Could someone help me to understand what geometric engineering in quantum field theory is? I didn't find any introductary articles online. Thank you!

Edit : Here is my background. I am math major. I know representation theory of quantum groups, quiver moduli space and hall algebra and some of algebraic geometry. I know very little physics except some brief mention from articles. That's why I feel difficult to follow the original paper posted on arxiv. What I want to know basicly is 1) what they want to do 2)what they did 3)what part of math they used.

Ps: Thank Urs for your suggestions.