Hitchin Equation, Irregular Singularity, and $N=2$ Asymptotical Free Theories

In this paper, we study irregular singular solution to Hitchin’s equation and use it to describe four dimensional $N=2$ asymptotically free gauge theories. For $SU(2)$ $A$ type quiver, two kinds of irregular singularities besides one regular singularity are needed for the solution of Hitchin’s equation; We then classify irregular singularities needed for the general $SU(N)$ $A$ type quiver.