A. V. Litvinov

On AGT conjecture

In these notes we consider relation between conformal blocks and the Nekrasov partition function of certain

Correlation functions in conformal Toda field theory I

\mathcal{N} = 2

Classical conformal blocks and Painlevé VI

SYM theories proposed recently by Alday, Gaiotto and Tachikawa. We concentrate on

A differential equation for a four-point correlation function in Liouville field theory and elliptic four-point conformal blocks

\mathcal{N} = {2^*}

On differential equation on four-point correlation function in the conformal Toda field theory

theory, which is the simplest example of AGT relation.

On spectrum of ILW hierarchy in conformal field theory

Two-dimensional {sl}(n) quantum Toda field theory on a sphere is considered. This theory provides an important example of conformal field theory with higher spin symmetry. We derive the three-point correlation functions of the exponential fields if one of the three fields has a special form. In this case it is possible to write down and solve explicitly the differential equation for the four-point correlation function if the fourth field is completely degenerate. We give also expressions for the three-point correlation functions in the cases, when they can be expressed in terms of known functions. The semiclassical and minisuperspace approaches in the conformal Toda field theory are studied and the results coming from these approaches are compared with the proposed analytical expression for the three-point correlation function. We show, that in the framework of semiclassical and minisuperspace approaches general three-point correlation function can be reduced to the finite-dimensional integral.

Multipoint correlation functions in Liouville field theory and minimal Liouville gravity

A bstractWe study the classical c → ∞ limit of the Virasoro conformal blocks. We point out that the classical limit of the simplest nontrivial null-vector decoupling equation on a sphere leads to the Painlevé VI equation. This gives the explicit representation of generic four-point classical conformal block in terms of the regularized action evaluated on certain solution of the Painlevé VI equation. As a simple consequence, the monodromy problem of the Heun equation is related to the connection problem for the Painlevé VI.

Parafermionic polynomials, Selberg integrals and three-point correlation function in parafermionic Liouville field theory

Liouville field theory on a sphere is considered. We explicitly derive a differential equation for four-point correlation functions with one degenerate field . We also introduce and study a class of four-point conformal blocks which can be calculated exactly and represented by finite-dimensional integrals of elliptic theta-functions for an arbitrary intermediate dimension. We also study 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.

On spectrum of ILW hierarchy in conformal field theory II: coset CFT’s

The properties of completely degenerate fields in the conformal Toda field theory are studied. It is shown that a generic four-point correlation function that contains only one such field does not satisfy an ordinary differential equation, in contrast to the Liouville field theory. Some additional assumptions for other fields are required. Under these assumptions, we write such a differential equation and solve it explicitly. We use the fusion properties of the operator algebra to derive a special set of three-point correlation functions. The result agrees with the semiclassical calculations.

Coulomb integrals in Liouville theory and Liouville gravity

A bstractWe consider a system of Integrals of Motion in conformal field theory related to the