Irregular conformal block, spectral curve and flow equations

Understand

Irregular conformal block is motivated by the Argyres-Douglas type of N=2 super conformal gauge theory.

• We investigate the classical/NS limit of the irregular conformal block using spectral curve on a Riemann surface with irregular punctures, which is equivalent to the loop equation of irregular matrix model.
• The spectral curve is reduced to the second order (Virasoro symmetry, $SU(2)$ for the gauge theory) and third order ($W_3$ symmetry, $SU(3)$) differential equations of a polynomial with finite degree.
• The Virasoro and W symmetry generate flow equations in the spectral curve and determine the irregular conformal block, hence the partition function of the Argyres-Douglas theory ala AGT conjecture.