{"id":4056,"date":"2013-11-09T15:36:45","date_gmt":"2013-11-09T07:36:45","guid":{"rendered":"http:\/\/lttt.blog.ustc.edu.cn\/?p=4056"},"modified":"2013-11-09T15:36:45","modified_gmt":"2013-11-09T07:36:45","slug":"overview-of-moduli-spacesreview-of-g-bundles-and-connections","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=4056","title":{"rendered":"Overview of moduli spaces,review of G-bundles and connections"},"content":{"rendered":"

All the contents are from this wikisite<\/a>, which is aimed to have a E-version lecture notes of the seminar given by Prof. Mrowka<\/a>. All rights are reserved by the original wikisite<\/a>, any reprint should be indicate this.
\nMain Contributor<\/strong>:Christian<\/p>\n

\n

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:
\n
\n\\[
\n \\End(V)\/Iso(V) \\simeq Mat_{n \\times n}( \\mathbb{C})\/Gl_{n}(\\mathbb{C}) = \\mathcal{M}_{n}
\n\\] The Jordan canonical form gives a description of $\\mathcal{M}_{n}$ as a set.
\nIn 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}$.<\/p>\n

In the next section, we list some basic notions of principal $G$-bundles which we will need for this class.<\/p>\n

\\section{G-bundles}<\/p>\n

\\begin{defn}<\/p>\n

Let $G$ be a Lie group.A (right) smooth principal $G$-bundle is a smooth fiber bundle $ \\pi: P \\rightarrow X$ such that:
\n\\begin{enumerate}
\n\\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
\n\\[
\n\\phi: \\pi^{-1}(U) \\rightarrow U \\times G
\n\\] such that:
\n\\begin{eqnarray*}
\n \\phi(p \\cdot g)&= \\phi(p) \\cdot g \\\\
\n \\pi(p) &= \\pi_{U}(\\phi(p))
\n\\end{eqnarray*}
\nfor 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.
\n\\end{enumerate}
\n\\end{defn}<\/p>\n

Given $ g \\in G, q \\in P$, we set
\n\\begin{eqnarray*}
\n r_{g}: P &\\rightarrow P \\\\
\n p &\\rightarrow p\\cdot g
\n\\end{eqnarray*}
\n and
\n\\begin{eqnarray*}
\n\\iota_{q}: P &\\rightarrow P \\\\
\n h &\\rightarrow q\\cdot h
\n\\end{eqnarray*}
\nThe map $\\iota_{q}$ induces an isomorphism
\n\\begin{eqnarray*}
\n (\\iota_{q})_{\\ast}:\\mathfrak{g}\\rightarrow T_{p}P_{x}
\n\\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 $p$).<\/p>\n

\\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
\n \\[
\n P \\times_{G}F = ( P \\times F) \/ \\sim
\n\\] 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<\/p>\n

\\begin{eqnarray*}
\n \\tau : P \\times_{G}F &\\rightarrow X \\\\
\n \\{p,y\\} & \\rightarrow \\pi(p)
\n\\end{eqnarray*}
\nwhere $\\{p,y\\}$ 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}<\/p>\n

\\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
\n \\[
\n TP_{p}= \\mathcal{H}^{A}_{p}\\oplus VTP_{p}
\n\\] Hence
\n \\[
\n \\pi_{\\ast}: \\mathcal{H}^{A}_{p} \\rightarrow T_{x}X
\n\\]
\n is an isomorphism.It is also required that the distribution is preserved under the $G$-action
\n \\[
\n (r_{g})_{\\ast}(\\mathcal{H}^{A}_{p})= \\mathcal{H}^{A}_{p.g}
\n\\]
\n \\end{defn}
\nFor every $p \\in P$,we will denote by $ j^{A}_{p}$ the projection map
\n\\begin{eqnarray*}
\n TP_{p} \\rightarrow VTP_{p}
\n\\end{eqnarray*} with kernel $ \\mathcal{H}^{A}_{p}$.Then, given $g \\in G $, we have \\begin{eqnarray*}
\n j^{A}_{pg} \\circ (r_{g})_{\\ast} = (r_{g})_{\\ast} \\circ j^{A}_{p}
\n\\end{eqnarray*}
\nGiven 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.<\/p>\n

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 :
\n\\[
\n \\omega_{mc}(v)= (L_{g^{-1}})_{\\ast}(v)
\n\\] where $v \\in T_{g}G $ and $L_{g^{-1}}$ is left translation by $g^{-1}$.<\/p>\n

\\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<\/p>\n

\\[
\n \\omega_p=(\\iota_{p})^{-1}_{\\ast}\\circ j^{A}_{p}: TP_{p} \\rightarrow \\mathfrak{g}
\n\\] $\\omega$ is clearly smooth.Given $g \\in G$ and $v \\in T_{p}P$, we have \\begin{eqnarray*}<\/p>\n

\\omega_{pg}((r_{g})_{\\ast}( v ))&=&(\\iota_{pg})^{-1}_{\\ast}\\circ j^{A}_{pg}((r_{g})_{\\ast}( v))\\\\
\n &=&( L_{g^-1})_{\\ast}\\circ (\\iota_{p})^{-1} \\circ (r_{g})_{\\ast} \\circ j^{A}_{p}(v)\\\\
\n &=&( ( L_{g^-1})_{\\ast}\\circ (\\iota_{p})^{-1} \\circ (r_{g})_{\\ast})\\circ j^{A}_{p}(v)\\\\
\n &=& (ad_{g^{-1}}\\circ (\\iota_{p})^{-1})\\circ j^{A}_{p}(v)\\\\
\n &=& ad_{g^{-1}}( \\omega_{p}(v))
\n\\end{eqnarray*}
\nHence, the first condition is satisfied and ,by doing another simple computation, we can show that $\\omega$ also has the second property.<\/p>\n

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 $\\omega$.<\/p>\n

Covariant derivatives.Finally, a connection $A$ on $ \\pi: P \\rightarrow X$ induces a connection on the associated vector bundle $ad(P) $.<\/p>\n

\\[
\n \\nabla^{A} :C^{\\infty}(X,ad(P)) \\rightarrow C^{\\infty}(X,ad(P) \\otimes T^{\\ast}X )
\n\\] 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
\n \\[
\n \\nabla^{A}_{v}(s)_{(x)}=\\{p,(\\iota_{p})^{-1}_{\\ast} ( [\\tilde{v},\\tilde{s}]_{p})\\}
\n\\] where $p \\in P_{x}$.Note that<\/p>\n

\\[
\n\\pi_{\\ast} [\\tilde{v},\\tilde{s}]= [\\pi_{\\ast}(\\tilde{v}),\\pi_{\\ast}(\\tilde{s})]=[v,0]=0
\n\\]
\nHence $[\\tilde{v},\\tilde{s}] \\in VTP \\simeq \\mathfrak{g}$. 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 :
\n\\[
\n[\\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}]
\n\\]
\nand
\n\\[
\n[\\tilde{v},\\tilde{hs}]=h[\\tilde{v},\\tilde{s}]+\\tilde{v}(h)\\tilde{s}
\n\\]
\nThis implies that $ \\nabla^{A}_{v}$ is indeed a connection.( It is $C^{\\infty}$- linear in $v$ and satisfies the Leibniz rule).<\/p>\n

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 $C^{\\infty}(X,ad(P)\\otimes T^{\\ast}X )= \\Omega^{1}(X,ad(P))$.<\/p>\n

\\subsection{ The Gauge group}
\n\\begin{defn} An automorphism of a smooth principal $G$-bundle $P \\rightarrow X$ is a smooth map $ u \ud83d\ude1b \\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 $P$.
\n\\end{defn}<\/p>\n

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,
\n\\[
\n \\mathcal{G}_{P} \\simeq \\mathcal{M}(X,G) \\simeq C^{\\infty}(X,Ad(P))
\n\\]
\nIndeed, 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 $\\psi(p \\cdot g)= g^{-1}\\psi(p)g$.<\/p>\n

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 $C^{\\infty}(X,Ad(P))$.<\/p>\n

Induced action on the space of connections.The group $ \\mathcal{G}_{P} $ acts on $\\mathcal{A}_P$
\n\\begin{eqnarray*}
\n \\mathcal{A}_P \\times \\mathcal{G}_{P} \\rightarrow \\mathcal{A}_P\\\\
\n (A,u) \\rightarrow u\\cdot A
\n\\end{eqnarray*}
\nIf we view the connection $A$ as a distribution, then
\n\\[
\n \\mathcal{H}^{u\\cdot A}_{p}=(u_{\\ast})^{-1}(\\mathcal{H}^{ A}_{u(p)})
\n\\]
\nfor 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:
\n\\[
\n\\nabla^{A}(s)= \\tilde{u}^{-1}(\\nabla^{A}(\\tilde{u}(s))
\n\\]
\n where $\\tilde{u}: ad(P) \\rightarrow ad(P) $ is the automorphism induced by u.<\/p>\n

We will be studying the quotient space
\n\\[
\n \\mathcal{A}_{P}\/ \\mathcal{G}_{P}= \\mathcal{B}_P
\n \\]
\n We will show that $ \\mathcal{B}_{P}$ is a Hausdorff space. In fact, certain completions of this space are smooth Banach or Hilbert orbifolds.<\/p>\n","protected":false},"excerpt":{"rendered":"

All the contents are from this wikisite, …<\/p>\n

Overview of moduli spaces,review of G-bundles and connections<\/span> \u9605\u8bfb\u66f4\u591a »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[16],"tags":[],"class_list":["post-4056","post","type-post","status-publish","format-standard","hentry","category-gauge-group-and-4-dim-topology"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/4056","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4056"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/4056\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4056"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4056"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4056"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}