Irregular Conformal States and Spectral Curve: Irregular Matrix Model Approach

We present recent developments of irregular conformal conformal states. Irregular vertex operators and their adjoint in a new formalism are used to define the irregular conformal states and their inner product instead of using the colliding limit procedure. Free field formalism can be augmented by screening operators which provide more degrees of freedom. The inner product is conveniently given as the partition function of an irregular matrix model. (Deformed) spectral curve is the loop equation of the matrix model at Nekrasov–Shatashivili limit. We present the details of analytic structure of the spectral curve for Virasoso symmetry and its extensions, $W$-symmetry and super-symmetry.