# 09-03 Notes

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*
\section{ 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 $ \R^{3}$.

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

\section{G-bundles}

\begin{defn}

Let $G$ be a Lie group.A (right) smooth principal $G$-bundle is a smooth fiber bundle $ \pi: P \rightarrow X$ such that: \begin{enumerate}

\item There is a smooth free right action $ P \times G \rightarrow P $ with respect to which $\pi$ is invariant. \item 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{eqnarray*}

\phi(p \cdot g)= \phi(p) \cdot g \\ \pi(p) = \pi_{U}(\phi(p)) \end{eqnarray*}

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. \end{enumerate} \end{defn}

Given $ g \in G, q \in P$, we set \begin{eqnarray*}

r_{g}: P \rightarrow P \\ p \rightarrow p\cdot g

\end{eqnarray*} and \begin{eqnarray*} \iota_{q}: P \rightarrow P \\

h \rightarrow q\cdot h

\end{eqnarray*} The map $\iota_{q}$ induces an isomorphism \begin{eqnarray*}

(\iota_{q})_{\ast}:\mathfrak{g}\rightarrow T_{p}P_{x}

\end{eqnarray*} 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 $p$).

\subsection{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{eqnarray*} \tau : P \times_{G}F \rightarrow X \\ \{p,y\} \rightarrow \pi(p)

\end{eqnarray*} where $\{p,y\}$ is the equivalence class of $ (p,y) \in P \times F$.Here are some important examples of this construction. \begin{enumerate} \item 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)$. \item 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 $ad(P)$. \end{enumerate}

\subsection{Connections}
\begin{defn}
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}

\] \end{defn}For every $p \in P$,we will denote by $ j^{A}_{p}$ the projection map \begin{eqnarray*}

TP_{p} \rightarrow VTP_{p}

\end{eqnarray*} with kernel $ \mathcal{H}^{A}_{p}$.Then, given $g \in G $, we have \begin{eqnarray*}

j^{A}_{pg} \circ (r_{g})_{\ast} = (r_{g})_{\ast} \circ j^{A}_{p}

\end{eqnarray*}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 $g^{-1}$.

\begin{lem} 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: \begin{enumerate} \item $\omega_{pg}((r_{g})_{\ast}( v ))=ad_{g^{-1}}( \omega_{p}(v)) $ \item Given any $p \in P$,$( \iota_{p} )^{\ast}\omega=\omega_{mc}$ \end{enumerate} \end{lem} 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{eqnarray*}

\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{eqnarray*} 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 $C^{\infty}(X,ad(P)\otimes T^{\ast}X )= \Omega^{1}(X,ad(P))$.

\subsection{ The Gauge group} \begin{defn} 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 \textbf{gauge group} and we will denote it by $\mathcal{G}_{P}$.This group has a left action on $P$. \end{defn}

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{eqnarray*}

\mathcal{A}_P \times \mathcal{G}_{P} \rightarrow \mathcal{A}_P\\ (A,u) \rightarrow u\cdot A

\end{eqnarray*} 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.