4D Gauge Theory Reading Seminar
I’m helping to organise a reading seminar for MT 2022. The idea is to see a proof of Donaldson’s theorem and compare the theory of ASD instantons with Seiberg-Witten theory, the latter being easier but the former being more relevant to gauge theory in higher dimensions.
The rough schedule for the talks will be
- Definitions, introduction to Gauge theory and constructing manifold invariants.
- Yang Mills functional, ASD equations. Chern-Weil theory. U(1) example.
- and structures. Dirac Operators. Seiberg-Witten equations.
- ADHM construction of Instantons on . Compactness problems.
- Fredholm Theory, Moduli spaces, Uhlenbeck’s theorem and Gauge fixing.
- Intersection forms and Donaldson’s Theorem.
- Overview of the proof of Donaldson’s Theorem.
- Gauge Theoretic Invariants in Algebraic Geometry and Symplectic Geometry.
I plan to write notes after each talk and add them to this page. If you find any mistakes below, please let me know.
1 Introdution to Gauge Theory and Constructing Manifold Invariants.
The goal of this reading group is to give a self contained overview of gauge theory in 4 dimensions. The talks will gover both the theory of ASD instantons and the Seiberg-Witten (SW) equations. The aim of the reading group it to see a proof of Donaldson’s theorem on the possible intersection forms of smooth four-manifolds, and to compare the theories of ASD instantons and the SW equations.
In this first talk we will review the basic setup of gauge theory – principal bundles and the action of the gauge group on various forms – and the general procedure on setup can be used to construct manifold invariants. In the first talk there will be nothing specific to dimension 4. This talk is largely based on the lecture notes Haydys - Introduction to Gauge Theory, and is mostly just definitions.
1.2 Principal Bundles, Connections and Curvature
Throughout this talk we fix a manifold . This is the manifold that we want to define the invariants of.
Definition 1.1. Let be a Lie group. A principal -bundle is a smooth manifold with a smooth (right) action, such that
and P is locally trivial. That is, for each , there is an open set containing such that
which is equivariant, so , . We say that has structure group
Definition 1.2. Given a principal -bundle , the gauge group, , of is the set of equivariant bundle isomorphisms. That is
Then we have that
and so . Hence, if , then the map defines a gauge transformation. In physics terminology, this is called a global symmetry - an example of this is a change of phase in the theory of electromagnetism. A local symmetry in physics is just a regular gauge transformation.
We now study the tangent space of a principal bundle . This leads to the notion of a vertical subspace and connections.
Definition 1.4. Let be defined by
Taking the union of the image for each defines a subbundle with fibres isomorphic to . This is the vertical subspace.
We now define the vector bundles associated to representations of the structure group. An alternative way of setting up gauge theory is to start with some vector bundle and consider the frame bundle of that vector bundle. In this case the two descriptions agree.
Definition 1.5. Let be a principal -bundle and let be a -representation. Then the associated vector bundle is the quotient
Let . Then this induces a bundle isomorphism , given by
Note that , so this is well defined.
We now define a bijection between certain sections of , the trivial bundle over and sections of the associated bundle .
Proposition 1.6. Let . Then there exists an , such that
for all . More over this defines a bijection .
Proof. Note that for to satisfy (2), we must have that , that is . If we pick , then by definition
for some unique , since . Hence we can define . One the other hand, if , then is well defined. □
Remark 1.7. We may view gauge transformations as sections of the associated bundle , where
We can make a similar identification of forms on that take values in the vector bundle . For this we need a further definition.
Definition 1.8. A form taking values in the trivial bundle over is basic if for all , . Here is as in (1).
Definition 1.9. A form is equivariant if
where the left hand side is the pull back by the multiplication by the action of and the left hand side is applied on the trivial fibre.
Proposition 1.10. There is a bijection between basic equivariant forms and forms in the associated bundle
Proof. This follows from noting that, given such an , defining via
is well defined. □
We can now define connections on a principal bundle . Let denote the Lie algebra of , considered as a -representation via the adjoint action.
Definition 1.11. Let be a principal -bundle. A connection on , is a -equivariant one form taking values in such that
for all , where is as in (1).
Remark 1.12. Note that , viewed as a map , has rank and so defines a horizontal subspace . This is -invariant () and defining such an is equivalent to defining .
Proof. The difference of two connections is basic. The result then follows from Proposition 1.10. □
We can consider the action of on the space of connections.
Proof. We can view as the composition
We have that the pushforward
To see this, note that the pushforward is linear and that
Hence we have
since is a connection. □
We can also use to define a covariant derivative. To do this, define, for , the operator
The action of on is given by the derivative of the representation . This can be extended to an operator
in the standard way. We define
for and .
This derivative operator defines a derivative operator on the associated bundles. This follows from the following proposition.
Proof. Equivariance of follows from the fact that , and are -equivariant. To prove that is basic it is sufficient to consider
Hence, if one has a vector bundle with a derivative operator, then this defines a connection on the frame bundle. □
Definition 1.16. Let be a connection. The curvature of , is defined to be the map
given by . This is a tensor in the sense that depends linearly on and we have that under gauge transformations, .
Proof. We have
Note that we can view the values of as elements of , under the image . □
1.3 Constructing Manifold Invariants
We now conclude by considering how the above machinery can be used to construct manifold invariants. In this set up we consider an infinite dimensional manifold (for example, the space of connections, or a connection with some other data) and consider an action by a lie group (for example the gauge group) on the right. We then want to consider the quotient
We want this to be a nice space, in particular another manifold. Hence we may want to restrict to where the action of on is free, and consider the quotient .
We now pick a representation of . This is, for example, , where the action of the gauge group is given by conjugation. Our gauge theoretic equations are then an -equivariant map
For the ASD equations, this is just given by taking the self dual part of the curvature.
To this choice , we can associate a moduli space , which we define to be
We hope that this is a finite dimensional manifold. This will follow from picking a good candidate for (elliptic with positive index). To use the regular value theorem, we might have to perturb to a sufficiently generic situation - for example consider for some small .
If were -dimensional and compact, then one would be tempted to count . In the ASD case, one can define an orientation of , and it turns out that only the signed count (or counting mod 2) will give an invariant. The definition of a signed count can be seen as the pairing
where is the orientation.
For , to define integer invariants, we can pick cohomology classes and then define the invariants as
In the case of Seiberg-Witten theory, this is done as follows. One picks a point and considers the based gauge group , where
This gives a short exact sequence
and so the based moduli space
Is a principal -bundle over . Hence we can use powers of the characteristic classes of this bundle to pick .
2 Yang Mills functional, ASD equations. Chern-Weil theory. U(1) example.
2.1 Characteristic Classes and Classifying Space
Suppose, as often happens in life, we are given a fibration , with fibre . For example a vector bundle or principal bundle from Section 1. One tool for studying these types of fibrations (in particular, to classify them) is through characteristic classes. That is, given a such a fibration, we want to construct elements that depend on the bundle.
These classes should satisfy a naturally requirement under pullback. That is, if , then , where is the pullback bundle.
Remark 2.1. Characteristic classes are purely topological. In the theory of classifying spaces, one typically works in the category of CW-complexes.
If the type of fibration being studied is sufficiently nice (as in the case of principal bundles) then one can find a classifying space for the fibration. This is a fibration
with fibre that satisfies the following universal property: if is another fibration with fibre , then there exists a unique (up to homotopy) such that
Given a classifying space, one can then pick a class , and take to define the characteristic classes of .
Remark 2.2. Let be the 1st cohomology group with coefficients in the group . This can be defined along the lines of C˘ech cohomology theory, i.e. by picking an open cover and constructing collections of local functions for all , such that and in . The C˘ech cohomology is then defined as the set of such functions, modulo the equivalence relation whenever
for all , where . The first cohomology is then defined as the inductive limit of when the cover is refined. This is not a group in general, but we have a functor.
where is the category of complexes with maps up to homotopy. One way of describing a principal bundle is to give explicit trivialisations on a cover of . Hence its fairly clear that each element corresponds to a principal -bundle. The theory of classifying spaces is essentially the observation that this functor is representable, so there is a classifying space such that there is a natural isomorphism
where denotes the homotopy classes of maps .
Example 2.3 (Chern Classes). Let be a rank complex vector bundle. The classifying space for such bundles is , which is the Grassmanian of -planes in . This can be defined by considering the square
where is the Grassmanian (space of planes in ) and is the tautological vector bundle, where to each plane , the fibre of the bundle at is just . One then takes the inductive limit to get the bundle . It turns out that the bundle is the classifying space for Hermitian vector bundles. The cohomology of may be written as
with . These are used to define the chern classes.
Example 2.4 (-line bundles over ). To justify (5) we can consider the case for line bundles over . One can prove that
where generates . The proof follows by induction and Poincaré duality.
2.2 Chern-Weil Point of View
If is a compact Lie group, then as described in Section 1 we can consider principal -bundles , where is some manifold. Chern-Weil theory gives an explicit way to construct characteristic classes of .
We start by picking a connection , and consider the curvature which we can view as a two form on taking values in the adjoint bundle , as in Proposition 1.15. Let denote the invariant polynomials. These are maps
which are of the form where is a representation of and for all . In the case of , the characteristic polynomial
gives generators of . We have the following claim.
Lemma 2.5. Let . Then is a well defined differential form. Moreover, and if is a different connection, is exact.
Proof. First consider as an element of . Then, as is equivariant, defines a basic two form taking values in the trivial line bundle over . The bundle associated to the trivial representation is trivial, and so this defines a differential form on . The fact that follows from the Bianchi identity and that . The final claim follows from the identity
where is a connection and . □
Hence, given a principal -bundle we can associate a cohomology class to by picking a connection and calculating the cohomology class of .
Remark 2.6. There are two limitations of the Chern-Weil approach, compared to the more general approach of classifying spaces. The first is that we can only get classes of even degree. The second is that these quantities are valued in the real cohomology, not the integral cohomology.
We do however have the following.
Proposition 2.7. In real cohomology, the Chern-Weil approach coincides with the topological approach.
Proof. This follows from the naturality properties. For example , (where ) and both theories have the same normalisation
2.3 Line Bundles
Let be a complex surface and be a holomorphic line bundle with hermitian connection . A Chern connection is a connection on such that and . If is a local holomorphic section and , we can define a connection by
We can compute the curvature of , and it is given by
2.4 Yang-Mills functional and ASD equations on Hermitian Vector Bundles
Throughout this section, let be a compact, oriented, Riemannian four manifold. The hodge star operator, restricted to two forms, is such that
Hence is diagonalisable and has two eigenspaces, with eigenvalues . Moreover changing the orientation of swaps the eigenspaces, so these have the same dimension. Hence applying this fibrewise we have the bundle decomposition
and the two forms also decompose. We write this as . This also induces a splitting on the cohomology
which follows from Hodge theory and a Weitzenböck formula. Hence, we have refined betti numbers by taking the dimension of these.
Given a hermitian vector bundle , with metric we can define
via the map , where is the -dual of .1 This leads to a decomposition
Definition 2.8. A connection is anti-self-dual (ASD) if . Note that is the projection so is anti-self-dual if and only if
Definition 2.9. Let . Then we can define the norm
where is given by wedging the form part and composing the endomorphisms. The functional
is called the Yang-Mills functional.
Proposition 2.10. A connection is ASD if and only if is an absolute minimiser of the Yang-Mills functional.
Proof. This follows from a direct calculation. Note that
The quantity is topological, given by , which proves the result. □
Remark 2.11. We note the following about the ASD equations. First, is conformally invariant since in 4 dimensions, is conformally invariant. Second, the ASD condition is gauge invariant, and is a first order PDE which is nonlinear when the gauge group is non-abelian. The linearisation of the map
when is ASD is given by . If , then and so the energy of the ASD instantons is given by . Note that this implies that the possible energies of ASD instantons is quantised, since and can be described by elements in the integral cohomology groups.
2.5 Complex Surfaces
To give some examples of ASD instantons we can use some algebraic geometry. Let be a Kähler surface and let be a hermitian vector bundle.
Proof. This follows from writing out the ASD equations in local holomorphic coordinates. □
Note that implies that , where . And hence if is ASD then is integrable.
Definition 2.13. A holomorphic vector bundle is hermitian Yang-Mills if
where is the curvature of the connection defined by .
Remark 2.14. Taking the trace of (6) one obtains a necessary condition, for the existence of HYM metrics. The Donaldson-Uhlenbech-Yau theorem gives a sufficient condition (slope stability) for the existence.
2.6 Generalisation to other Gauge Groups
In the above we were considering the ASD equations with structure group . In practice one tends to use different structure groups. Typically or . For compact Lie groups we can pick a -invariant inner product on the Lie algebra, so we have an invariant quadratic form with which to define characteristic classes. In the case, the relevant characteristic classes is the Pontryagin classes
It turns out that . Since , the double cover of , connections on bundles are locally the same as connections on bundles since the lie algebras are the same.
2.7 Interpretations of the ASD condition
We finish this section by looking at different ways to interpret the ASD condition. Note that the formal adjoint of is given by
Hence, if is ASD,
since . These are Maxwell’s equations. Another way of seeing this is that the Maxwell’s equations are the Euler Lagrange equations for the Yang-Mills functional. Hence if , then ASD instantons are solutions to Maxwell’s equations, and for other gauge groups, one gets solutions to non-abelian gauge theories, for example the standard model takes . Here the connections represent gauge bosons, so in the case the connections are photons.