This brief instroduction is based on David Tong’s TASI Lectures on Solitons Lecture 1:Instantons.

We’ll first talk about the instantons arise in SU(N) Yang-Mills theory and then explain the connection between them and supersymmetry. By the end we’ll try to explain how string theory jumps in this whole business.

Instantons are nothing but a special kind of solution for the pure SU(N) Yang-Mills theory with action $S=\frac{1}{2 e^2}\int d^4x Tr F_{\mu\nu}F^{\mu\nu}$. Motivated by semi-classical evaluation of path integral, we search for finite action solutions to the Euclidean equations of motion, $\mathcal{D}_{\mu} F^{\mu\nu}=0$. In order to have a finite action, we need field potential $A_{\mu}$ to be pure gauge at the boundary $\partial R^4=S^3$, i.e. $A_{\mu}=ig^{-1}\partial_{\mu}g$.

Then the action will be given roughly by a surface integral which depends on the third fundamental group of SU(N), given by $\Pi_3(SU(N))=k$, k is usually called the charge of the instanton. For the original action to be captured by this number k, we need to have self-dual or anti self-dual field strength.

A specific solution of $k=1$ for SU(2) group can be given by $A_{mu}=\frac{\rho^2(x-X)_{\nu}}{(x-X)^2((x-X)^2+\rho^2)}\bar{\eta_{\mu\nu}}^i(g\sigma^i g^{-1})$ where $Xs$ are coordinate parameter $\rho$ is scale parameter and together with the three generator of the group, we have 8 parameters called collective coordinates. $\eta$ is just some matrix to interwine the group index $i$ with the space index $\mu$.

For a given instanton charge $k$ and a given group SU(2), an interesting question is how many independent solutions we have. The number is usually counted by given a solution $A_{\mu}$and we try to find how many infinitisiaml perturbation of this solution $\delta_\alpha A_\mu$, known as zero modes, $\alpha$ is the index for this solution space, usually called moduli space.

When we consider a Yang-Mills theory with an instanton background instead of a pure Yang-Mills theory. We’d like to know if we still have non-trivial solutions, and especially if these solutions will give rise to even more collective coordinates. This is where fermion zero modes and supersymmetry comes in. For $\mathcal{N}=2$ or 4 supersymmetry in $D=4$, it’s better to promote the instanton to be a string in 6 dimentiona or a 5 brane in 10 dimensions respectively. The details of how to solve the equation will be beyond the scope of this introduction, and We’ll refer the reader to the original lecture notes by David Tong.