During the spring semester there will be a course from the physics department with the above title. You can find more info here. I went today to the first lecture and it looks good.

In this series of post I would like to scratch the surface of an enormous iceberg called “instanton physics”. First I would like to mention some references:

1. arXiv:0802.1862 – Lectures on instantons by Vandoren and van Nieuwenhuizen,
2. arXiv:hep-th/0206063 – The calculus of many instantons by Dorey, Hollowood, Khoze and Mattis,
3. arXiv:hep-th/0004186 – Yang-Mills- and D-instantons by Belitsky, Vandoren and van Nieuwenhuizen.

Most of these posts are going to be very loyal to the first item, meaning that I will only discuss one instanton cases. As the title suggests, the second item deals with the case of many instantons.

So what is an instanton? A Yang-Mills instanton is a solution to the classical field equations in Euclidean space that give a finite action. Next you might ask, why finite action? Recall that for (classical) Yang-Mills the path integral has the form

A finite action then gives the leading contribution to . We can distinguish between regular instantons, ones that have a singularity at Euclidean infinity, and singular instanton, which don’t have singularity at infinity but at some point in space It turns out that we can map singular to regular solutions by a singular gauge transformation.

Later we will consider systems in the background of an instanton. We can achive this by the usual ways of minimal coupling,

When we have such a background one has to be careful with zero modes. These are solutions of the linearized field equations that are normalizable. Alternatively, zero modes are eigenfunctions of the quantum operator with zero eigenvalue. The quantum operator is (I think) the operator that appears in the action when one integrates by part the Lagrangian. For example,

In this case the quantum operator corresponds to . Zero modes have their own measure in the path integral and sometimes they are the only contribution (e.g. in supersymmetric theories the non-zero modes cancel).

Let us be a bit precise. Let us consider Yang-Mills gauge theory in 4 Euclidean space dimensions with gauge group SU(N). The Lie algebra has generators that are traceless, anti-hermitian matrices with the normalization

The action for Yang-Mills theory is

with the field strength given by

The classical field equations for are found from the Euler-Lagrange equations:

Since instantons are solutions to this equation but have finite action, we expect the field strength to vanish very far away from the origin. Since appears quadratic in the action, it should vanish faster than . The statement that the field strength vanishes leads us to looking for gauge potentials that are pure gauge, that is, they have the form

[To be continued...]

Last week saw the first seminar of this fall semester. We are meeting on Wednesdays, 2:15 PM at Eitan’s office (Math 2-122).

This week’s seminar (October 15th) will be about Instantons from the physics side.

Exactly a week ago I took my oral exam and passed, and soon it will be Eitan’s turn. This explains why things have been a bit quite around here. The bad news is that things will remain quite for a bit longer.

The good news is that we are planning to start again the meetings during the fall with topological field theory! I know nothing about this, erm, field. It will take some time to read and prepare something. But I already started to look into some notes on the arXiv:

• A mini-course on topological strings. hep-th/0504147
• Aspects of Chern-Simons Theory. hep-th/9902115

I found a rather nice article by Witten on the weak interactions and gauge symmetry breaking. It makes use of the terminology Eitan and Brandon have been presenting. The article can be found here.

Anybody has problems with the seminar being tomorrow (Wednesday) at 3:30 PM? The topic will be Electroweak and a bit of Yang-Mills.

Consider the matrix group , i.e. matrices such that where , or equivalently for any events in Minkowski spacetime. This group has 4 connected components coming from and or . The component containing the identity is called the proper, orthochronous Lorentz group . Physically it contains all rotations, and boosts (Lorentz tranformations) and so .

We can cover by the simply connected group , i.e. complex matrices with . First we identify Minkowski spacetime with the space of Hermitian matrices, i.e. matrices such that , in such a way that if is the Hermitian matrix identified with the event then . Then we can define a covering map by identifying with . We have that since
. It can be shown that is a 2-1 homomorphism of Lie groups.

Now, there are two important irreducible representations for on , the “spin ” representations given by multiplication and multiplication by the adjoint . The Dirac representation is the direct sum of these representations .

Let be the orthonormal frame bundle for spacetime. Its fibers are ordered orthonormal bases of , or equivalently isometries . There is a right action of given by right composition which makes the frame bundle an -bundle. We say that is space and time orientable iff has 4 components and a choice of component is a space and time orientation. Then the restriction is an -bundle.

The solder form is an -valued 1-form on given by . The torsion of a connection on is . It turns out that there is a unique connection whose torsion is . This is the Levi-Civita connection .

A spin structure on is a manifold and a smooth map such that is an -bundle with . We can define a connection on by where is the isomorphism of Lie algebras induced by .

Now consider sections of the vector bundle associated to by the Dirac representation. Dirac’s idea was to introduce an operator such that , i.e. the Dirac operator is the “square root” of the d’Alembert operator. A full understanding of the Dirac operator requires Clifford algebras, i.e the algebra generated over Minkowski space modulo . It turns out that the smallest representation of this Clifford algebra is 4-dimensional which is why we need a 4-dimensional representation of as well. Then we can define the Dirac operator as where is the connection associated to and we inner product them somehow.

In more detail for the d’Alembertian on Minkowski spacetime, , define

We can work out that .

Then we demand that the Dirac equation holds, . This gives us a theory of a spin- particle, an electron or positron, but we have not yet coupled it to electromagnetism.

We can splice a -bundle with a -bundle . Define and by . This is a -bundle with . Given connections on , we can define a connection on by with given by .

Splice together our -bundle with and also splice with . Consider the representation of on given by combining the Dirac representation with multiplication by . We get an associated vector bundle with an associated connection and Dirac operator . A charged electron coupled to electromagnetism is then a section for which the Dirac equation holds.

Suppose we had a principal -bundle with a connection with curvature .

The Lie algebra is just the set of imaginary numbers with trivial Lie bracket . The local potential is a real-valued 1-form defined by . The local field strength is defined by .

A change of gauge is given by with . We see that local connections are related by , so that local potentials are related by . Local curvatures are related by , so that local field strengths are related by . This means that the field strength is globally defined on .

By the Bianchi identity we have so , so the homogeneous Maxwell equation comes along for free. We can get the inhomogeneous Maxwell equation by requiring that .

Now, consider the action on given by multiplication . Associated to our principal bundle we get a vector bundle with fiber with an induced connection locally given by . We will write sections of the associated bundle as . We can define the d’Alembert operator . If we require the Klein-Gordon equation, , then we have a theory of a charged spin-0 particle coupled to electromagnetism.

In order to couple electromagnetism to more interesting particles like Dirac’s electron, we need to incorporate spin somehow.

We will assume is a group of matrices. A principal -bundle is a smooth surjection of manifolds with a free transitive right action of on such that and for any there is an open set with and a diffeomorphism called a “local trivialization” such that . Local trivializations correspond to the physical notion of “choice of gauge”.

Intuitively, is a manifold composed of copies of the group parametrized by the base space . A good example is the boundary of the Mobius strip which can be thought of as a -bundle over .

A useful notion is that of a local section with an open set with such that . It can be shown that there is a canonical 1-1 correspondence between local sections and local trivializations .

Define transition functions by where . This is well defined since . Transition functions correspond to the physical notion of “change of gauge”. We can relate any two local sections by .

Let be the Lie algebra for . A connection is a -valued 1-form on such that if If and is the tangent field on given by , then . Also we require that .

We define local connections on by . Local connections are related by .

We define curvature meaning . We can define local curvature by . Local curvatures are then related by . The Bianchi identity says .

We are now in a position to define electrodynamics on a principal bundle.

Maxwell’s equations in relativistically covariant form are

Since we can define a 2-form . We can also define a 1-form . Then we can re-express Maxwell’s equations using exterior differentiation and the Hodge star.

The continuity equation then follows from the inhomogeneous Maxwell equation. We expect from the homogeneous Maxwell equation that . In fact this is only true locally. This means that for every event in our spacetime there is an open set with and a 1-form on with . This follows from Poincare’s lemma.

We cannot say the exists globally. For instance if , the area form of the unit sphere in spherical coordinates, then since by antisymmetry of wedge product of 1-forms. Also, taking to be the unit sphere, we know that . However, by Stokes’ Theorem, if then . So, we cannot have globally.

Physically we interpret this as a magnetic monopole with magnetic charge and worldline, the time axis, . Mathematically, what is happening is that the complement of the time axis has nontrivial topology. Specifically its second de Rham cohomology is nontrivial. Intuitively, there is a kind of 2-dimensional, spherical “hole” in the complement of the time axis.

In addition to being nonglobal, the potential is defined only up to addition of a closed 1-form since . We would like to find a global mathematical object corresponding to the potential which doesn’t depend on our “choice of gauge”. This is our motivation for understanding connections on principal bundles.