcal{F}(\Lambda)$ from the set of smooth, oriented, closed $d$-manifolds to the set of finitely generated modules over $\Lambda$. This functor is respected on boundaries – for $\Sigma\in \mathcal{M}_d$, if $\partial \Sigma\neq 0$ then $\Phi(\Sigma)\in \Phi(\partial \Sigma)$. This functor satisifes the following axioms:

• Functorial with respect to (orientation-preserving) diffeomorphisms $f:\Sigma\to \Sigma'$: There exist isomorphisms $f(\Phi):\Phi(\Sigma)\to \Phi(\Sigma')$ and $\Phi(fg)=\Phi(f)\Phi(g)$ for $g:\Sigma'\to \Sigma''$.
• Involutory: $\Phi(\bar{\Sigma})=\Phi(\Sigma)^*$ (a reverse of orientation gives the dual module).
• Multiplicative: For $\Sigma=\Sigma_1\cup \Sigma_2$ we have $\Phi(\Sigma)=\Phi(\Sigma_1)\otimes \Phi(\Sigma_2)$.

These properties extend to the boundaries as well, which follows from the initial requirement that $\Phi(\Sigma)\in \Phi(\partial \Sigma)$.

If you are not familiar with rings and modules, I’ll give you the short story. A ring $R$ is an additive group with multiplication defined on it (so $(\mathbb{Z},+)$ where you can also multiply elements) and a module $X$ over a ring is an additive group $X$ on which you can multiply elements of the ring. Good examples of modules are vector spaces, since $\vec{v}+\vec{u}$ is a vector, and so is $a\vec{v}$ for $a$ in a ring (say $\mathbb{C}$). In physics we generally take the modules in the above definition to be Hilbert spaces – which requires that the ring be a field ($\mathbb{R}$, $\mathbb{Z}$, or $\mathbb{C}$) and there be a complete norm on the elements of the module.

Finite Gauge Groups on Riemann Surfaces

The connection on a Lie group $G$ is an element of the tangent space at 0 – which is the Lie algebra $\mathfrak{g}$. For finite groups, the tangent space at the identity is a point so the connections are $A=1$ and are therefore flat for all $G$. They are differentiated only by topology, in the following way: the classifying map $\gamma:M\to BG$ induces a homomorphism on fundamental groups $\pi_1(M)\to \pi_1(BG)$. The fundamental group of the classifying space is equal to the group itself, so the $G$-bundles are classified by a homomorphism

$\gamma_*:=\lambda:\pi_1(M)\to G$.

The partition function is then the sum over all such bundles (since each bundle is a complete state), weighed by an action $W=\exp(2\pi i S)$:

$Z(M)=\frac{1}{|G|}\sum_{\lambda}W(\gamma)$

Here we take $W(\gamma)=h(\partial B)$ for the extension of $M$ to the chain $B$, as discussed in the previous post. Dijkgraaf and Witten call this “$\langle \gamma^*\alpha,[M]\rangle$” since they are identifying the differential character $h$ with its curvature $\alpha$. Now the dimensions of the Hillbert space $\mathcal{H}$ will be given by the number of terms in the partition function. As a practice example, let’s do $\Sigma=\Sigma_g\times \mathbb{S}^1$ where $\Sigma_g$ is a Riemann surface of genus $g$. By picking $\alpha=0$ we have $h=1$ and the partition function will just sum over the representations $\lambda: \pi_1(\Sigma_g\times \mathbb{S}^1)\to G$. $\pi_1(\Sigma_g)$ is generated by pairs of elements $(g_i,h_i)$ for $1\leq g \leq i$ such that $\Pi_i [g_i,h_i]=1$ (see figure).

There is an additional element $k$ which is due to the $\mathbb{S}^1$ part. In the sum, there will be some representations of the form $k\lambda(\delta)=\lambda(\delta)$ for $\delta \in \pi_1(\Sigma_g)$ and $k\in G$ – elements which satisfy this condition are in the stabilizer subgroup $N_{\lambda}\subset G$.  The order (number of copies in $G$) of this subgroup is $|N_\lambda|$, so the size of the Hilbert space is

$\Phi(\Sigma_g\times \mathbb{S}^1)=\frac{|N_\lambda|}{|G|}$

This is exactly what we expect from  quantization – we expect the degrees of freedom to be the moduli space $\mathcal{M}_g$ of $G$ bundles $\mathcal{M}_g=Hom(\pi_1(\Sigma_g),G)/G$, with size $|\mathcal{M}_g|=| N_\lambda|/|G|$.

What about when $\alpha\neq 1$? Well, writing down the explicit answer is not so easy, but finding out some properties that the action satisfies can be instructive. Since we are working with Riemann surfaces, we have the uniformization theorem: there are only three classes of conformally inequivalent Riemann surfaces, a disk (parabolic), a complex plane (flat), or a sphere (spherical). Further, each of these can be generated by considering 2-spheres with punctures:

• $\mathbb{S}^2$ with zero punctures is the Riemann sphere.
• $\mathbb{S}^2$ with 1 puncture is the complex plane (two punctures is also conformally flat)
• $\mathbb{S}^2$ with 3 punctures (which Dijkgraaf and Witten call $Y$) is hyperbolic (a pair of pants!).

So we have all three categories of Riemann surfaces by considering punctured spheres up to 3, and in fact the thrice-punctured version can be used to generate any Riemann surface of genus $g\geq 2$. For instance, by sewing two copies of $Y$ together you get $g=2$. Thus, we can construct any $\Sigma_g\times \mathbb{S}^1$ by sewing together copies of $Y\times \mathbb{S}^1$. Since the boundary of $Y$ is three spheres, we have

$\partial (\Sigma_g\times \mathbb{S}^1)=(\mathbb{S}^1\times \mathbb{S}^1)\cup(\mathbb{S}^1\times \mathbb{S}^1)\cup(\mathbb{S}^1\times \mathbb{S}^1)=\Sigma_1\cup\Sigma_1\cup\Sigma_1$

and our functors are maps $\Phi(\partial (Y\times\mathbb{S}^1)): \mathcal{H}_{\Sigma_1}\times\mathcal{H}_{\Sigma_1}\times\mathcal{H}_{\Sigma_1}\to \mathbb{C}$. A quick note about notation; Dijkgraaf and Witten call this a “path-integral”, which is natural since it represents a transition between the boundaries of a manifold $Y\times \mathbb{S}^1$.

How can we find this map on $Y\times \mathbb{S}^1$? Well,  we need to sum over the representations $\lambda$, but notice that $\pi_1(Y)$ is generated by $g_1,g_2,g_3$ which satisfy $g_1\cdot g_2\cdot g_3=1$ (you have three loops based at the same point going around each hole, so you can deform “around the back of the sphere” to the trivial loop), and $\pi_1(\mathbb{S}^1)$ is just generated by an element $h$. Since $g_3=g_2^{-1}\cdot g_1^{-1}$ we have that our action can only depend on three variables, $W(\gamma)=c_h(g_1,g_2)$. To understand this $c_h$ a little better, consider constructing the sphere with four holes $\mathbb{S}^2_4$ from $Y$. Dijkgraaf and Witten do this diagrammatically:

I think this is pretty clear – gluing a hole means setting the element of the fundamental group equal, taking into account orientations (and the above unity relation is satisfied at each vertex). By chasing around the relations you can see that $g_4=g_3{^-1}g_2^{-1}g_3^{-1}$. By the axioms of TQFT this gluing corresponds to multiplication in the partition function $c(g_1,g_2)c(g_1,g_2,g_3)$. However, this gluing can be done in another way; this is easiest to see by shrinking the $g_1g_2$ line to a four-vertex, and then pulling it apart vertically:

Although it looks like I have some deeper knowledge here, I am really just enforcing the group multiplication at the vertices when playing with these diagrams. One should also check that this is not just a relabeling of the first one, which is pretty easy to see. Anyway, this means that our action satisfies

$c_h(g_1,g_2)c_h(g_1,g_2,g_3)=c_h(g_1,g_2,g_3)c_h(g_2,g_3)$

But notice what happens if we think of our action as a function $c_h:\pi_1(\mathbb{S}^2_4)\to\mathbb{C}$, and apply the coboundary operator to it:

$\delta c_h(g_1,g_2,g_3)=c_h(g_1,g_2)^{-1}c_h(g_2,g_3)c_h(g_1g_2,g_3)^{-1}c_h(g_1,g_2g_3)=1$

The equality to unity comes from applying the relation above (this is actually being relaxed with notation; we had previously defined $c_h$ as “a function of the independent generators”, so it didn’t have a specific number of slot to put things in. Now, we see this is because it lives in the cohomology complex). Thus, our action $c_h$ is a 2-cocycle in $H^2(\mathbb{S}^2_4,\mathbb{C})$. Under the representation, these elements all live in the gauge group $G$, so we have a group 2-cocycle $c_h$ associated to each stabilizer subgroup $N_h\subset G$.

At some level this is the best that can be done with finite groups without further restrictions. We will see in the next post how also making the spacetime finite – that is, a lattice – will finally give us some concrete examples to play with.