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

Parametric dependence of irregular conformal block

Irregular conformal block is an important tool to study a new type of conformal theories, which can be constructed as the colliding limit of the regular conformal block. The irregular conformal block is realized as the $\beta$-deformed Penner matrix model whose partition function is regarded as the inner product of the irregular modules. The parameter dependence of the inner product is obtained explicitly using the loop equation with close attention to singularities in the parameter space. It is noted that the exact singular structure of the parameter space in general can be found using a very simple and powerful method which uses the flow equations of the partition function together with the hierarchical structure of the singularity. This method gives the exact expression to all orders of large $N$ expansion without using the explicit contour integral of the filling fraction.Comment: 34pages, 8figure