On AGT conjecture

In these notes we consider relation between conformal blocks and the Nekrasov partition function of certain $\mathcal{N}=2$ SYM theories proposed recently by Alday, Gaiotto and Tachikawa. We concentrate on $\mathcal{N}=2^{*}$ theory, which is the simplest example of AGT relation.Comment: References adde

The Higgs and Coulomb/Confining Phases in "Twisted-Mass" Deformed CP(N-1) Model

We consider non-supersymmetric two-dimensional CP(N-1) model deformed by a term presenting the bosonic part of the twisted mass deformation of N=2 supersymmetric version of the model. Our deformation has a special form preserving a Z_N symmetry at the Lagrangian level. In the large mass limit the model is weakly coupled. Its dynamics is described by the Higgs phase, with Z_N spontaneously broken. At small masses it is in the strong coupling Coulomb/confining phase. The Z_N symmetry is restored. Two phases are separated by a phase transition. We find the phase transition point in the large-N limit. It lies at strong coupling. As was expected, the phase transition is related to broken versus unbroken Z_N symmetry in these two respective phases. The vacuum energies for these phases are determined too.Comment: 20 pages, 3 figures, reference adde

On scaling fields in ZNZ_N Ising models

We study the space of scaling fields in the $Z_N$ symmetric models with the factorized scattering and propose simplest algebraic relations between form factors induced by the action of deformed parafermionic currents. The construction gives a new free field representation for form factors of perturbed Virasoro algebra primary fields, which are parafermionic algebra descendants. We find exact vacuum expectation values of physically important fields and study correlation functions of order and disorder fields in the form factor and CFT perturbation approaches.Comment: 2 Figures, jetpl.cl

Vertex Operator Extension of Casimir W A(N) Algebras

We give an extension of Casimir of Casimir $\cal{WA_N}$ algebras including a vertex operator which depends on non-simple roots of $A_{N-1}$.Comment: 7 pages,no figures,TeX file,(to appear in Mod.Phys.Lett.A

Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks

Liouville field theory on a sphere is considered. We explicitly derive a differential equation for four-point correlation functions with one degenerate field $V_{-\frac{mb}{2}}$. We introduce and study also a class of four-point conformal blocks which can be calculated exactly and represented by finite dimensional integrals of elliptic theta-functions for arbitrary intermediate dimension. We study also the bootstrap equations for these conformal blocks and derive integral representations for corresponding four-point correlation functions. A relation between the one-point correlation function of a primary field on a torus and a special four-point correlation function on a sphere is proposed