View source for Seiberg-Witten equations

Equations constituting a breakthrough in work on the topology of four-dimensional manifolds (cf. also [[Four-dimensional manifold|Four-dimensional manifold]]). The equations, which were introduced in [[#References|[a1]]] have their origins in physics in earlier work of N. Seiberg and E. Witten [[#References|[a2]]], [[#References|[a3]]].

One of the advances provided by the Seiberg–Witten equations concerns Donaldson polynomial invariants for four-dimensional manifolds (see also below).

If one chooses an oriented, compact, closed, [[Riemannian manifold|Riemannian manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s1200801.png" />, then the data needed for the Seiberg–Witten equations are a [[Connection|connection]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s1200802.png" /> on a line bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s1200803.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s1200804.png" /> and a  "local spinor field"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s1200805.png" />. The Seiberg–Witten equations are then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s1200806.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s1200807.png" /> is the Dirac operator and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s1200808.png" /> is made from the gamma-matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s1200809.png" /> according to

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008010.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008011.png" /> is called a  "local spinor"  because global spinors need not exist on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008012.png" />; however, orientability guarantees that a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008013.png" /> structure does exist and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008014.png" /> is the appropriate section for this <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008015.png" /> structure. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008016.png" /> is just a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008017.png" /> Abelian connection, and so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008018.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008019.png" /> being the self-dual part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008020.png" />.

==Example.==
The equations clearly provide the absolute minima for the action

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008021.png" /></td> </tr></table>

If one uses a Weitzenböck formula to relate the Laplacian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008022.png" /> (cf. also [[Laplace operator|Laplace operator]]) to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008023.png" /> plus curvature terms, one finds that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008024.png" /> satisfies

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008025.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008026.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008027.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008028.png" /> is the [[Scalar curvature|scalar curvature]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008030.png" /> is the [[Chern class|Chern class]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008031.png" />.

The action now looks like one for monopoles; indeed, in [[#References|[a1]]], Witten refers to what are now called the Seiberg–Witten equations as the  "monopole equations" . But now suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008032.png" /> is positive and that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008033.png" /> is a solution to the Seiberg–Witten equations; then the left-hand side of this last expression is zero and all the integrands on the right-hand side are positive, so the solution must obey <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008035.png" />. It turns out that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008036.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008037.png" /> (see below for a definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008038.png" />), then a perturbation of the metric can preserve the positivity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008039.png" /> but perturb <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008040.png" /> to be simply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008041.png" />, rendering the connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008042.png" /> flat (cf. also [[Flat form|Flat form]]). Hence, in these circumstances, the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008043.png" /> is the trivial one. This means that one has a new kind of vanishing theorem in four dimensions ([[#References|[a1]]], 1994): No four-dimensional manifold with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008044.png" /> and non-trivial Seiberg–Witten invariants admits a metric of positive scalar curvature.

==Polynomial invariants.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008045.png" /> be a smooth, simply-connected, orientable Riemannian four-dimensional manifold without boundary and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008046.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008047.png" /> connection which is anti-self-dual, so that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008048.png" /></td> </tr></table>

Then the space of all gauge-inequivalent solutions to this anti-self-duality equation, the moduli space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008049.png" />, has a dimension, given by the integer

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008050.png" /></td> </tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008051.png" /> is the instanton number, which gives the topological type of the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008052.png" />. The instanton number is minus the second Chern class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008053.png" /> of the bundle on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008054.png" /> is defined. This means that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008055.png" /></td> </tr></table>

The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008056.png" /> is defined to be the rank of the positive part of the intersection form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008057.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008058.png" />; the intersection form being defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008059.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008060.png" /> denoting the cup product.

A Donaldson invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008061.png" /> is a symmetric integer polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008062.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008063.png" />-homology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008064.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008065.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008066.png" /></td> </tr></table>

Given a certain mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008067.png" /> (cf. [[#References|[a4]]], [[#References|[a5]]]),

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008068.png" /></td> </tr></table>

then, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008070.png" /> represents a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008071.png" />, one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008072.png" /> by writing

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008073.png" /></td> </tr></table>

The evaluation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008074.png" /> on the right-hand side of the above equation means that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008075.png" /></td> </tr></table>

so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008076.png" /> is even dimensional, this is achieved by requiring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008077.png" /> to be odd.

Now, the Donaldson invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008078.png" /> are differential topological invariants rather than topological invariants, but they are difficult to calculate as they require detailed knowledge of the instanton moduli space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008079.png" />. However, they are non-trivial and their values are known for a number of four-dimensional manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008080.png" />. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008081.png" /> is a complex [[Algebraic surface|algebraic surface]], a positivity argument shows that that they are non-zero when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008082.png" /> is large enough. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008083.png" /> can be written as the connected sum

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008084.png" /></td> </tr></table>

where both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008086.png" /> have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008087.png" />, then they all vanish.

Turning now to physics, it is time to point out that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008088.png" /> can also be obtained (cf. [[#References|[a6]]]) as the correlation functions of twisted supersymmetric topological field theory.

The action <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008089.png" /> for this theory is given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008090.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008091.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008092.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008093.png" /> is the curvature of a connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008095.png" /> are a collection of fields introduced in order to construct the right supersymmetric theory; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008096.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008097.png" /> are both spinless while the multiplet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008098.png" /> contains the components of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s12008099.png" />-form, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080100.png" />-form and a self-dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080101.png" />-form, respectively.

The significance of this choice of multiplet is that the instanton deformation complex used to calculate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080102.png" /> contains precisely these fields.

Even though <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080103.png" /> contains a metric, its correlation functions are independent of the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080104.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080105.png" /> can still be regarded as a topological field theory. This is because both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080106.png" /> and its associated energy-momentum tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080107.png" /> can be written as BRST commutators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080109.png" /> for suitable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080110.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080111.png" />.

With this theory it is possible to show that the correlation functions are independent of the gauge coupling and hence one can evaluate them in a small coupling limit. In this limit, the functional integrals are dominated by the classical minima of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080112.png" />, which for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080113.png" /> are just the instantons

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080114.png" /></td> </tr></table>

It is also required that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080115.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080116.png" /> vanish for irreducible connections. If one expands all the fields around the minima up to quadratic terms and does the resulting Gaussian integrals, the correlation functions may be formally evaluated.

A general correlation function of this theory is now given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080117.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080118.png" /> denotes the collection of fields present in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080119.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080120.png" /> is a polynomial in the fields.

Now, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080121.png" /> has been constructed so that the zero modes in the expansion about the minima are the tangents to the moduli space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080122.png" />. This suggest that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080123.png" /> integration can be done as follows: Express the integral as an integral over modes, then all the non-zero modes may be integrated out first leaving a finite-dimensional integration over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080124.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080125.png" /> denotes the compactified moduli space). The Gaussian integration over the non-zero modes is a Boson–Fermion ratio of determinants, a ratio which supersymmetry constrains to be of unit modulus since Bosonic and Fermionic eigenvalues are equal in pairs.

This amounts to expressing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080126.png" /> as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080127.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080128.png" /> denotes an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080129.png" />-form over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080130.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080131.png" />. If the original polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080132.png" /> is judiciously chosen, then calculation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080133.png" /> reproduces the evaluation of the Donaldson polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080134.png" />. It is now time to return to the Seiberg–Witten context.

There is a set of rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080135.png" />, known as the Seiberg–Witten invariants, which can be obtained by combining the Donaldson polynomials into a generating function. To do this one assumes that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080136.png" /> have the property that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080137.png" /></td> </tr></table>

A simply-connected manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080138.png" /> whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080139.png" /> have this property is said to be of simple type. This property makes it useful to define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080140.png" />, by writing

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080141.png" /></td> </tr></table>

The generating function, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080142.png" />, is given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080143.png" /></td> </tr></table>

According to P.B. Kronheimer and T.S. Mrowka [[#References|[a7]]], [[#References|[a8]]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080144.png" /> can be expressed in terms of a finite number of classes (known as basic classes) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080145.png" /> with rational coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080146.png" /> (called the Seiberg–Witten invariants), resulting in the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080147.png" /></td> </tr></table>

Hence, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080148.png" /> of simple type the polynomial invariants are determined by a (finite) number of basic classes and the Seiberg–Witten invariants.

Returning now to the physics, one finds that the [[Quantum field theory|quantum field theory]] approach to the polynomial invariants relates them to properties of the moduli space for the Seiberg–Witten equations, rather than to properties of the instanton moduli space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080149.png" />.

The moduli space for the Seiberg–Witten equations generically has dimension

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080150.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080151.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080152.png" /> are the [[Euler characteristic|Euler characteristic]] and [[Signature|signature]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080153.png" />, respectively. This vanishes when

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080154.png" /></td> </tr></table>

and then the moduli space, being zero dimensional, is a collection of points. There are actually only a finite number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080155.png" /> of these, and so they form a set

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080156.png" /></td> </tr></table>

Each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080157.png" /> has a sign <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080158.png" /> associated with it, coming from the sign of the determinant of the elliptic operator whose index gave the dimension of the moduli space, cf. [[#References|[a1]]]. The sum of these signs is a topological invariant, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080159.png" />, i.e.

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080160.png" /></td> </tr></table>

Using this information, one can pass to a formula of [[#References|[a1]]] for the generating function which, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080161.png" /> of simple type, reads (though note that the bundle denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080162.png" /> here corresponds to the square of the bundle denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080163.png" /> in [[#References|[a1]]]):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080164.png" /></td> </tr></table>

with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080165.png" /></td> </tr></table>

and the sum over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080166.png" /> on the right-hand side of the formula is over (the finite number of) line bundles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080167.png" /> that satisfy

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080168.png" /></td> </tr></table>

in other words, it is a sum over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080169.png" /> with zero-dimensional Seiberg–Witten moduli spaces.

Comparison of the two formulas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080170.png" /> (the first mathematical in origin and the second physical) allows one to identify the Seiberg–Witten invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080171.png" /> and the Kronheimer–Mrowka basic classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080172.png" /> as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080173.png" />; also, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080174.png" /> must satisfy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080175.png" /> as had been suggested already.

The physics underlying these topological results is of great importance, since many of the ideas originate there. It is known from [[#References|[a6]]] that the computation of the Donaldson invariants may use the fact that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080176.png" /> gauge theory is asymptotically free. This means that the ultraviolet limit, being one of weak coupling, is tractable. However the less tractable infrared or strong coupling limit would do just as well to calculate the Donaldson invariants, since these latter are metric independent.

In [[#References|[a2]]], [[#References|[a3]]] this infrared behaviour is determined and it is found that, in the strong coupling infrared limit, the theory is equivalent to a weakly coupled theory of Abelian fields and monopoles. There is also a duality between the original theory and the theory with monopoles, which is expressed by the fact that the (Abelian) gauge group of the monopole theory is the dual of the maximal torus of the group of the non-Abelian theory.

Recall that the Yang–Mills gauge group in the discussion above is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080177.png" />. This infrared equivalence of [[#References|[a2]]], [[#References|[a3]]] means that the achievement of [[#References|[a1]]] is to successfully replace the counting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080178.png" /> instantons used to compute the Donaldson invariants in [[#References|[a6]]] by the counting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080179.png" /> monopoles. Since this monopole theory is weakly coupled, everything is computable now in the infrared limit.

The theory considered in [[#References|[a2]]], [[#References|[a3]]] possesses a collection of quantum vacua labelled by a complex parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080180.png" />, which turns out to parametrize a family of elliptic curves (cf. also [[Elliptic curve|Elliptic curve]]). A central part is played by a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080181.png" /> on which there is a modular action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120080/s120080182.png" />. The successful determination of the infrared limit involves an electric-magnetic duality and the whole matter is of considerable independent interest for quantum field theory, quark confinement and string theory in general.

If one allows the four-dimensional manifold $M$ to have a boundary $Y$, then one induces certain three-dimensional Seiberg–Witten equations on the three-dimensional manifold $Y$, cf. [[#References|[a5]]], [[#References|[a7]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Witten,   "Monopoles and four-manifolds"  ''Math. Res. Lett.'' , '''1'''  (1994)  pp. 769–796 {{ZBL|0867.57029}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Seiberg, E. Witten,   "Electric-magnetic duality, monopole condensation, and confinement in $N=2$ supersymmetric Yang–Mills theory"  ''Nucl. Phys.'' , '''B426'''  (1994)  pp. 19–52  (Erratum: B430 (1994), 485-486)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Seiberg, E. Witten,   "Monopoles, duality and chiral symmetry breaking in $N=2$ supersymmetric QCD"  ''Nucl. Phys.'' , '''B431'''  (1994)  pp. 484–550</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  S.K. Donaldson, P.B. Kronheimer,   "The geometry of four manifolds" , Oxford Univ. Press  (1990)</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  S.K. Donaldson, "The Seiberg–Witten equations and 4-manifold topology"  ''Bull. Amer. Math. Soc.'' , '''33'''  (1996)  pp. 45–70</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  E. Witten, "Topological quantum field theory"  ''Comm. Math. Phys.'' , '''117'''  (1988)  pp. 353–386</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  P.B. Kronheimer,   T.S. Mrowka, "Recurrence relations and asymptotics for four manifold invariants"  ''Bull. Amer. Math. Soc.'' , '''30'''  (1994)  pp. 215–221</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  P.B. Kronheimer,   T.S. Mrowka, "The genus of embedded surfaces in the projective plane"  ''Math. Res. Lett.'' , '''1'''  (1994)  pp. 797–808</TD></TR>
</table>