Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation

In this work the correspondence between the semiclassical limit of the DOZZ quantum Liouville theory on the torus and the Nekrasov-Shatashvili limit of the N=2* (Omega-deformed) U(2) super-Yang-Mills theory is used to propose new formulae for the accessory parameter of the Lame equation. This quantity is in particular crucial for solving the problem of uniformization of the one-punctured torus. The computation of the accessory parameters for torus and sphere is an open longstanding problem which can however be solved if one succeeds to derive an expression for the so-called classical Liouville action. The method of calculation of the latter has been proposed some time ago by Zamolodchikov brothers. Studying the semiclassical limit of the four-point function of the quantum Liouville theory on the sphere they have derived the classical action for the Riemann sphere with four punctures. In the present work Zamolodchikovs idea is exploited in the case of the Liouville field theory on the torus. It is found that the Lame accessory parameter is determined by the classical Liouville action on the one-punctured torus or more concretely by the torus classical block evaluated on the saddle point intermediate classical weight. Secondly, as an implication of the aforementioned correspondence it is obtained that the torus accessory parameter is related to the sum of all rescaled column lengths of the so-called "critical" Young diagrams extremizing the instanton "free energy" for the N=2* gauge theory. Finally, it is pointed out that thanks to the known relation the sum over the "critical" column lengths can be expressed in terms of a contour integral in which the integrand is built out of certain special functions.Comment: 41 pages, published versio

Investment and Regulation in Telecommunications

This article presents the difficulties associated with the implementation of the regulatory goal of promoting investment and innovation within the area of sector specific regulation in telecoms. The encouragement of efficient investment is one of the major goals reflected in the EC and domestic legal rules on telecoms access as well as price- and rate of return regulation. The law and the interplay of the interests of incumbents and alternative operators create a fertile soil for the emergence of various regulatory concepts of stimulating investment and facility-based competition. Considered here are the concepts most frequently referred to in this context including: the notion of new and emerging markets, the ladder of investment theory, sunset clauses and dynamic pricing policies. However, most of these concepts had little influence on regulatory practice so far, seeing as telecoms regulation is mostly directed at service competition and effective utilisation of existing infrastructures. This fact is the result of national regulators balancing their various regulatory goals in the existing technical and economic environment of the sector. The approach of the Polish regulatory authority towards these concepts constitutes an example of this reality. The urgent need to establish a new policy for next generation networks and access, bringing new technologies and business models to the sector, will have to induce more recognition for some concepts presented in this article.telecommunication, regulation, investment, regulatory holidays, ladder of investment, sunset clause, price regulation, telecommunications access

Solving Heun's equation using conformal blocks

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 parameter determined by the 4-point classical block on the sphere, and (ii) a pair of the Floquet type linearly independent solutions. A key point in a derivation of the above result is the classical asymptotic of the 5-point degenerate blocks in which the so-called heavy and light contributions decouple. In the present work the semi-classical heavy-light factorization of the 5-point degenerate conformal blocks is studied. In particular, a mechanism responsible for the decoupling of the heavy and light contributions is identified. Moreover, it is shown that the factorization property yields a practical method of computation of the Floquet type Heun's solutions. Finally, it should be stressed that tools analyzed in this work have a broad spectrum of applications, in particular, in the studies of spectral problems with the Heun class of potentials, sphere-torus correspondence in 2d CFT, the KdV theory, the connection problem for the Heun equation and black hole physics. These applications are main motivations for the present work.Comment: 28 pages, revised and extended versio

Classical irregular block, N=2 pure gauge theory and Mathieu equation

Combining the semiclassical/Nekrasov-Shatashvili limit of the AGT conjecture and the Bethe/gauge correspondence results in a triple correspondence which identifies classical conformal blocks with twisted superpotentials and then with Yang-Yang functions. In this paper the triple correspondence is studied in the simplest, yet not completely understood case of pure SU(2) super-Yang-Mills gauge theory. A missing element of that correspondence is identified with the classical irregular block. Explicit tests provide a convincing evidence that such a function exists. In particular, it has been shown that the classical irregular block can be recovered from classical blocks on the torus and sphere in suitably defined decoupling limits of classical external conformal weights. These limits are "classical analogues" of known decoupling limits for corresponding quantum blocks. An exact correspondence between the classical irregular block and the SU(2) gauge theory twisted superpotential has been obtained as a result of another consistency check. The latter determines the spectrum of the 2-particle periodic Toda (sin-Gordon) Hamiltonian in accord with the Bethe/gauge correspondence. An analogue of this statement is found entirely within 2d CFT. Namely, considering the classical limit of the null vector decoupling equation for the degenerate irregular block a celebrated Mathieu's equation is obtained with an eigenvalue determined by the classical irregular block. As it has been checked this result reproduces a well known weak coupling expansion of Mathieu's eigenvalue. Finally, yet another new formulae for Mathieu's eigenvalue relating the latter to a solution of certain Bethe-like equation are found.Comment: 47 pages, 3 figure

Hubble Space Telescope survey of the Perseus Cluster - I: The structure and dark matter content of cluster dwarf spheroidals

We present the results of a Hubble Space Telescope (HST) Advanced Camera for Surveys (ACS) study of dwarf galaxies in the core of the rich nearby Perseus Cluster, down to M_V=-12. We identify 29 dwarfs as cluster members, 17 of which are previously unstudied. All the dwarfs we examine are remarkably smooth in appearance, and lack internal features. Based on these observations, and the sizes of these dwarfs, we argue that some of the dwarfs in our sample must have a large dark matter content to prevent disruption by the cluster potential. We derive a new method, independent of kinematics, for measuring the dark matter content of dEs, based on the radius of the dwarf, the projected distance of the dwarf from the cluster centre, and the total mass of the cluster interior to it. We find that the mass-to-light ratios of these dwarfs are comparable to those of the Local Group dSphs, ranging between 1 and 120.Comment: accepted for publication by MNRA