# 't Hooft many instanton solutions

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I'm study 't Hooft many instanton solutions of self-duality equation. In this method $A^a_\mu=-\bar{\eta}^{a}_{\mu\nu}\partial^\nu \ln{\Phi}$. After substitution in self-duality equation I've proven that

$$\Phi^{-1}\square{\Phi}=0$$

We can write the solution of this equation as $\Phi=1+\sum^q_{i=1}\frac{\lambda_i}{(x-x_i)^2}$.

I want to prove that this solution has instanton number equal to $q$. As I understand I need to calculate integral $\int F_{\mu \nu} F^{\mu \nu}$ and show that it is $\sim q$, but it's too tricky to me. Can you explain how to show it?

This post imported from StackExchange Physics at 2015-04-22 17:05 (UTC), posted by SE-user xxxxx
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