2.6 Second group cohomology
Let us formalize what we have learned above for a general group \(G\). We assume that for each \(g\in G\), we have a symmetry transformation \(T_g\) acting on \(\mathbb{P}\mathcal{H}\) as symmetry transformations satisfying the multiplication law: \[\begin{equation} \tag{2.38} T_g T_{g'} = T_{gg'} \end{equation}\]
By Wigner’s theorem, for each \(g\), we also have \(U_g\). The multiplication law of \(T_g\) only guarantees \[\begin{equation} \tag{2.39} U_g U_{g'} = e^{\mathrm{i}\alpha(g,g')}U_{gg'}, \end{equation}\] for some phase \(\alpha(g,g')\), because the correspondence between \(T_g\) and \(U_g\) is only up to a phase. Here we introduce the terminology from algebraic topology. A map from \(n\)-th power of \(G\) to some abelian group \(M\) is called \(M\)-valued \(n\)-cochain, denoted by \[\begin{equation} \tag{2.40} C^n(G,M) := Map(G^n,M). \end{equation}\] Note that a cochain is not demanded to be a group homomorphism; it is just a map. \(\alpha\) is a \(\mathbb{R}/2\pi\mathbb{Z}\) (\(\cong U(1)\)) valued 2-cochain. In this lecture, \(M\) is almost always \(\mathbb{R}/2\pi\mathbb{Z}\), so we will not explicitly declare the value domain onwards.
The 2-cochain \(\alpha\) is actually not completely arbitrary. The constraint comes form the associativity of the unitary operators: \((U_{g_1}U_{g_2})U_{g_3} = U_{g_1}(U_{g_2}U_{g_3})\), which leads \[\begin{equation} \tag{2.41} \delta_3 \alpha (g_1,g_2,g_3) := -\alpha(g_1,g_2)-\alpha(g_1g_2,g_3) + \alpha(g_1,g_2g_3) + \alpha(g_2,g_3) = 0. \end{equation}\] Here we have defined the derivative \(\delta_3: C^2(G) \to C^3(G)\) on 2-cochains. A 2-cochain satisfies this condition is called a cocycle, and the set of cocycle is denoted by8 \[\begin{equation} \tag{2.42} Z^2(G) := \mathrm{Ker}(\delta_3) = \{\alpha\in C^2(G)\mid \delta_3 \alpha = 0\}. \end{equation}\]
Exercise 2.1 Show that the particular cochain \(\alpha \in C^2(\mathbb{Z}_2\times\mathbb{Z}_2)\) defined in (2.28) is indeed a cocycle.
Not all the (nonzero) elements of \(C^2(G)\) are interesting: some of them might be eliminated by redefinition of the unitary operators \(U_g\): \[\begin{equation} \tag{2.43} U_g \mapsto e^{\mathrm{i}\beta(g)}U_g, \end{equation}\] with \(\beta \in C^1(G)\). The phase \(\alpha(g,g')\) gets shifted by this redefinition as: \[\begin{equation} \tag{2.44} \alpha(g_1,g_2) \mapsto \alpha(g_1,g_2) - \delta_2 \beta (g_1,g_2) := \alpha(g_1,g_2) + \beta(g_1g_2)-\beta(g_1) - \beta(g_2). \end{equation}\] The map \(\delta_2: C^1(G) \to C^2(G)\) is also called the derivative. An image of \(\delta_2\) is called a coboundary, denoted by \[\begin{equation} \tag{2.45} B^2(G) := \mathrm{Im}(\delta_2) = \{\delta_2 \beta\mid \beta \in C^1(G)\}. \end{equation}\] The crucial feature of the derivative is that it vanishes when composed: \[\begin{equation} \tag{2.46} \delta_3\circ \delta_2 = 0. \end{equation}\] Therefore we have \(B^2(G)\subset Z^2(G)\). Now, interesting projective phases are those not in \(B^2(G)\). This motivates us the define the group cohomology group9 \[\begin{equation} \tag{2.47} H^2(G) := Z^2(G)/B^2(G). \end{equation}\]
In this lecture we do not have time to learn how to compute the cohomology group, but the examples we have seen are \[\begin{align} \tag{2.48} H^2(SO(3)) &\cong \mathbb{Z}_2,\\ H^2(\mathbb{Z}_2\times\mathbb{Z}_2) &\cong \mathbb{Z}_2. \end{align}\]
Exercise 2.2 Prove (2.46).
Exercise 2.3 Show that \(H^2(\mathbb{Z}_2)=0\). (If you want, also calculate \(H^2(\mathbb{Z}_2,\mathbb{Z}_2)\).)
The derivative \(\delta_{n+1}\) for general \(n\)-cochain can be defined, but we will postpone this until next chapter, other than \(\delta_2\) which is defined just below.↩︎
Do not confuse this with the cohomology group of the group as a topological space. The two, the group cohomology of a group and the geometric cohomology of the same, can be denoted by the same symbol \(H(G)\), but they are different. They are confusing especially when \(G\) is continuous. Rather, the group cohomology is identified with the (simplicial/cellular) cohomology of an infinite-dimensional topological space called the classifying space \(BG\).↩︎