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\begin{document}

\title{Basics of Seiberg-Witten theory}
\author{Tim Perutz\thanks{Comments and corrections gratefully received at tim.perutz@ic.ac.uk}}
\date{Imperial College Junior Geometry Seminar, May 2002}
\maketitle

The difficulty in summarising Seiberg-Witten theory is that 
the Seiberg-Witten equations are complicated. 
The data they involve---$\spinc$-structures, complex spinors, coupled Dirac 
operators---are themselves tricky to understand, never mind the 
analysis required to understand their solutions. 
But, while the equations appear awkward, the moduli spaces of 
Seiberg-Witten solutions modulo gauge are compact, oriented, 
typically zero-dimensional manifolds, and this makes Seiberg-Witten invariants considerably 
easier to compute than Donaldson's instanton invariants.
Witten's paper of November 1994 \cite{W} suppresses detailed calculations, but the
basic arguments are beautifully clear; it is this style, rather 
than the useful but impenetrable thicket of detail provided by Nicolaescu \cite{N}, 
for instance, that I would like to emulate in this lecture. I have also made considerable use 
of Donaldson's review \cite{Do}. For an elegant discussion of spinors consult \cite{De}.

\section{$\spinc$-structures}
The compact Lie group $\spinc(n)$ is defined to be
\begin{equation}
\spinc(n)=\frac{\mathrm{Spin}(n)\times \mathrm{U}(1)}{\pm(1,1)} 
\end{equation}
where $\mathrm{Spin}(n)$ is the unique Lie group which double covers $\mathrm{SO}(n)$,
realised as the following subset of the Clifford algebra $\mathrm{Cliff}(n)$ associated to
the negative of the Euclidean norm $\|\cdot\|$ on $\R^n$:
$$\mathrm{Spin}(n)=
	\{ x_1\cdots x_{2k} \in \mathrm{Cliff}(n): \; x_i\in\R^n;\; \left\| x_i\right\| =1 \}.$$ 
We have
\begin{align*}
	\spinc(3)&\cong(\mathrm{SU}(2)\times\mathrm{U}(1))/\pm(1,1)\cong\mathrm{U}(2);\\
	\spinc(4)&\cong(\mathrm{SU}(2)\times\mathrm{SU}(2)\times\mathrm{U}(1))/\pm(1,1,1)\\
	&\cong \{(A,B) \in \mathrm{U}(2)\times\mathrm{U}(2): \det(A)=\det(B) \}.
\end{align*}
A {\bf Spin}$^c${\bf -structure} on an oriented Riemannian $n$-manifold $(X,g)$ is a lift of 
the structure group from $\mathrm{SO}(n)$ to $\spinc(n)$ under the natural $S^1$-fibration  
$\pi:\spinc(n)\to \mathrm{SO}(n)$. 
Thus if $\{U_\alpha\}$ is a cover of $X$, all of whose
multiple intersections are contractible, then a $\spinc$-structure is encoded by a \v{C}ech
1-cocycle $\tilde{g}_{\alpha\beta}: U_\alpha\cap U_\beta \to \spinc(n)$ such that the maps
$g_{\alpha\beta}=\pi\circ \tilde{g}_{\alpha\beta}$ are the transition functions for $(TX,g)$.
The lifted transition functions define a principal bundle $P_{\spinc(2n)}$ and a homomorphism 
of principal bundles $P_{\spinc(2n)}\to P_{\mathrm{SO}(2n)}$ which restricts fibrewise 
to the covering map $\spinc(2n)\to\mathrm{SO}(2n)$.
\begin{itemize}
\item
A {\bf Spin-structure}, that is, a lift of the structure group to $\mathrm{Spin}(n)$, 
induces a $\spinc$-structure via the map 
$$\mathrm{Spin}(n)\to\spinc(n): \quad x \to \pm(x,1). $$ 
\item
An almost complex structure---a lift of the structure group from $\mathrm{SO}(2k)$ to 
$\mathrm{U}(k)$---also yields a canonical $\spinc$-structure, because the natural map
$\mathrm{U}(k)\to\mathrm{SO}(2k)$ factors through $\spinc(2k)$. 
\item
Hirzebruch and Hopf proved that any smooth oriented manifold of dimension 
$\leq 4$ admits a $\spinc$-structure (see \cite{GS});
in contrast, a smooth oriented 4-manifold admits a $\mathrm{Spin}$-structure iff its 
intersection form is even, iff $w_2=0$.
\end{itemize}
Let $\mathfrak{s}$ be a $\spinc$-structure with 1-cocycle 
$\tilde{g}_{\alpha\beta}=(\pm h_{\alpha\beta},\pm\zeta_{\alpha\beta})$. Then 
$z_{\alpha\beta}:=(\pm\zeta_{\alpha\beta})^2$ defines a $\mathrm{U}(1)$-valued
1-cocycle, and hence a complex line bundle $L$. 
We call $L$ the {\bf determinant line bundle} 
$\det(\mathfrak{s})$ of the $\spinc$-structure $\mathfrak{s}$. Write $c_1(\mathfrak{s})$
for its {\bf first Chern class} $c_1(L)\in H^2(X;\Z)$, which is represented by the 
$\Z$-valued 2-cocycle 
$n_{\alpha\beta\gamma}=\frac{1}{2\pi i}\log (z_{\alpha\beta}z_{\beta\gamma}z_{\beta\gamma})$; 
it reduces mod 2 to $w_2(X)$.

We regard $\spinc$-structures as {\bf equivalent} if their associated 1-cocycles 
$\tilde{g}_{\alpha\beta}$ and $\tilde{h}_{\alpha\beta}$ are cohomologous, in the sense that 
$\epsilon_\beta \tilde{g}_{\alpha\beta}=\tilde{h}_{\alpha\beta}\epsilon_{\alpha}$ 
for a 0-cocycle $\epsilon_\alpha: U_\alpha \to (1,\pm1)\subset \spinc(n)$, 
and write $\spinc(X)$ for the set of equivalence classes.
There is a free, transitive right action of the group $\check{H}^1(X;S_1)$ on $\spinc(X)$.
Thus, if $\spinc(X)\neq\emptyset$ then we can identify it \emph{non-canonically} with 
$\check{H}^1(X;S^1)$, and hence with the group of smooth line bundles. Given 
$\mathfrak{s}_1,\mathfrak{s}_2\in \spinc(X)$ we can write 
$\mathfrak{s}_2=\mathfrak{s}_1\otimes L$ for a line bundle $L$, so that 
$\det \mathfrak{s}_2=\det \mathfrak{s}_1\otimes L^2$ and crucially 
\begin{equation}
	c_1(\mathfrak{s}_2)=c_1(\mathfrak{s}_1)+2c_1(L).
\end{equation}
If $\mathfrak{s}$ is the \emph{canonical} $\spinc$-structure of an almost complex structure, 
then $\det(\mathfrak{s})$ is $K_X^{-1}$, the dual of the canonical line bundle, and we can write a
general $\spinc$-structure as $K_X^{-1}\otimes L$.

\section{Spinor bundles}
A basic fact about real Clifford algebras is that there is a $\Z/2$-graded $\C$-vector space
$\s_{2n}=\splus_{2n}\oplus \sminus_{2n}$ and an isomorphism
of $\Z/2$-graded (or `super') algebras
$$ \cl: \mathrm{Cliff}(2n) \otimes \C \to \mathrm{End}(\s_{2n}). $$
In fact, a choice of almost complex structure $J$ on $V=\R^{2n}\otimes \C$ effects a splitting
$V=V^{1,0}\oplus V^{0,1}$ of $V$ into $\pm i$-eigenspaces for $J$, and 
we can take 
$$\s_{2n}=\Lambda^{0,\,\bullet}V=\Lambda^\bullet V^{0,1},$$ 
with the $\Z/2$-grading induced by the natural $\Z$-grading
by degree. In particular, $\dim_\C \splus_{2n}=\dim_\C\sminus_{2n}=2^{n-1}$. 
Via the inclusion map
$i:\mathrm{Spin}(2n)\hookrightarrow \mathrm{Cliff}^+(2n)$ we obtain a representation
$$\rho: \spinc(2n)\to \mathrm{Aut}(\s_{2n}): \quad \pm(x,z)\mapsto z\,\cl(i(x)) $$ 
which decomposes into representations 
$\rho_\pm: \spinc(2n)\to\mathrm{Aut}(\s_{2n}^\pm)$. 

Consider an oriented Riemannian $2n$-manifold $(X,g)$ and a $\spinc$-structure with
principal bundle $P_{\spinc(2n)}$. Form the associated bundles
$$ \s_{\mathfrak{s}} = P_{\spinc(2n)}\times_{\rho}\s_{2n}, \quad 
	\s^\pm_{\mathfrak{s}}=P_{\spinc(2n)}\times_{\rho_{\pm}}\s_{2n}^\pm. $$
Sections of these bundles are called (positive or negative) {\bf complex spinors}. 

On a 4-manifold $(X,g)$ we can be more explicit. Working over a chart $U$, we have
\begin{equation} \label{splus}
\splus_{\mathfrak{s}}(U)=\Lambda^{0}(T^*U) \oplus \Lambda^{0,2}(T^*U),
\quad \sminus_{\mathfrak{s}}(U)=\Lambda^{1,0}(T^*U).
\end{equation}
Note first that
$$ \Lambda^2 \splus_{\mathfrak{s}} \cong \det(\mathfrak{s}) 
\cong \Lambda^2 \sminus_{\mathfrak{s}}, $$
an observation which it is enough to verify locally. Next, we can use the metric
to identify $\Lambda^{0,2}$ with the bundle $\Lambda^2_+$ of complexified self-dual 2-forms:
\begin{equation}\label{splus2}
	\splus_{\mathfrak{s}}=\Lambda^{0}(T^*U) \oplus \Lambda^2_+(T^*U).
\end{equation}
Thus locally a positive spinor decomposes into two pieces, a function $X\to \C$ 
and a complexified self-dual 2-form. In the case of the canonical $\spinc$-structure of an
almost complex 4-manifold formulae (\ref{splus}) and (\ref{splus}) hold globally, 
but in general we must twist by sections of a complex line bundle.

\section{Dirac operators}
Let $\nabla$ denote the Levi-Civita connection on $(X^{2n},g)$, 
regarded as a connection on the principal frame bundle $P_{\mathrm{SO}(2n)}$. 
Given a $\mathrm{Spin}$-structure on $X$ we can identify the Lie algebras of $\mathrm{Spin}(2n)$ 
and $\mathrm{SO}(2n)$ and so 
obtain a connection on  $P_{\mathrm{Spin}(2n)}$, hence also on the associated spinor bundles. 
If we only have a $\spinc$-structure $\mathfrak{s}$ then we need a bit more information, since  
$\spinc(2n)$ has Lie algebra $\mathfrak{so}(2n)\oplus\mathfrak{u}(1)$. 
However, if we take in addition a unitary connection $A$ on $\det(\mathfrak{s})$ 
then we do obtain a connection $\nabla^A$
on the complex spinor bundle $\s_{\mathfrak{s}}$, which respects the $\Z/2$-decomposition.
Locally this connection looks like
\begin{equation}\label{nablaA}
 \nabla^A = d++\frac{1}{2}iA+\frac{1}{2}\sum_{i<j}{\Omega_{ij}e_i e_j}, 
\end{equation}
where the $e_i$ are an orthonormal frame field for $TX$, and they act by Clifford 
multiplication, explained in a moment.
The {\bf Dirac operator} $ D_A: \Gamma(\s_{\mathfrak{s}}) \to \Gamma(\s_{\mathfrak{s}})$ is now defined by
$$ D_A s = \sum{e_i(\nabla^A_{e_i} s)}. $$
This needs some thinking through: $\nabla^A s$ lives in $\Gamma(T^*X \otimes \s_{\mathfrak{s}})$,
and the covariant derivative $\nabla^A_{e_i}s$ contracts this to a section of $\s_{\mathfrak{s}}$.
Now over each point $x\in X$, $e_i$ can be considered as an element of $\mathrm{Cliff}(2n)$, 
hence as an endomorphism of $\s$; thus $e_i(\nabla^A_{e_i} u)\in \Gamma(\s_{\mathfrak{s}})$. 
The jargon is that $e_i$ acts by {\bf Clifford multiplication}
$\cl: TM\otimes \s_{\mathfrak{s}}\to\s_{\mathfrak{s}}$. 
A coordinate-free expression for $D_A$ is
\begin{align*} D_A=\cl\circ\nabla_A: \; \Gamma(\s_{\mathfrak{s}})
			& \stackrel{\nabla_A}{\longrightarrow}\Gamma(T^*M\otimes\s_{\mathfrak{s}})\\
			& \stackrel{\text{metric}}{\longrightarrow}\Gamma(TM\otimes\s_{\mathfrak{s}})\\
			& \stackrel{\cl}{\longrightarrow}\Gamma(\s_{\mathfrak{s}}). 
\end{align*}
The Dirac operator satisfies a Weitzenb\"ock identity due to Lichnierowicz 
(`one of the most fruitful calculations in differential geometry'---SKD): 
\begin{equation} \label{weitz}
D_A^* D_A = (\nabla^A)^*\nabla^A+\frac{s}{4}+\frac{1}{2}\cl(F_A^+),
\end{equation}
where $s$ is the scalar curvature of $g$, and the $i\R$-valued curvature 2-form $F_A$ can act
by Clifford multiplication by identifying $\frac{1}{2}x\wedge y$ with the Clifford product $xy$.
One can check that $D_A$ exchanges sections of the subbundles $\s^\pm_{\mathfrak{s}}$; in fact, 
usually the notation $D_A$ is used for the map 
$\Gamma(\splus_{\mathfrak{s}})\to \Gamma(\sminus_{\mathfrak{s}})$.
I remark without elaboration that $D_A$ is a generalised Laplacian, that is
its symbol map (re)defines a Clifford algebra stucture.
%\newpage
\section{The Seiberg-Witten equations}
Fix the following data:
\begin{itemize}
\item		A smooth, oriented\footnote{I mean orient\emph{ed}, not just 
		orient\emph{able}, i.e. we fix an isomorphism $H^4(X;\Z)\to \Z$.}, 
		Riemannian 4-manifold $(X,g)$; we won't assume compactness yet. 
\item		A $\spinc$-stucture $\mathfrak{s}$ on $X$, with determinant
		line bundle $L=\det(\mathfrak{s})$.
\item		A hermitian metric $h$ on $L$.
\item		A closed, imaginary 2-form $\eta$, the {\bf perturbation parameter}.
\end{itemize}
The $\spinc$-structure $\mathfrak{s}$ defines complex spinor bundles $\s_{\mathfrak{s}}^\pm$.
Define the {\bf configuration space} $\mathcal{A}_{\mathfrak{s}}$ to be the
set of pairs $(\psi, A)$ where 
\begin{itemize}
\item		$\psi$ is a section of $\splus_{\mathfrak{s}}$ 
		(a `spinor field' on $X$);
\item		$A$ is a unitary connection on $L$ (a `potential').
\end{itemize}
For each such $A$ there is a Dirac operator 
$D_A: \Gamma(\splus_{\mathfrak{s}})\to\Gamma(\sminus_{\mathfrak{s}})$.
The {\bf Seiberg-Witten equations} are:

\vspace{5mm}
\noindent
\fbox{%
\begin{minipage}{138mm}
\vspace{-\abovedisplayskip}
\begin{align*} 
& D_A \psi =0  && \text{(Dirac equation)} \\
&\cl(F_A^+ +\eta^+)=\frac{1}{2}q(\psi)  && \text{(monopole equation)}
\end{align*}
\end{minipage}}
\vspace{5mm}

The Dirac equation takes place in $\Gamma(\sminus_{\mathfrak{s}})$, the monopole equation in 
$\mathrm{End}(\splus_{\mathfrak{s}})$.
Here $q(\psi)\in \mathrm{End}(\splus_{\mathfrak{s}})$ is the traceless symmetric endomorphism
$$ q(\psi) = \overline{\psi}\otimes\psi-\frac{1}{2}|\psi|^2 \mathrm{Id},$$
or in matrix form
$$ q\left	(\begin{array}{c}
			\alpha\\
			\beta
		\end{array}\right)=
\left(\begin{array}{cc}
		\frac{1}{2}(|\alpha|^2-|\beta|^2)	&	\overline{\alpha}{\beta}	\\
		\overline{\beta}\alpha			&	\frac{1}{2}(|\beta|^2-|\alpha|^2)
	\end{array}\right).$$
\subsection{Gauge symmetry}
The Seiberg-Witten equations have more symmetry than meets the eye: they are 
preserved by a (left) action of the $\mathrm{U}(1)$ 
{\bf gauge group} $ \mathcal{G}=\mathrm{Map}(X,\mathrm{U}(1))$. 
Let's try to understand how this group acts.

First, $\mathrm{U}(1)$ acts by scalar multiplication on the vector spaces $\s^\pm_4$, 
so (from their definition as associated bundles) we get a fibre-preserving action of 
$\mathcal{G}$ on $\s^\pm_{\mathfrak{s}}$. Hence $\mathcal{G}$ acts on sections 
$\Gamma(\s^\pm_{\mathfrak{s}})$. Now things get more subtle: $\mathcal{G}$ naturally 
acts by conjugation on the set of connections on $\s_\mathfrak{s}$; 
given our unitary connection $A$ on $L$, we obtain a covariant derivative $\nabla^A$ on 
$\s_\mathfrak{s}$, and $\gamma\cdot \nabla^A:=\gamma \nabla^A \gamma^{-1}$. 
But this connection is also induced from a unitary connection on $L$, namely 
$A-2(d\gamma)\gamma^{-1}$. The factor of 2 looks strange if you're used to Yang-Mills 
theory---to see where it comes from look at equation \ref{nablaA}. 
Finally, we have a formula for the action on configuration 
space $\mathcal{A}_{\mathfrak{s}}$:
$$ \gamma \cdot (\psi,A)= (\gamma \psi, A-2d\gamma.\gamma^{-1}); $$  
using this formula you can deduce the invariance of the Seiberg-Witten equations.
The {\bf Seiberg-Witten moduli space} associated to our data is defined to be
$$ \mathcal{M}_{\mathfrak{s}}=\{\text{Seiberg-Witten solutions}\}/\mathcal{G}
\subset \mathcal{A}_{\mathfrak{s}}/\mathcal{G}. $$
\subsection{Involution}
There is an additional symmetry of the equations, this time beween solutions with perturbation 
parameter $\eta$ and those with parameter $-\eta$. It is induced by the involution 
$(h_{\alpha\beta},z_{\alpha\beta})\mapsto(h_{\alpha\beta},\overline{z_{\alpha\beta}})$ 
on $\spinc(X)$.
%\newpage
\section{Properties of the equations}
Suppose $(\psi,A)$ is an $L^2$-solution to the Seiberg-Witten equations on $X$ (e.g. any 
solution when $X$ is compact). By the Weitzenb\"ock formula (\ref{weitz}),
$$ 0 = D_A^* D_A \psi = (\nabla^A)^*\nabla^A\psi +\frac{s}{4}\psi+\frac{1}{2}F_A^+(\psi) $$
(recall $s$ is scalar curvature).
Taking inner products with $\psi$, integrating, and simplifying using the monopole equation,
it's straightforward to show
\begin{equation}\label{scal}
 \int_X{\left( 
	|\nabla^A \psi|^2+\frac{1}{4}|\psi|^4-(\eta^+(\psi),\psi)
 \right)\mathrm{vol}_g}=-\int_X{\frac{s}{4}|\psi|^2 \mathrm{vol}_g}.
\end{equation}
\subsection{Non-negative scalar curvature}
Suppose $s\geq 0$ everywhere, and take $\eta=0$. Then the LHS of (\ref{scal}) is non-negative 
while the RHS is non-positive, hence both are zero. So $\psi\equiv 0$ and $F_A^+ \equiv 0$. 
We will discuss solutions of 
this form later; the point is that, when $b_2^+ >0$, for generic metrics they do not exist.
We will conclude from this that on a closed 4-manifold with 
$b_2^+>0$ admitting a metric of non-negative scalar curvature the Seiberg-Witten invariants
vanish identically. Both this statement and its contrapositive are useful in 4-manifold 
topology.
\subsection{Finiteness}
Now suppose $s_0:=-\min_X(s)\geq 0$ on the compact manifold $X$. 
Again, take a solution $(\psi,A)$ to the $\eta=0$ equations. By (\ref{scal}) we have
$$ \int_X{(|\nabla^A\psi|^2 + \frac{1}{4}|\psi|^4) \mathrm{vol}_g}
	\leq \frac{s_0}{4} \int_X{|\psi|^2\mathrm{vol}_g}
	\leq \frac{s_0}{4}\sqrt{\mathrm{Vol}(X)\int_X{|\psi^4|\mathrm{vol}_g}} $$
where the second inequality is an application of Cauchy-Schwartz. Throw away the 
$|\nabla^A\psi|^2$ term and rearrange, to obtain the bound
$$ \int_X{|\psi|^4}\leq s_0^2\mathrm{Vol}(X). $$
This in turn yields an $L^2$ upper bound on $F_A^+$, and since by Chern-Weil theory we have 
$$ c_1^2(\mathfrak{s})[X]=\frac{1}{4\pi^2}\int_X{F_A^2}
=\frac{1}{4\pi^2}\int_X{(|F_A^+|^2-|F_A^-|^2)\mathrm{vol}_g}, $$
$c_1^2(\mathfrak{s})[X]$ is bounded above. 
We will see later from an index calculation (\ref{index}) that
for generic metrics there are no solutions when 
$c_1^2(\mathfrak{s})[X]<2\chi(X)+3\tau(X)$, where $\chi$ and $\tau$ are Euler 
characteristic and signature. So $c_1^2(\mathfrak{s})$ is bounded in $H^4(X;\Z)$, hence 
$c_1(\mathfrak{s})$ is bounded in $H^2(X;\Z)$, hence there are only finitely many 
$\spinc$-structures on $X$
for which the $\eta=0$ Seiberg-Witten equations admit solutions.

\section{The gauge theory strategy}
In gauge theory one considers the space $\mathcal{M}_{X,E}(g)$ 
of solutions modulo gauge transformations to a 
PDE on a vector bundle $E$ over a smooth oriented manifold $X$. Solutions to the PDE are 
invariant under a group of bundle automorphisms, the {\bf gauge group}.
We are interested just in the
orientation-preserving diffeomorphism type of $X$; however, the PDE will
involve some auxiliary data $g$ (metrics on $X$ and $E$, perturbation parameters, \dots) 
which are not preserved by 
orientation-preserving diffeomorphisms. The idea, then, is this:
\begin{itemize}
\item If the gauge group acts freely, the configuration space divided by the gauge group action
	will be a smooth Banach manifold, containing the moduli space $\mathcal{M}(g)$. 
	There may be {\bf reducible solutions}, which have nontrivial stabilizer. 
	We must arrange that for generic\footnote{A property holds generically in a metric space 
	if it holds on a countable intersection of open, dense subsets, itself dense by
	Baire's theorem.} 
	data $g$ these reducibles do not hit $\mathcal{M}(g)$.
	More than this, we need to show that reducibles do not occur in {\bf generic 1-parameter
	families} of data $g$. This will allow us to construct `cobordisms' between moduli
	spaces $\mathcal{M}(g_1)$ and $\mathcal{M}(g_2)$ in such a way as to show that 
	$\mathcal{M}(g)$ does not depend on $g$.
\item If the equations are {\bf elliptic} then away from reducibles their linearizations 
	will define Fredholm maps
	between Banach manifolds. For generic choices of the data $g$, 
	the moduli space $\mathcal{M}(g)$ will then be a smooth
	finite-dimensional manifold (Sard-Smale theorem); reducible solutions
	would introduce singularities. The dimension of $\mathcal{M}(g)$, for generic $g$, is 
	the {\bf index} of the equations; this can be computed using the Atiyah-Singer 
	index theorem.
\item In some cases the moduli spaces will be compact; if not, one must construct some 
	appropriate compactification, typically as a completion in the ambient space. The
	`ends' of the compactification often have a natural geometric description.
\item One must define an orientation for the moduli space. Writing $T$ for the linearization 
	of the operators defining the PDE, 
	$\mathrm{ker} (T) \oplus (\mathrm{coker}(T))^*$ defines an element of the real K-theory on $X$, the 
	{\bf index bundle} $\mathrm{ind} (T)$; 
	its top exterior power is actually a real line bundle $\det \mathrm{ind}(T)$, 
	and Donaldson gave a recipe for producing an orientation of the moduli space from an 
	orientation of $\det\mathrm{ind} (T)$. 
\end{itemize}
This list is to some extent an idealisation; one must consider variants, such as isolated 
reducibles appearing in generic 1-parameter families; and to get the arguments to work one 
must usually allow solutions in certain Sobolev spaces, then use a `bootstrapping' argument
to show that this doesn't affect the topology of the moduli space.
However, it does represent the basic strategy used in a number of problems---one of course, 
being Seiberg-Witten theory:
\begin{itemize}
\item		Instantons (Donaldson theory);
\item		Pseudo-holomorphic curves in symplectic manifolds 
		(Gromov, Gromov-Witten invariants);
\item		Floer homology (Lagrangian, instanton, Seiberg-Witten, etc.) and 	3-manifold 
		invariants (Casson, etc.).
\end{itemize}
The moduli spaces in Donaldson or Seiberg-Witten theory can be used in two basic ways. First,
to give non-existence proofs, e.g. for smooth, closed, oriented 4-manifolds with negative
definite, non-diagonal intersection form, by deriving a contradiction from the topology of
the moduli space. Second, to define invariants of such manifolds. The strategy is then
\begin{itemize}
\item		If the moduli space $\mathcal{M}$ is a 0-dimensional compact manifold then it is a 
		finite set;
		one counts the points with signs obtained by comparing the canonical orientation 
		of a finite set with the orientation constructed on $\mathcal{M}$,
		and obtains an invariant
		$$  \sum_{x\in\mathcal{M}}{\varepsilon_x},\quad \varepsilon_x=\pm 1 . $$
\item		In higher dimensions, construct a map $\mu: H_2(X;\Z) \to H^2(\mathcal{M};\Z)$.
		If $\mathcal{M}$ has even dimension $2d$ then define invariants as
		$$ \langle \mu([\Sigma_1])\cup\dots\cup\mu([\Sigma_d]),[\mathcal{M}] \rangle. $$ 
		We will only discuss a cheap version of this, using a fixed class 
		$U\in H^2(\mathcal{M};\Z)$.
\end{itemize}
\section{The Seiberg-Witten case}
{\bf Reducibles.} If the gauge group $\mathcal{G}$ fails to act freely on $(\psi,A)$
then clearly $\psi=0$; then $F_A^+ +\eta^+=0$, and the stabilizer of $(\psi,A)$ is 
$\mathrm{U}(1)$. Consider the unperturbed case $\eta=0$ (so we
have an `abelian instanton'). Now $F_A/2\pi i$ represnts $c_1(L)$, so it lies in the integral
lattice in $H^2(X;\Z)$, and $F_A^+=0$, so it lies in the anti-self-dual subspace. But if 
$b_2^+\geq 1$ then for generic metrics an integral, anti-self-dual harmonic form is zero,
and for $b_2^+\geq 2$ the same is true for generic 1-parameter families of metrics.
So to get absolute invariants we must assume $b_2^+\geq 2$, while for $b_2^+=1$ we will get
`chambers' in the space of metrics, giving different invariants.
Note how useful it was that we considered pairs $(\psi,A)$, not just connections $A$.

{\bf Index.} Witten shows that the linearization of the Seiberg-Witten equations is elliptic,
of index 
$$\mathrm{ind}(d^*\oplus d^+)+\mathrm{ind}(D_A),$$ 
where both terms are given by standard formulae.
The result is that the `expected' dimension of $\mathcal{M}_{\mathfrak{s}}$ is
\begin{equation}\label{index}
	\frac{1}{4}\{c_1(L)^2-(2\chi(X)+3\tau(X)\} 
\end{equation}
where $\chi$ and $\tau$ denote Euler characteristic and signature. If the index is negative 
the moduli space will generically be empty.
One feature of this formula is immediately striking: on an almost complex 4-manifold,
$c_1^2(TX)[X]=2\chi+3\tau$, so conveniently the expected dimension for the canonical 
$\spinc$-structure is zero.

{\bf Compactness.} The Seiberg-Witten moduli spaces are always compact, unlike the instanton 
moduli spaces.
Witten remarks that the lack of compactness in the instanton theory can be explained in the 
following way: the equations are conformally invariant; there exists a nontrivial $L^2$ instanton on 
flat $\R^4$; such a solution may be embedded into a small region of any 4-manifold, giving a
highly localised approximate solution; this may sometimes be perturbed to an exact solution.
There may therefore be a sequence of instanton solutions shrinking to zero size, and such a
sequence has no convergent subsequence (think of approximations to the delta distribution). 
In contrast, we saw in out discussion of non-negative scalar curvature that the Seiberg-Witten 
equations admit no nontrivial $L^2$ solutions on flat $\R^4$.
One can prove compactness by establishing an $L^\infty$ bound on the curvature $F_A$ using the 
maximum principle, then applying elliptic theory.

{\bf Orientation.} 
In Seiberg-Witten theory the orientation on $\det\mathrm{ind}(T)$ arisies from an orientation of the vector
space $H^2_+(X;\R)\oplus H^1(X;\R)$.

\subsection{Invariants}
In the 0-dimensional case we make a signed count of solutions, setting
$$ \mathbf{SW}_X(\mathfrak{s})= 
\sum_{x\in\mathcal{M}}{\varepsilon_x},\quad \varepsilon_x=\pm 1. $$
In higher dimensions we can use the pullback square
$$\begin{CD}
	i^*\mathbb{U_{\mathfrak{s}}} 		@>>> 		\mathbb{U_{\mathfrak{s}}}	\\
	@VVV						@VVV					\\
	\mathcal{M}_\mathfrak{s}@>{i}>> 	\mathcal{A}_{\mathfrak{s}}/\mathcal{G}
\end{CD}$$
where $\mathbb{U_{\mathfrak{s}}}
=\mathcal{A}_{\mathfrak{s}}/\mathrm{Stab_\mathcal{G}(\mathrm{pt})}$. 
Since $\mathbb{U_{\mathfrak{s}}}\to \mathcal{A}_{\mathfrak{s}}/\mathcal{G}$ is a line bundle away from
reducibles, we can take first Chern classes and so obtain an element 
$U=c_1(i^*\mathbb{U_{\mathfrak{s}}})\in H^2(\mathcal{M_\mathfrak{s}};\Z)$. Now define
$$\mathbf{SW}_X(\mathfrak{s})=\langle(1-U)^{-1},[\mathcal{M}_\mathfrak{s}]\rangle. $$
Note that this can be nonzero only when $\dim \mathcal{M}_\mathfrak{s}$ is even.
When $b_2^+>1$ $\mathbf{SW}_X(\mathfrak{s})$ is an invariant; when $b_2^+=1$ it takes two 
values, depending on $\eta$ and $g$, and we may consider it crudely 
as an unordered pair of integers.

A less sensitive but more readily computable invariant is
$$\mathrm{Bas}(X)=\{ c_1(\det(\mathfrak{s})): \mathfrak{s}\in\spinc(X); \; 
\mathbf{SW}_X(\mathfrak{s})\neq 0 \}\subset H^2(X;\Z),$$
the set of Seiberg-Witten {\bf basic classes}.
In all known cases where $b_2^+>1$, the basic classes correspond to 0-dimensional moduli 
spaces, but when $b_2^+=1$ this is frequently not the case. 

\section{Seiberg-Witten invariants of K\"ahler surfaces: Witten's factorization method}
In this section $(X,\omega)$ is a compact K\"ahler surface with canonical line bundle $K_X$.
We'll also assume $b_2^+ >1$ with respect to the canonical orientation $\omega^2>0$: 
since $b_2^+=2p_g+1$ we can then take as perturbation parameter 
a {\bf holomorphic 2-form} $\eta \in H^0(X;K_X)$.
As we've seen, the canonical $\spinc$-structure $\mathfrak{s}_0$ has determinant line bundle
$K_X^{-1}$, and spinor bundles
$$ \splus_0 = \Lambda^{0,0}T^*X\oplus\Lambda^{0,2}T^*X=\underline{\C}\oplus K_X^{-1},
\quad \sminus_0=\Lambda^{0,1}T^*X. $$
A general $\spinc$-structure can now be obtained by twisting: 
$\mathfrak{s}=\mathfrak{s}_0\otimes L$;
$\det(\mathfrak{s})=K_X^{-1}\otimes L^2$. 
The spinor bundles $\s^\pm_L:=\s^\pm_{\mathfrak{s}}$ are now
$$ \splus_L = L\oplus(L\otimes K_X^{-1}), \quad \sminus_L=L\otimes\Lambda^{0,1}T^*X. $$
Using this splitting we write an even spinor $\psi\in\Gamma(\splus_L)$ as 
$$\psi=\alpha+\beta, \quad \alpha\in \Gamma(L), 
\;\beta \in \Gamma(L\otimes K_X^{-1})=\Gamma(L\otimes \Lambda^{0,2}). $$
The space of complexified self-dual 2-forms can be identified with 
$\Lambda^{2,0}\oplus\C.\omega\oplus\Lambda^{0,2}$, and using this decomposition 
Witten rewrites the $\eta$-perturbed monopole equation as
$$ F^{2,0}= \overline{F^{0,2}}=\overline{\alpha}\beta -\eta, 
\quad iF^{1,1}\cdot\omega = |\beta|^2-|\alpha|^2,$$
where $F=F_B$ is the curvature of a connection $B$ on $L$.
The Dirac operator $D_A$ can be written in the form 
$\sqrt{2}(\overline{\partial}_B+\overline{\partial}^*_B)$. 
The coupled Dolbeault operator 
$\overline{\partial}_B$ is a map $\Gamma(L)\to \Gamma(L\otimes\Lambda^{0,1})$---it kills 
$\Gamma(L\otimes\Lambda^{0,2})$---while its dual $\overline{\partial}^*_B$ is a map 
$\Gamma(L\otimes\Lambda^{0,2})\to\Gamma(L\otimes\Lambda^{0,1})$. So the Dirac equation is
$$ \overline{\partial}_B\alpha + \overline{\partial}^*_B \overline{\beta}=0. $$
Therefore
$$ F^{0,2}\alpha = \overline{\partial}_B\overline{\partial}_B\alpha 
= -\overline{\partial}_B\overline{\partial}^*_B \overline{\beta}, $$
so when $\eta=0$ we have 
$\overline{\partial}_B\overline{\partial}^*_B \overline{\beta}+|\alpha|^2\beta=0 $ and
$$ \int_X{(|\overline{\partial}^*_B|^2+|\alpha|^2|\beta|^2)\mathrm{vol}_g}. $$
Hence $F^{0,2}=\overline{\alpha}\beta=0$. By the Newlander-Nirenberg theorem, 
$\overline{\partial}_B$ defines a holomorphic structure on $L$. In particular, $c_1(L)$ is of
type $(1,1)$.

We now drop the assumption $\eta=0$, but we \emph{can} assume 
$\int_X{F\wedge\eta}=\int_X{F^{0,2}\wedge\eta}=0$, because basic classes must have type 
$(1,1)$. A similar procedure now shows 
$$ \int_X{(|\overline{\partial}^*_B|^2+|\alpha\overline{\beta}-\eta|^2)\mathrm{vol}_g}=0.  $$
So we find:
\begin{itemize}
\item		$F^{0,2}=0$. Thus $B$ defines a holomorphic structure on $L$. 
\item		$\overline{\partial}_B\alpha=0$, so $\alpha$ is a holomorphic section of $L$.
\item		$\overline{\partial}_B^*\beta=0,$ so $\overline{\beta}$ is a holomorphic section
		of $L^*\otimes K_X$. 
\item 	$\alpha\overline{\beta}=\eta$: solutions describe a {\bf holomorphic factorization}
		of the perturbation parameter.
\end{itemize}
Conversely, given such data we can go backwards and find a unique solution to the equations.
So the slogan is
$$\textit{Everything reduces to algebraic geometry!}$$
A consequence of this factorization is that solutions of the monopole equation are 
\emph{isolated}---think of their divisors---and so the moduli space is at most 0-dimensional. 
>From the dimension formula, we find that a basic class $x$ satisfies 
$$ x^2 = K_X^2. $$
Of course, a $\spinc$-structure with non-empty 0-dimensional moduli space may not define a 
basic class: the signs attached to the points of the moduli space could cancel out.
Writing the Poincar\'e dual of $\eta$ as $\sum{r_i [C_i]}$ for disjoint connected curves $C_i$
and $r_i\geq 0$,
we can see from the factorization that if the line bundle $L$ corresponds to a basic class
then the divisor of a section of $L$ has the form $\sum{s_i [C_i]}$ with $0\leq s_i\leq r_i$.
Integrating the K\"ahler form $\omega$ on the $C_i$ we see that if $\kappa\in\mathrm{Bas}(X)$ 
then
$$ 0 \leq |\kappa\cdot[\omega]|=\left|\sum{(2s_i-r_i)\int_{C_i}{\omega}}\right|
	\leq \sum {r_i \int_{C_i}{\omega}}=K\cdot\omega. $$ 
Equality occurs iff all $s_i=0$ or $s_i=r_i$, which means either $L$ or $K_X \otimes L^*$ is 
trivial, i.e. $\kappa=K_X^{\pm 1}$. In either case there is a single solution, so
(conflating line bundles with $\spinc$-structures) we have 
$\mathbf{SW}_X(K_X^{\pm 1})=\pm 1. $ 

To summarise:

\vspace{5mm}
\noindent
\fbox{%
\begin{minipage}{138mm}
%\vspace{-\abovedisplayskip}
$(X,\omega)$ K\"ahler, $p_g(X)>0$, $\kappa\in\mathrm{Bas}(X)$. Then $\kappa^2=K_X^2$ and 
\begin{align*}
&\kappa \in H^2(X;\Z)\cap H^{1,1}(X); && 0 \leq |\kappa\cdot\omega|\leq K_X\cdot\omega;\\
&|\kappa\cdot\omega|=K_X\cdot\omega\Leftrightarrow\kappa=\pm K_X ;
		&& \mathbf{SW}_X(K_X^{\pm 1})=\pm 1.
\end{align*}
\end{minipage}}
\vspace{5mm}

An immediate consequence is that if $K_X$ is torsion (which is true for tori, K3s, 
Enriques and hyperelliptic surfaces) we have
$$ \mathrm{Bas}(X)=\{ \pm K_X\}. $$
The same is true for minimal surfaces with $K_X$ ample (`general type'): you can prove this in 
a couple of lines using the Hodge index theorem and the fact $K^2>0$. Consequently
orientation-preserving diffeomorphisms between manifolds of these types preserve
the canonical class, up to sign\footnote{Take care: an orientation-preserving diffeomorphisms 
$f: X\to Y$ induces $f^*: \spinc(Y) \to \spinc(X)$, and 
$\mathbf{SW}_Y=\pm\mathbf{SW}_X\circ f^*$. However, in general $f^*$ will not be compatible 
with our identification of $\spinc$-structures with cohomology classes.}. 
On minimal elliptic surfaces the set of basic classes
is larger, and distinguishes various diffeomorphism types.
If $\tilde{X}$ is a blow-up of $X$ at a point, with exceptional 
divisor Poincar\'e dual to $E$, then
$$\mathrm{Bas}(\tilde{X})=\text{proper transform}(\mathrm{Bas}(X))\pm E.$$ 
A final point: we used the canonical orientation on $X$, determined by setting $\omega^2>0$.
However, if $b_2^+$ and $b_2^-$ were both $>1$ we could have used the opposite orientation.
We would have found that the Seiberg-Witten invariants all vanished. It seems to be typical
of smooth orientable 4-manifolds that the Seiberg-Witten invariants are nontrivial in at most
one orientation.

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\end{document}