Classical Virasoro irregular conformal block

Virasoro irregular conformal block with arbitrary rank is obtained for the classical limit or equivalently Nekrasov-Shatashvili limit using the beta-deformed irregular matrix model (Penner-type matrix model for the irregular conformal block). The same result is derived using the generalized Mathieu equation which is equivalent to the loop equation of the irregular matrix model.Comment: 18 pages; v2: comments and references added, version to appear in JHE

Super-spectral curve of irregular conformal blocks

We use super-spectral curve to investigate irregular conformal states of integer and half-odd integer rank. The spectral curve is the loop equation of supersymmetrized irregular matrix model. The case of integer rank corresponds to the colliding limit of supersymmetric vertex operators of NS sector and half-odd integer to the Ramond sectors. The spectral curve is simply integrable at Nekrasov-Shatashvili limit and the partition function (inner product of irregular conformal state) is obtained from the superconformal structure manifest in the spectral curve. We present some explicit forms of the partition function of integer (NS sector) and of half-odd ranks (Ramond sector)

Holstein-Primakoff Realizations on Coadjoint Orbits

We derive the Holstein-Primakoff oscillator realization on the coadjoint orbits of the $SU(N+1)$ and $SU(1,N)$ group by treating the coadjoint orbits as a constrained system and performing the symplectic reduction. By using the action-angle variables transformations, we transform the original variables into Darboux variables. The Holstein-Primakoff expressions emerge after quantization in a canonical manner with a suitable normal ordering. The corresponding Dyson realizations are also obtained and some related issues are discussed.Comment: 14 pages, Revtex, A minor revision is mad

Nekrasov and Argyres-Douglas theories in spherical Hecke algebra representation

AGT conjecture connects Nekrasov instanton partition function of 4D quiver gauge theory with 2D Liouville conformal blocks. We re-investigate this connection using the central extension of spherical Hecke algebra in q-coordinate representation, q being the instanton expansion parameter. Based on AFLT basis together with interwiners we construct gauge conformal state and demonstrate its equivalence to the Liouville conformal state, with careful attention to the proper scaling behavior of the state. Using the colliding limit of regular states, we obtain the formal expression of irregular conformal states corresponding to Argyres-Douglas theory, which involves summation of functions over Young diagrams.Comment: 22 pages; v2: minor modifications, published versio