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This paper is based on my presentation at RIMS workshop on "Theory of Integrable Systems and Its Applications in Various Fields" held in Kyoto on 19--21, August 2015. The aim of the present paper is to give a short account of recent studies…

We show how to deal with screening charges involving fractional powers of free fields. This enables us to use the free field Wakimoto construction to obtain complete expressions for integral representations of conformal blocks for N-point…

We present the details of a recently discovered representation of conformal four-point ladder integrals as thermal one-point functions in scalar field theories. We show that the conformal ladder integrals can be constructed from the…

We study conformal blocks of conformal field theories with a W3 symmetry algebra in the limit where the central charge is large. In this limit, we compute the four-point block as a special case of an sl3-invariant function. In the case when…

The decomposition of correlation functions into conformal blocks is an indispensable tool in conformal field theory. For spinning correlators, non-trivial tensor structures are needed to mediate between the conformal blocks, which are…

The best way to represent generic conformal blocks is provided by the free-field formalism, where they acquire a form of multiple Dotsenko-Fateev-like integrals of the screening operators. Degenerate conformal blocks can be described by the…

We construct modular linear differential equations (MLDEs) w.r.t. subgroups of the modular group whose solutions are Virasoro conformal blocks appearing in the expansion of a crossing symmetric 4-point correlator on the sphere. This uses a…

In this paper we consider parafermionic Liouville field theory. We study integral representations of three-point correlation functions and develop a method allowing us to compute them exactly. In particular, we evaluate the generalization…

We develop a general technique for computing functional integrals with fixed area and boundary length constraints. The correct quantum dimensions for the vertex functions are recovered by properly regularizing the Green function. Explicit…

We study the momentum-space 4-point correlation function of identical scalar operators in conformal field theory. Working specifically with null momenta, we show that its imaginary part admits an expansion in conformal blocks. The blocks…

In this thesis we study two-dimensional conformal field theories with Virasoro algebra symmetry, following the conformal bootstrap approach. Under the assumption that degenerate fields exist, we provide an extension of the analytic…

In the paper, the family of conformal four-point ladder diagrams in arbitrary space-time dimensions is considered. We use the representation obtained via explicit calculation using the operator approach and conformal quantum mechanics to…

We consider the correlation functions of Coulomb branch operators in four-dimensional N=2 Superconformal Field Theories (SCFTs) involving exactly one anti-chiral operator. These extremal correlators are the "minimal" non-holomorphic local…

It is known that the classical limit of the second order BPZ null vector decoupling equation for the simplest two 5-point degenerate spherical conformal blocks yields: (i) the normal form of the Heun equation with the complex accessory…

We formulate a set of general rules for computing $d$-dimensional four-point global conformal blocks of operators in arbitrary Lorentz representations in the context of the embedding space operator product expansion formalism…

Few years ago Zamolodchikov and Zamolodchikov proposed an expression for the 4-point classical Liouville action in terms of the 3-point actions and the classical conformal block. In this paper we develop a method of calculating the…

We study semiclassical correlation functions in Liouville field theory on a two-sphere when all operators have large conformal dimensions. In the usual approach, such computation involves solving the classical Liouville equation, which is…

We solve the Riemann-Hilbert problem on the sphere topology for three singularities of finite strength and a fourth one infinitesimal, by determining perturbatively the Poincare' accessory parameters. In this way we compute the…

We consider 5-point functions in conformal field theories in d > 2 dimensions. Using weight-shifting operators, we derive recursion relations which allow for the computation of arbitrary conformal blocks appearing in 5-point functions of…

We discuss how point transformations can be used for the study of integrability, in particular, for deriving classes of integrable variable-coefficient differential equations. The procedure of finding the equivalence groupoid of a class of…