10-01 Notes
Topology of the Classifying Space of the Gauge Group
Our goal is to understand the algebraic topology of the solutions to the Yang Mills equations on a closed, oriented surface \(\Sigma\). Following Atiyah and Bott, our strategy will be to do Morse theory for the Yang-Mills functional \(\mathcal{E}(A) = \int_M |F_A|^2 dvol\). This will allow us to obtain information about the critical points, i.e. Yang-Mills solutions, from the algebraic topology of the domain of \(\mathcal{E}\), which is either \(\mathcal{A}/\mathcal{G}\) or simply \(\mathcal{A}\) with the note that \(\mathcal{E}\) is invariant under the \(\mathcal{G}\) action. Because \(\mathcal{A}\) is contractible and the action of \(\mathcal{G}\) is almost free, the space \(\mathcal{A}/\mathcal{G}\) is similar to \(B\mathcal{G}\). Thus a first step is to understand the topology of \(B\mathcal{G}\).
Let \(M\) be a closed, oriented manifold (which later we will specialize to be of dimension \(2\)). Let \(P\) be a principal \(U(n)\)-bundle over \(M\); that is, the frame bundle of a rank \(n\) vector bundle \(E\) over \(M\). Let \(\mathcal{G}\) denote the gauge group of \(P\) (sometimes denoted \(\mathcal{G}_P\) for clarity). We study the classifying space \(B\mathcal{G}\).
Proposition 1. (Last Time) \(B\mathcal{G} \simeq [M,BG]_P\)
Note that we use \([X,Y]\) to denote the space of maps from \(X\) to \(Y\), not (as usual) the set of homotopy classes of maps from \(X\) to \(Y\), which is for us \(H_0([X,Y])\), the set of connected components of \([X,Y]\). The subscript \(P\) denotes the component of \([M,BG]\) whose elements \(\phi\) satsify \(\phi^*EG \cong P\) (that is, pull back \(EG \rightarrow BG\) to a bundle isomorphic to \(P \rightarrow M\)). Because \(BG\) is the classifying space for \(G\), there is a unique component for which this is the case.
Thus we wish to understand the topology of \([M,BG]\). When \(BG\) is an Eilenberg-Maclane space \(K(\pi,\ell)\), this is given by a theorem of Thom:
Theorem 1. (Thom1) \([M,K(\pi,\ell)] \simeq \Pi_{i=0}^\ell K(H^{\ell-i}(M,\pi),i)\)
For example, when \(G = U(1)\), we have \(BU(1) \simeq K(\mathbb{Z},2) \simeq \mathbb{C} P^\infty\). Thus \([M,BG] \simeq [M,K(\mathbb{Z},2)] \simeq K(H^2(M,\mathbb{Z}),0) \times K(H^1(M,\mathbb{Z}),1)\times K(H^0(M,\mathbb{Z}),2)\)
\(\simeq H^2(M,\mathbb{Z}) \times (S^1)^{b_1(M)} \times (\mathbb{C} P^\infty)^{b_0(M)}.\)
The first term is simply the discrete group \(H^2(M,\mathbb{Z})\) itself. The second is a torus of dimension \(b_1(M)\) because \(H^1(M,\mathbb{Z})\) is always free and \(S^1\) is a \(K(\mathbb{Z},1)\). Because \(\mathbb{C} P^\infty\) is a \(K(\mathbb{Z},2)\), we have that the third term is \(\mathbb{C} P^\infty\) to the number of path components of \(M\).
Thus if we consider a principal \(U(1)\) bundle \(P \rightarrow M\) over a connected manifold \(M\), then \[B\mathcal{G} \simeq [M,BU(1)]_P \simeq (S^1)^{b_1(M)}\times \mathbb{C} P^\infty.\] This follows from the above with the remark that in Theorem [thom] it is evident that all components are homotopy equivalent and we have simply taken a component from our above description.
Geometric description in the \(U(1)\) Case
When \(G = U(1)\), we have that \(\mathcal{G} = [M,S^1]\) Endow \(M\) with a Riemannian metric. We consider
\(\mathcal{G}^h = \left\{ u:M\rightarrow S^1\ |\ d^*(u^{-1} du) = 0\right\}.\)
That is, we consider the space of harmonic maps from \(M\) to \(S^1\).
When the target is \(S^1\), harmonic maps are reasonably simply understood. Harmonic is a local condition. Thinking of \(S^1\) as \(\mathbb{R}/2\pi\mathbb{Z}\), if we locally in an open set \(U =subset M\) lift \(u|_U\) to a map \(\log(u|_U): U \rightarrow \mathbb{R}\), the map \(u|_U\) is harmonic if and only if \(\log(u|_U)\) is. To see this, note that \(\Delta \log(u|_U) = d^*d \log(u|_U) = d^*(u^{-1}du)\). (Alternatively if we think of \(S^1\) as the unit disk in \(\mathbb{C}\), the lift to \(\mathbb{R}\) is a function \(f\) such that \(u|_U = e^{if}\).)
Proposition 2. \(\mathcal{G}^h \hookrightarrow \mathcal{G}\) is a homotopy equivalence.
Proof. We define a deformation retraction. Let \(u \in \mathcal{G}\). Note that \(u^{-1}du \in i\Omega^1(M)\) is closed as locally it is given by \(d \log{u|_U}\). Let \(h \in \Omega^1(M)\) be the unique harmonic representative of \([-iu^{-1}du]\). Then there is a unique function \(f: M \rightarrow \mathbb{R}\) such that \(u^{-1}du = ih + idf\) and \(\int_M f \ dvol = 0\).
Note that \(f\) is smoothly determined by \(u\). To see this, note first that \(h\) is smoothly determined by \(u\) because we have the composition \(u \mapsto u^{-1}du \mapsto [u^{-1}du] \mapsto h\) from \(\mathcal{G}\) to \(H^1(M;\mathbb{R})\) and then to \(\mathcal{H}^1(M)\). To see that \(f\) is smoothly determined, we use the short exact sequence \(0 \rightarrow \mathbb{R} \rightarrow [M,\mathbb{R}] \rightarrow B^1(M;\mathbb{R}) \rightarrow 0\). This has a splitting given by \(\frac{1}{vol(M)}\int_M dvol : [M,\mathbb{R}] \rightarrow \mathbb{R}\) and thus we get \(f\) smoothly from \(h\) and \(u\) by the map \(B^1(M;\mathbb{R}) \rightarrow [M,\mathbb{R}]\) taking \(-iu^{-1}du - h\) to \(f\).
We now consider the retraction \(r_t: \mathcal{G} \rightarrow \mathcal{G}\) given by \(r_t(u) = ue^{-itf}\). Note that \(r_0(u) = u\) and \(r_1(u) = ue^{-if}\). We have \[r_1(u)^{-1} d r_1(u) = u^{-1} e^{if} d \left(u e^{-if}\right) = u^{-1} e^{if} \left(e^{-if} du - idf e^{-if}\right) = u^{-1} du - idf = h,\] as desired. ◻
The space \(\mathcal{G}^h\) is more readily understood.
Proposition 3. \(\mathcal{G}^h \cong U(1) \times H^1(M,\mathbb{Z})\) as topological groups.
Proof. We have \(\Phi: \mathcal{G}^h \rightarrow i\mathcal{H}(M) \cong iH^1(M,\mathbb{R})\) by \(u \mapsto u^{-1}du = ih \mapsto i[h] = [u^{-1}du]\). This has image \(2\pi i H^1(M;\mathbb{Z})\). To see this, note that this has the same image as \(\mathcal{G} \rightarrow iH^1(M,\mathbb{R})\) given by \(u \mapsto [u^{-1}du]\). If we have a loop \(\gamma: S^1 \rightarrow M\) then \(\int_{S^1} \gamma^*(u^{-1}du) = \int_{S^1} \left(u \circ\gamma\right)^*(id\theta) = 2\pi i\ \deg(u|_\gamma)\), so the image lies inside \(2\pi iH^1(M;\mathbb{Z})\). To show surjectivity, we obtain an element of \(\mathcal{G}\) from an element of \(2\pi i H^1(M;\mathbb{Z})\) as follows: We choose a basepoint \(m \in M\). Given \(\alpha \in 2\pi iH^1(M;\mathbb{Z})\) and \(x \in M\), we let \(\gamma\) be a path from \(m\) to \(x\) and let \(u(x) = e^{\int_\gamma \alpha}\). This is well-defined because the integral of \(\alpha\) around any closed loop is in \(2\pi i \mathbb{Z}\).
The kernel of the \(\Phi\) is evidently the constant maps. Thus we have a short exact sequence \[0 \rightarrow U(1) \rightarrow \mathcal{G}^h \rightarrow 2\pi i H^1(M,\mathbb{Z}) \rightarrow 0.\] We have already given in the previous paragraph a splitting of this exact sequence \(2\pi i H^1(M,\mathbb{Z}) \rightarrow \mathcal{G}^h\). Thus we have a semi-direct product. The conjugation action of \(U(1)\) on \(2\pi iH^1(M,\mathbb{R})\) must be trivial, however, because the latter is discrete. Thus we have \(\mathcal{G}^h \cong U(1) \times H^1(M,\mathbb{Z})\). ◻
This gives us \(B\mathcal{G} \simeq B\mathcal{G}^h \simeq BU(1) \times B(\mathbb{Z}^{b_1(M)}) \simeq \mathbb{C} P^\infty \times (S^1)^{b_1(M)},\) reproducing the result we obtained in the previous section for \(U(1)\).
Equivariant Topology and the Borel Construction
Instead of considering the quotient space \(\mathcal{A}/\mathcal{G}\), we consider the equivariant topology of the space \(\mathcal{A}\) with its \(\mathcal{G}\) action. The idea is to take the homotopy quotient of \(\mathcal{A}\) by \(\mathcal{G}\). That is, we find a \(G\)-space homotopy equivalent to \(\mathcal{A}\) (via maps respecting the \(G\)-action) on which the action of \(\mathcal{G}\) is free, and take the quotient.
We have the Borel construction. Let \(X\) be a topological space with a \(G\)-action, for \(G\) a topological group. Let \(X_G = X\times_G EG = (X\times EG)/G\). The action of \(G\) on \(X\times EG\) is free because it is free on \(EG\), and the inclusion of \(X\) into \(X\times EG\) is a \(G\)-map which is a homotopy equivalence, because \(EG\) is contractible. Note that if \(X\) and \(EG\) are (e.g. Hilbert) manifolds (as is the case when \(G\) is a Lie group), then so is \(X_G\).
We define \(H_G^*(X) = H^*(X_G)\), the equivariant cohomology. This is much better behaved than \(H^*(X/G)\). Equivariant homology and homotopy groups are defined analogously.
We have maps \(\alpha: X_G \rightarrow X/G\) and \(\beta: X_G \rightarrow BG\) induced by the projections. The fibers of \(\alpha\) are \(\alpha^{-1}(x) = EG/\textrm{Stab}(x) \simeq B\textrm{Stab}(x)\). The fibers of \(\beta\) are simply \(X\), so \(\beta\) expresses \(X_G\) as an \(X\) bundle over \(BG\).
This gives the equivariant cohomology \(H_G^*(X)\) the structure of a bi-module over \(H^*(X/G)\) and \(H^*(BG)\) via cupping with \(\alpha^*\) and with \(\beta^*\). We note that when the action of \(G\) is free, \(X_G \simeq X/G\) and the action of \(H^*(BG)\) is trivial.
Equivariant Cohomology of the Gauge Group Action
We consider the gauge group action of \(\mathcal{G}\) on \(\mathcal{A}\). We have the fibration \(\mathcal{A} \rightarrow \mathcal{A}_\mathcal{G} \rightarrow B\mathcal{G}\) as in the previous section. But note that \(\mathcal{A}\) is an affine space and thus contractible. Thus \(\mathcal{A}_\mathcal{G} \simeq B\mathcal{G}\) and so we have \(H^*_\mathcal{G}(\mathcal{A}) = H^*(B\mathcal{G})\). We wish to compute this. Restricting to the case \(M = \Sigma\), a closed, oriented surface, we have that \(B\mathcal{G} = [\Sigma, BU(n)]_P\). For \(n > 1\), the space \(BU(n)\) is not an Eilenberg-Maclane space, but we will see that it has the same rational homotopy type as a product of Eilenberg-Maclane spaces.
The cohomology of \(BU(n)\) is generated by the Chern classes: \(H^*(BU(n),\mathbb{Z}) \cong \mathbb{Z}[c_1,\ldots, c_n]\), where \(c_i\) is of degree \(2i\). Thus we have \(\mathbf{c} = \prod_{i=1}^n c_i: BU(n) \rightarrow \prod_{i=1}^n K(\mathbb{Z},2i)\). The cohomology of \(K(\mathbb{Z},n)\) is complicated over \(\mathbb{Z}\), but over \(\mathbb{Q}\) it is given by \(H^*(K(\mathbb{Z},n), \mathbb{Q}) = \mathbb{Q}[x_n]\), where this is understood as the graded commutative polynomial ring; that is, the usual polynomial ring if \(n\) is even and \(\mathbb{Q}[x_n]/x_n^2\) if \(n\) is odd. Thus \(H^*(\prod_{i=1}^n K(\mathbb{Z},2i),\mathbb{Q}) \cong \mathbb{Q}[x_2,x_4,\ldots,x_{2n}]\) and so \(\mathbf{c}\) is an isomorphism on rational cohomology.
Thus the map \([\Sigma,BU(n)] \stackrel{\mathbf{c}}{\rightarrow} [\Sigma,\prod_{i=1}^n K(\mathbb{Z},2i)]\) is also an isomorphism on rational cohomology. Letting \(g\) be the genus of \(\Sigma\), we have \([\Sigma,K(\mathbb{Z},2i)] \simeq K(\mathbb{Z},2i) \times K(\mathbb{Z},2i-1)^{2g}\times K(\mathbb{Z},2i-2)\). Thus \(B\mathcal{G}\) is rationally homotopy equivalent to the appropriate product of \(K(\mathbb{Z},2k)\)’s (leaving off \(K(\mathbb{Z},0)\) because \(B\mathcal{G} = [\Sigma,BU(n)]_P\), a component of \(\Sigma,BU(n)\)).
References
R. Thom. L’homologie des espaces fonctionnels. Colloque de topologie algébrique, Louvain, 1956, pp. 29-39. MR0089408 (19,669h)↩︎
