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We explore the RG flow between neighboring minimal CFT models with $W_3$ symmetry. After computing several classes of OPE structure constants we were able to find the matrices of anomalous dimensions for three classes of RG invariant sets of local fields. Each set from the first class consists of a single primary field, the second one of three primaries, while sets in the third class contain six primary and four secondary fields. We diagonalize their matrices of anomalous dimensions and establish the explicit maps between UV and IR fields (mixing coefficients). While investigating the three point functions of secondary fields we have encountered an interesting phenomenon, namely violation of holomorphic anti-holomorphic factorization property, something that does not happen in ordinary minimal models with Virasoro symmetry solely. Furthermore, the perturbation under consideration preserves a non-trivial subgroup of $W$ transformations. We have derived the corresponding conserved current explicitly. We used this current to define a notion of anomalous $W$-weights in perturbed theory: the analog for matrix of anomalous dimensions. For RG invariant sets with primary fields only we have derived a formula for this quantity in terms of structure constants. This allowed us to compute anomalous $W$-weights for the first and second classes explicitly. The same RG flow we investigate also with the domain wall approach for the second RG invariant class and find complete agreement with the perturbative approach., Comment: 50 pages, a reference added, published version

We shall interpret the null hexagonal Wilson loop (or, equivalently, six gluon scattering amplitude) in 4D ${\cal N}=4$ Super Yang-Mills, or, precisely, an integral representation of its matrix part, via an ADHM-like instanton construction. In this way, we can apply localisation techniques to obtain combinatorial expressions in terms of Young diagrams. Then, we use our general formula to obtain explicit expressions in several explicit cases. In particular, we discuss those already available in the literature and find exact agreement. Moreover, we are capable to determine explicitly the denominator (poles) of the matrix part, and find some interesting recursion properties for the residues, as well., Comment: 40 pages, 7 figures

The hyperbolic Anosov C-systems have exponential instability of their trajectories and as such represent the most natural chaotic dynamical systems. Of special interest are C-systems which are defined on compact surfaces of the Lobachevsky plane of constant negative curvature. An example of such system has been introduced in a brilliant article published in 1924 by the mathematician Emil Artin. The dynamical system is defined on the fundamental region of the Lobachevsky plane which is obtained by the identification of points congruent with respect to the modular group, a discrete subgroup of the Lobachevsky plane isometries. The fundamental region in this case is a hyperbolic triangle. The geodesic trajectories of the non-Euclidean billiard are bounded to propagate on the fundamental hyperbolic triangle. In this article we shall expose his results, will calculate the correlation functions/observables which are defined on the phase space of the Artin billiard and demonstrate the exponential decay of the correlation functions with time. We use Artin symbolic dynamics, the differential geometry and group theoretical methods of Gelfand and Fomin., Comment: 22 pages, 4 figures, references added

We study several aspects of the $N=1$ super Liouville theory. We show that certain elements of the fusion matrix in the Neveu-Schwarz sector related to the structure constants according to the same rules which we observe in rational conformal field theory. We collect some evidences that these relations should hold also in the Ramond sector. Using them the Cardy-Lewellen equation for defects is studied, and defects are constructed., 28 pages, comment and reference added

We compute the light asymptotic limit of $A_{n-1}$ Toda conformal blocks by using the AGT correspondence. We show that for certain class of CFT blocks the corresponding Nekrasov partition functions in this limit are simplified drastically being represented as a sum of a restricted class of Young diagrams. In the particular case of $A_{2}$ Toda we also compute the corresponding conformal blocks using conventional CFT techniques finding a perfect agreement with the results obtained from the Nekrasov partition functions., 19 pages, 3 figures

We specify Gaiotto's proposal for the RG domain wall between some coset CFT models to the case of two minimal N=1 SCFT models $SM_p$ and $SM_{p-2}$ related by the RG flow initiated by the top component of the Neveu-Schwarz superfield $\Phi_{1,3}$ . We explicitly calculate the mixing coefficients for several classes of fields and compare the results with the already known in literature results obtained through perturbative analysis. Our results exactly match with both leading and next to leading order perturbative calculations., Comment: 19 pages

The Lagrangian of the Liouville theory with topological defects is analyzed in detail and general solution of the corresponding defect equations of motion is found. We study the heavy and light semiclassical limits of the defect two-point function found before via the bootstrap program. We show that the heavy asymptotic limit is given by the exponential of the Liouville action with defects, evaluated on the solutions with two singular points. We demonstrate that the light asymptotic limit is given by the finite dimensional path integral over solutions of the defect equations of motion with a vanishing energy-momentum tensor., Comment: 50 pages, typos corrected, references added, comments and explanations added