TITLE: What is an "Instanton" in classical gauge theory? (to a mathematician)
QUESTION [15 upvotes]: There's already a question about the same topic but I think its aim is different.
Classical (non-quantum) gauge theory is a completely rigorous mathematical theory. It can be phrased in completely differential-geometric terms (where the main players are bundle with connections on a manifold). 
I think I have a basic understanding of what gauge theory is about and what various words mean in this context (yang mils, potential, energy, etc...). However I have still not managed to figure out what "Instanton" means in this context. 

What is an Instanton?

Is it something special to Yang-Mils theory? Is it something special to Quantum Gauge theory? Are there any mathematical interpretations/applications for Instantons?

REPLY [6 votes]: Generally speaking, you could say they are a special type of solution to the field equations of gauge theories.  More specifically, an instanton is a classical solution in a classical Euclidean field theory with finite non-zero action.
The name is due to the fact that they happen for an 'instant' (a point) of Euclidean time and so they are important in the path-integral formulation of a theory which uses Euclidean signature and as critical points.  As other commentators have mentioned, in QFT we are generally talking about the Yang-Mills instanton (where a Yang-Mills theory is a QFT with a non-abelian gauge group).  For a mathematical interpretation, the instanton solution of the Euclidean Yang-Mills equation leads an $SU(2)$ fibre bundle over $S^{4}$ but it can also be proved that any finite action solution of the Euclidean Yang-Mills equations leads to a fibre bundle over the four-sphere (see Uhlenbeck, 1979).
If we consider the best-known example, we take a pure Yang-Mills theory with symmetry group $SU(2)$ in $R^{4}$ with Euclidean signature.  The equations of motion (ie. the Yang-Mills equations) are $D*F=0$ and $DF=0$.  When we introduce the condition called the anti-self-duality equation, these equations reduce to ODEs for the gauge potential $A$.  If we make an ansatz for the solution to the anti-self-duality equation which only differs from a pure gauge by a function of $r$ at infinite radius, we guarantee that our solutions have finite, non-zero action.  Our ansatz for the gauge field is such that it becomes a pure gauge as $r$ tends to infinity and the associated field strength disappears, meaning that the action is finite.
If we choose the appropriate gauge transformation and use this to evaluate the field strength, we then obtain an equation whose full-solution is the 'instanton potential' which is regular on all of $R^{4}$.  The action is equal to $-8 
 \pi^{2}/g^{2}$, so it is obviously finite.   See Gockeler and Schucker (1987) for more on the fibre bundle interpretation: this looks at other related fibre bundle structures such as the Dirac monopole.  The fibre bundle structure of the Dirac monopole is extremely similar (basically the same) as the Yang-Mills instanton fibre bundle.  (I only add this as seems like you were after a more interesting mathematical interpretation).
Following this logic we can attempt to construct a 'gravitational instanton' ie. we look for a metric with Euclidean signature described locally by an orthonormal frame and solve the Einstein field equations without matter.  We end up with a solution such that $g$ and $f$ tend to $1$ as $r$ tends to infinity: this is similar to the Yang-Mills case where the Yang-Mills instanton potential becomes a pure gauge as $r$ tends to infinity.  This is similar to the way that gravity ends up being 'analogous' to other gauge theories, rather than directly comparable, since gravity does not quantize well.