3.4 Third group cohomology

3.4.1 Projective phases on Hilbert spaces on intervals

Consider a 1+1-dimensional theory with boundary conditions $\mathcal{B}_i$ and symmetry $G$. For each pair of boundary conditions $(\mathcal{B}_L,\mathcal{B}_R)$, we have the Hilbert space $\mathcal{H}_{\mathcal{B}_L,\mathcal{B}_R}$ (for a fixed length of the interval). A boundary condition $\mathcal{B}$ does not necessarily fixed by the symmetry action, but rather mapped to another boundary condition $g\mathcal{B}$. Accordingly, the unitary in general maps a Hilbert space to another Hilbert space¹⁸ ¹⁹: \[\begin{equation} \tag{3.22} U_g|_{\mathcal{B}_L,\mathcal{B}_R} : \mathcal{H}_{\mathcal{B}_L,\mathcal{B}_R} \to \mathcal{H}_{g\mathcal{B}_L,g\mathcal{B}_R}. \end{equation}\] Then, the composition rule of $U_g's$ might involve projective phases: \[\begin{equation} \tag{3.23} U_{g_1}U_{g_2}|_{\mathcal{B}_L,\mathcal{B}_R} = e^{\mathrm{i}\tilde{\alpha}(g_1,g_2,\mathcal{B}_L,\mathcal{B}_R)}U_{g_1g_2}|_{\mathcal{B}_L,\mathcal{B}_R}. \end{equation}\] We further assume that the phase is local; it should be a sum of contributions from left and right boundaries, and the two contributions should obey the same rule: \[\begin{equation} \tag{3.24} \tilde{\alpha}(g_1,g_2,\mathcal{B}_L,\mathcal{B}_R) = \hat{\alpha}(g_1,g_2,\mathcal{B}_L) - \hat{\alpha}(g_1,g_2,\mathcal{B}_R) \end{equation}\] for some function $\hat\alpha$. The negative sign is because the orientation is reversed between the left and right boundaries; in other words the contribution from the right boundary should be the time-reversal of the one from the left. In particular, we assume $\tilde{\alpha}(g_1,g_2,\mathcal{B},\mathcal{B}) = 0$. As this can also be considered as the IR cutoff of $\mathbb{R}$ when the interval is long, this should indeed vanish according to the assumptions we made in Section 3.2.

We also consider the ambiguity of the unitary. In this case we also demand that the phase redefinition of the unitary is local and should looks like \[\begin{equation} \tag{3.25} U_g|_{\mathcal{B}_L,\mathcal{B}_R} \mapsto e^{\mathrm{i}(\hat\beta(g,\mathcal{B}_L)-\hat\beta(g,\mathcal{B}_R))} U_g|_{\mathcal{B}_L,\mathcal{B}_R} \end{equation}\] for some function $\hat\beta$.

3.4.2 Group cohomology

Now let us see that the projective phase (3.23), subject to the splitting (3.24) and the ambiguity (3.25), defines a group cohomology. First, we fix a simple boundary condition ²⁰ $\mathcal{B}_0$, define a 3-cochain $\alpha \in C^3(G)$ by \[\begin{equation} \tag{3.26} \alpha_{\mathcal{B}_0}(g_1,g_2,g_3) = \tilde{\alpha}(g_1,g_2,g_3\mathcal{B}_0,\mathcal{B}_0) = \hat\alpha(g_1,g_2,g_3\mathcal{B}_0) - \hat\alpha(g_1,g_2,\mathcal{B}_0). \end{equation}\] Although this definition depends on the simple boundary condition $\mathcal{B}_0$, the cohomology class of $\alpha_{\mathcal{B}_0}$, which will be defined shortly, is independent of the choice of the boundary condition. This fact will be explained in the later part of this section. Therefore the cohomology class of $\alpha_{\mathcal{B}_0}$ is a property of the theory itself, not of the boundary condition. For this reason we omit the subscript of $\alpha_{\mathcal{B}_0}$ unless necessary.

From this definition, we have \[\begin{equation} \tag{3.27} \tilde{\alpha}(g_1,g_2,g_3\mathcal{B}_0,g_4\mathcal{B}_0) = \alpha(g_1,g_2,g_3) - \alpha(g_1,g_2,g_4). \end{equation}\] Now let us examine the associativity of $U_{g_1}U_{g_2}U_{g_3}|_{g_4\mathcal{B}_0,\mathcal{B}_0}$. One way of composing it is \[\begin{align} \tag{3.28} U_{g_1}(U_{g_2}U_{g_3})|_{g_4\mathcal{B}_0,\mathcal{B}_0} &= e^{\mathrm{i}(\alpha(g_2,g_3,g_4))}U_{g_1}U_{g_2g_3}|_{g_4\mathcal{B}_0,\mathcal{B}_0}\\ &= e^{\mathrm{i}(\alpha(g_2,g_3,g_4)+\alpha(g_1,g_2g_3,g_4))}U_{g_1g_2g_3}|_{g_4\mathcal{B}_0,\mathcal{B}_0}, \end{align}\] while the other is \[\begin{align} \tag{3.29} (U_{g_1}U_{g_2})|_{g_3g_4\mathcal{B}_0,g_3\mathcal{B}_0}U_{g_3}|_{g_4\mathcal{B}_0,\mathcal{B}_0} &= e^{\mathrm{i}(\alpha(g_1,g_2,g_3g_4)-\alpha(g_1,g_2,g_3))}U_{g_1g_2}U_{g_3}|_{g_4\mathcal{B}_0,\mathcal{B}_0}\\ &= e^{\mathrm{i}(\alpha(g_1,g_2,g_3g_4)-\alpha(g_1,g_2,g_3)+\alpha(g_1g_2,g_3,g_4))}U_{g_1g_2g_3}|_{g_4\mathcal{B}_0,\mathcal{B}_0}. \end{align}\] Thus, the associativity demands \[\begin{align} \tag{3.30} 0 = (\delta_4 \alpha) (g_1,g_2,g_3,g_4) = &\alpha(g_2,g_3,g_4) + \alpha (g_1,g_2g_3,g_4) \\&- \alpha(g_1,g_2,g_3g_4) + \alpha(g_1,g_2,g_3) - \alpha(g_1g_2,g_3,g_4). \end{align}\] Also, the ambiguity (3.25) causes the shift to $\alpha$ by, \[\begin{equation} \tag{3.31} \alpha(g_1,g_2,g_3 )\to \alpha(g_1,g_2,g_3) + (\delta_3 \beta)(g_1,g_2,g_3) \end{equation}\] where $\beta(g_1,g_2) = \hat\beta(g_1,g_2\mathcal{B}_0) - \hat\beta(g_1,\mathcal{B}_0)$ and $\delta_3$ is the one we defined in (2.41): \[\begin{equation} \tag{3.32} (\delta_3 \beta)(g_1,g_2,g_3) := -\beta(g_1,g_2)-\beta(g_1g_2,g_3)+\beta(g_1,g_2g_3)+\beta(g_2,g_3). \end{equation}\]

Exercise 3.1 Check that $\delta_4\circ \delta_3 = 0$.

Now we can define the 3rd cohomology in the same we did for the second cohomology: \[\begin{equation} \tag{3.33} H^3(G) = \mathrm{Ket}(\delta_4)/\mathrm{Im}(\delta_3). \end{equation}\] This is the invariant information from the projective phases associated to the Hilbert spaces on intervals.

Exercise 3.2 Calculate $H^3(\mathbb{Z}_2)$.

3.4.3 Boundary condition independence

We again emphasize that it is crucial that the cohomology class of $\alpha_{\mathcal{B}}$ is independent of $\mathcal{B}$, and thus the anomaly defined in this way is a property of the theory itself. To show this independence, consider the associativity of $U_{g_1}U_{g_2}U_{g_3}|_{\mathcal{B}_1,\mathcal{B}_2}$ when the simple boundary conditions $\mathcal{B}_1$ and $\mathcal{B}_2$ are not necessarily related by any symmetry action. From the locality (3.24), this leads to the equation \[\begin{equation} \tag{3.34} \hat\alpha(g_1,g_2,g_3\mathcal{B}_1) + \hat\alpha(g_1g_2,g_3,\mathcal{B}_1) - \hat\alpha(g_1,g_2g_3,\mathcal{B}_1) - \hat\alpha(g_2,g_3,\mathcal{B}_1) = \text{($\mathcal{B}_1 \to \mathcal{B}_2$)}. \end{equation}\] From the definition of the cochain $\alpha_{\mathcal{B}}$ in (3.26), we conclude that the two cocycles $\alpha_{\mathcal{B}_1}$ and $\alpha_{\mathcal{B}_2}$ are cohomologous: \[\begin{equation} \tag{3.35} \alpha_{\mathcal{B}_1} = \alpha_{\mathcal{B}_2} + \delta_3 \beta_{\mathcal{B}_1,\mathcal{B}_2}, \end{equation}\] where the 2-cochain $\beta_{\mathcal{B}_1,\mathcal{B}_2}$ is \[\begin{equation} \tag{3.36} \beta_{\mathcal{B}_1,\mathcal{B}_2}(g_1,g_2) = \hat\alpha(g_1,g_2,\mathcal{B}_1) - \hat\alpha(g_1,g_2,\mathcal{B}_2). \end{equation}\]

Of course, all the infinite-dimensional separable Hilbert spaces are isomorphic to each other but identifying all of the Hilbert spaces $\mathcal{H}_{\mathcal{B}_L,\mathcal{B}_R}$ by this universal isomorphism is just confusing and of no use.↩︎
When the theory is a TQFT, there is a multiplication between the states in $\mathcal{H}_{\mathcal{B}_L,\mathcal{B}_M}$ and $\mathcal{H}_{\mathcal{B}_M,\mathcal{B}_R}$ valued in $\mathcal{H}_{\mathcal{B}_L,\mathcal{B}_R}$. You can regard this structure as a category, whose objects are the boundary conditions, and morphisms are states on intervals (equivalently, topological boundary-changing operators). Then $U_g$ is a self-functor on this category, defining categorical representation of the (suspension of the) group $G$. However the composition rule of the functors is up to natural transformations, which are the projective phases. When the theory is a general QFT there is no canonical multiplication among the states but there is an infinite-dimensional family of multiplications. In such cases KO imagines that the boundary conditions would define a category over some operad (something like the swiss-cheese one), and we should be dealing with the representation theory valued in such categories (in 1+1d CFT one can use Witten’s OSFT product to define a usual category though). As far as the anomaly is concerned, the only thing for which we need the operad-category structure is to show the splitting (3.24). In higher dimensional QFTs, the category also has to be higher.↩︎
A boundary condition is called simple when it cannot be written as a direct sum of two boundary conditions. A non-simple boundary condition $\mathcal{B}_1\oplus \mathcal{B}_2$ is like having a external qubit sitting on the boundary which switches the two boundary conditions. Being external means that the state of the qubit cannot be affected by the dynamics of the system. So, e.g. in a spin system, the direct sum over the fixed boundary conditions (which is non-simple) is not the same as the free boundary condition (which is simple). For a non-simple boundary condition the projective “phase” can take different values for each of the simple components and actually be a diagonal matrix.↩︎