09-03 Notes

From Moduli

These lecture notes are based on notes from the 18.999 geometry seminar class taught by Tomasz Mrowka and from the IAS/Park city Mathematic series book Gauge theory and the topology of four-manifolds

Moduli spaces

A moduli space can be viewed as a geometric object which classifies the solutions of some problem. For example, there is the moduli space of \(n\)-tuples of points in \(I=[0,1]\),that is, points in \(I^{n}\) modulo symmetric transformations:\(I^{n}/ Sym_{n}\). Given a complex vector space \(V\) of dimension \(n\), we can look at the space of endomorphisms of \(V\) modulo isomorphisms: \[End(V)/Iso(V) \simeq Mat_{n \times n}( \mathbb{C})/Gl_{n}(\mathbb{C}) = \mathcal{M}_{n}\] The Jordan canonical form gives a description of \(\mathcal{M}_{n}\) as a set.

In this class, we will look at moduli spaces which arise from the study of connections on \(G\)-bundles such as the moduli space of instantons on a 4-dimension manifold (ADHM) or the moduli space of Yang-Mills-Higgs monopoles on \(\mathbb{R}^{3}\).

In the next section, we list some basic notions of principal \(G\)-bundles which we will need for this class.

G-bundles

Definition 1. Let \(G\) be a Lie group.A (right) smooth principal \(G\)-bundle is a smooth fiber bundle \(\pi: P \rightarrow X\) such that:

There is a smooth free right action \(P \times G \rightarrow P\) with respect to which \(\pi\) is invariant.
There exist \(G\)-equivariant local trivializations: For any \(x \in X\), there exist an open neighborhood \(U\) and a diffeomorphism \[\phi: \pi^{-1}(U) \rightarrow U \times G\] such that: \[\begin{aligned} \phi(p \cdot g)= \phi(p) \cdot g \\ \pi(p) = \pi_{U}(\phi(p)) \end{aligned}\] for all \(p \in P\) and \(g \in G\).\(U \times G\) is equipped with the standard right \(G\)-action and \(\pi_{U}:U \times G \rightarrow U\) is the projection map.

Given \(g \in G, q \in P\), we set \[\begin{aligned} r_{g}: P \rightarrow P \\ p \rightarrow p\cdot g \end{aligned}\] and \[\begin{aligned} \iota_{q}: P \rightarrow P \\ h \rightarrow q\cdot h \end{aligned}\] The map \(\iota_{q}\) induces an isomorphism \[\begin{aligned} (\iota_{q})_{\ast}:\mathfrak{g}\rightarrow T_{p}P_{x} \end{aligned}\] where \(\mathfrak{g}\) is the Lie algebra of \(G\), \(x= \pi(p)\) and \(P_{x}= \pi^{-1}(x)\) the fiber over \(x\). We will also denote \(T_{p}P_{x}\) by \(VTP_p\) (the space of vertical tangent vectors at \(p\)).

Associated bundles

Suppose \(\pi: P \rightarrow X\) is a smooth principal \(G\)-bundle and \(F\) a smooth manifold equipped with a smooth left \(G\)-action. Then,we can define a fiber bundle over \(X\) with fiber \(F\) as follows. We set \[P \times_{G}F = ( P \times F) / \sim\] where \((p,y)\sim (p.g,g^{-1} \cdot y)\) for all \(p \in P, y \in F\) and \(g \in G\). We define the map \[\begin{aligned} \tau : P \times_{G}F \rightarrow X \\ \{p,y\} \rightarrow \pi(p) \end{aligned}\] where \(\{p,y\}\) is the equivalence class of \((p,y) \in P \times F\).Here are some important examples of this construction.

Let \(F=G\) and let \(G\) act on itself via conjugation :\(g \cdot h = ghg^{-1}\).Then \(P \times_{G}G \rightarrow X\) is a smooth fiber bundle with fiber \(G\) which is usually denoted by \(Ad(P)\).
Suppose that \(F\) is a vector space \(V\) and \(\rho: G \rightarrow Gl(V)\) is a linear representation. Then \(P \times_{G}V \rightarrow X\) is a smooth vector bundle.In particular, if we consider the adjoint representation \(ad: G \rightarrow Gl(\mathfrak{g})\), then the corresponding smooth vector bundle is denoted by \(ad(P)\).

Connections

Definition 2. Suppose \(X\) is a smooth n-dimensional manifold and \(\pi: P \rightarrow X\) is a smooth principal \(G\)-bundle.Then, a connection \(A\) for this bundle is an n-dimensional horizontal distribution \(\mathcal{H}^{A}\): For every \(p \in P\), we have a decomposition \[TP_{p}= \mathcal{H}^{A}_{p}\oplus VTP_{p}\] Hence \[\pi_{\ast}: \mathcal{H}^{A}_{p} \rightarrow T_{x}X\] is an isomorphism.It is also required that the distribution is preserved under the \(G\)-action \[(r_{g})_{\ast}(\mathcal{H}^{A}_{p})= \mathcal{H}^{A}_{p.g}\]

For every \(p \in P\),we will denote by \(j^{A}_{p}\) the projection map \[\begin{aligned} TP_{p} \rightarrow VTP_{p} \end{aligned}\] with kernel \(\mathcal{H}^{A}_{p}\).Then, given \(g \in G\), we have \[\begin{aligned} j^{A}_{pg} \circ (r_{g})_{\ast} = (r_{g})_{\ast} \circ j^{A}_{p} \end{aligned}\]Given a connection on a \(G\)-bundle \(\pi: P \rightarrow X\), we can lift smooth paths on \(X\) to smooth paths on \(P\). Suppose \(\gamma : [ 0,1] \rightarrow X\) , and \(p \in P_{\gamma(0)}\).There is a unique smooth path \(\tilde{\gamma} : [ 0,1] \rightarrow X\) such that \(\tilde{\gamma}(0)=p\) and \(\tilde{\gamma}^{\prime}(t) \in \mathcal{H}_{\tilde{ \gamma}(t)}\) for all \(t \in [ 0,1]\).Hence,a connection gives us a notion of parallel transport.

A connection can also be defined via a 1-form. The Maurer-Cartan form \(\omega_{mc}\) is a 1-form on \(G\) with values in \(\mathfrak{g}\) defined by : \[\omega_{mc}(v)= (L_{g^{-1}})_{\ast}(v)\] where \(v \in T_{g}G\) and \(L_{g^{-1}}\) is left translation by \(g^{-1}\).

Lemma 1. A connection on a smooth principal \(G\)-bundle \(\pi: P \rightarrow X\) is equivalent to a 1-form \(\omega \in \Omega^{1}(P;\mathfrak{g})\) having the following properties:

\(\omega_{pg}((r_{g})_{\ast}( v ))=ad_{g^{-1}}( \omega_{p}(v))\)
Given any \(p \in P\),\(( \iota_{p} )^{\ast}\omega=\omega_{mc}\)

Proof:Suppose we are given a connection \(A\).For every \(p \in P\) , we set \[\omega_p=(\iota_{p})^{-1}_{\ast}\circ j^{A}_{p}: TP_{p} \rightarrow \mathfrak{g}\] \(\omega\) is clearly smooth.Given \(g \in G\) and \(v \in T_{p}P\), we have \[\begin{aligned} \omega_{pg}((r_{g})_{\ast}( v ))&= (\iota_{pg})^{-1}_{\ast}\circ j^{A}_{pg}((r_{g})_{\ast}( v))\\ &= ( L_{g^-1})_{\ast}\circ (\iota_{p})^{-1} \circ (r_{g})_{\ast} \circ j^{A}_{p}(v)\\ &= ( ( L_{g^-1})_{\ast}\circ (\iota_{p})^{-1} \circ (r_{g})_{\ast})\circ j^{A}_{p}(v)\\ &= (ad_{g^{-1}}\circ (\iota_{p})^{-1})\circ j^{A}_{p}(v)\\ &= ad_{g^{-1}}( \omega_{p}(v)) \end{aligned}\] Hence, the first condition is satisfied and ,by doing another simple computation, we can show that \(\omega\) also has the second property.

Conversely, suppose \(\omega \in \Omega^{1}(P;\mathfrak{g})\) has the properties mentioned above. Define a connection \(A\) by setting \(\mathcal{H}^{A}_{p}\) to be the kernel of the map \(\omega_p: TP_{p} \rightarrow \mathfrak{g}\) for every \(p \in P\).The second property implies that \(\omega_{p}\) induces an isomorphism \(VTP_{p} \simeq \mathfrak{g}\).Hence,\(\mathcal{H}^{A}\) is indeed a smooth distribution such that \(\pi_{\ast}: \mathcal{H}^{A}_{p} \rightarrow T_{x}X\) is an isomorphism for every \(p\).The condition\((r_{g})_{\ast}(\mathcal{H}^{A}_{p})= \mathcal{H}^{A}_{p.g}\) follows form the first property of \(\omega\).

Covariant derivatives.Finally, a connection \(A\) on \(\pi: P \rightarrow X\) induces a connection on the associated vector bundle \(ad(P)\). \[\nabla^{A} :C^{\infty}(X,ad(P)) \rightarrow C^{\infty}(X,ad(P) \otimes T^{\ast}X )\] It is defined as follows: Suppose \(v\) is a vector field on \(X\) and \(s: X \rightarrow ad(P)\) is a section of \(ad(P)\).Let \(\tilde{v}\) be the vector field on \(P\) defined by \(\pi_{\ast}(\tilde{v})= v\) and \(\tilde{v}_{p} \in \mathcal{H}^{A}_{p}\) for all \(p \in P\). We also define the vector field \(\tilde{s}\) on \(P\) by \(\tilde{s}_{p}=( \iota_{p})_{\ast}(\varepsilon)\) where \(s(\pi(p))= [p,\varepsilon]\).Hence \(\tilde{s}_{p} \in VTP_P\) for evey \(p \in P\). For \(x \in X\), we set \[\nabla^{A}_{v}(s)_{(x)}=\{p,(\iota_{p})^{-1}_{\ast} ( [\tilde{v},\tilde{s}]_{p})\}\] where \(p \in P_{x}\).Note that \[\pi_{\ast} [\tilde{v},\tilde{s}]= [\pi_{\ast}(\tilde{v}),\pi_{\ast}(\tilde{s})]=[v,0]=0\] Hence \([\tilde{v},\tilde{s}] \in VTP \simeq \mathfrak{g}\). Also \([\tilde{v},\tilde{s}]_{pg}=ad_{g^{-1}}( [\tilde{v},\tilde{s}]_{p})\).So \(\nabla^{A}_{v}(x)\) is well-defined. Finally, if \(h,f \in C^{\infty}(X)\), we have : \[[\widetilde{fv},\tilde{s}]=\pi^{\ast}f [\tilde{v},\tilde{s}]-\tilde{s}(\pi^{\ast}f)\tilde{v}=f[\tilde{v},\tilde{s}]-\pi_{\ast}(\tilde{s})(f)\tilde{v}=f[\tilde{v},\tilde{s}]\] and \[[\tilde{v},\tilde{hs}]=h[\tilde{v},\tilde{s}]+\tilde{v}(h)\tilde{s}\] This implies that \(\nabla^{A}_{v}\) is indeed a connection.( It is \(C^{\infty}\)- linear in \(v\) and satisfies the Leibniz rule).

We will denote by \(\mathcal{A}_P\) the space of connections on the bundle \(P \rightarrow X\).Any smooth principal \(G\) bundle has a connection.So \(\mathcal{A}_P\) is non-empty. Furthermore, one can show that \(\mathcal{A}_P\) is an affine space for the vector space \(C^{\infty}(X,ad(P)\otimes T^{\ast}X )= \Omega^{1}(X,ad(P))\).

The Gauge group

Definition 3. An automorphism of a smooth principal \(G\)-bundle \(P \rightarrow X\) is a smooth map \(u :P \rightarrow P\) such that \(u(p \cdot g)=u(p)\cdot g\) and \(\pi(u(p))=\pi(p)\) for all \(p\in P\) and \(g \in G\). The set of automorphisms of the \(G\)-bundle \(P \rightarrow X\) form a group called the gauge group and we will denote it by \(\mathcal{G}_{P}\).This group has a left action on \(P\).

Consider the fiber bundle \(Ad(P) \rightarrow X\) with fiber \(G\). The group structure on \(G\) induces a group structure on \(C^{\infty}(X,Ad(P))\),the space of smooth sections of \(Ad(P)\), via fiber-wise multiplication.Let \(\mathcal{M}(P,G)\) be the space of smooth maps \(\psi:P \rightarrow G\) such that \(\psi(p \cdot g)= g^{-1}\psi(p)g\) for all \(p\in P\) and \(g \in G\).The group structure on \(G\) alos induces a group sturucture on \(\mathcal{M}(P,G)\). Furthermore, \[\mathcal{G}_{P} \simeq \mathcal{M}(X,G) \simeq C^{\infty}(X,Ad(P))\] Indeed, suppose \(u:P \rightarrow P\) is an automorphism. Since u is a fiber-preserving map,there exists a unique smooth map \(\psi:P \rightarrow G\) such that \(u(p)=p\cdot \psi(p)\) for all \(p \in P\).The condition \(u(p \cdot g)=u(p)\cdot g\) implies that \(\psi(p \cdot g)= g^{-1}\psi(p)g\).

Conversely, given a map \(\psi:P \rightarrow G\) with the above property, we define \(u:P \rightarrow P\) by \(u(p)=p\cdot \psi(p)\).Hence, \(\mathcal{G}_{P} \simeq \mathcal{M}(X,G)\) as claimed. Finally, the group \(\mathcal{M}(X,G)\) can be natuarlly identified with \(C^{\infty}(X,Ad(P))\).

Induced action on the space of connections.The group \(\mathcal{G}_{P}\) acts on \(\mathcal{A}_P\)\[\begin{aligned} \mathcal{A}_P \times \mathcal{G}_{P} \rightarrow \mathcal{A}_P\\ (A,u) \rightarrow u\cdot A \end{aligned}\] If we view the connection \(A\) as a distribution, then \[\mathcal{H}^{u\cdot A}_{p}=(u_{\ast})^{-1}(\mathcal{H}^{ A}_{u(p)})\] for every \(p \in P\). If we represent \(A\) by a one-form \(\omega \in \Omega^{1}(P;\mathfrak{g})\), then \(u \cdot A\) is represented by \(u^{\ast}\omega\). Finally, in terms of the covariant derivative induced on ad(P), we have: \[\nabla^{A}(s)= \tilde{u}^{-1}(\nabla^{A}(\tilde{u}(s))\] where \(\tilde{u}: ad(P) \rightarrow ad(P)\) is the automorphism induced by u.

We will be studying the quotient space \[\mathcal{A}_{P}/ \mathcal{G}_{P}= \mathcal{B}_P\] We will show that \(\mathcal{B}_{P}\) is a Hausdorff space. In fact, certain completions of this space are smooth Banach or Hilbert orbifolds.