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

Maintaining (locus of) control? Assessing the impact of locus of control on education decisions and wages

This paper establishes that individuals with an internal locus of control, i.e., who believe that reinforcement in life comes from their own actions instead of being determined by luck or destiny, earn higher wages. However, this positive effect only translates into labor income via the channel of education. Factor structure models are implemented on an augmented data set coming from two different samples. By so doing, we are able to correct for potential biases that arise due to reverse causality and spurious correlation, and to investigate the impact of premarket locus of control on later outcomes. --locus of control,wages,latent factor model,data set combination

Classical conformal blocks, Coulomb gas integrals, and quantum integrable models

In this paper, we recall Richardson's solution of the reduced BCS model, its relationship with the Gaudin model, and the known implementation of these models in conformal field theory. The CFT techniques applied here are based on the use of the free field realization, or more precisely, on the calculation of saddle-point values of Coulomb gas integrals representing certain (perturbed) WZW conformal blocks. We identify the saddle-point limit as the classical limit of conformal blocks. We show that this observation implies a new method for calculating classical conformal blocks and can be further used in the study of quantum integrable models.Comment: 8 pages, based on a talk given at the XII. International Symposium on Quantum Theory and Symmetries, July 24-28 , 2023, Prague, Czech Republic; to be published in Journal of Physics: Conference Serie

A first measurement of the Proper Motion of the Leo II dwarf spheroidal galaxy

We use 14-year baseline images obtained with the Wide Field Planetary Camera 2 on board the Hubble Space telescope to derive a proper motion for one of the Milky Way's most distant dwarf spheroidal companions, Leo II, relative to an extragalactic background reference frame. Astrometric measurements are performed in the effective point spread function (ePSF) formalism using our own developed code. An astrometric reference grid is defined using 3,224 stars that are members of Leo II that are brighter than magnitude 25 in the F814W band. We identify 17 compact extra-galactic sources, for which we measure a systemic proper motion relative to this stellar reference grid. We derive a proper motion [\mu_{\alpha},\mu_{\delta}]=[+104+/-113,-33+/-151] microarcseconds/yr for Leo II in the heliocentric reference frame. Though marginally detected, the proper motion yields constraints on the orbit of Leo II. Given a distance of 230 Kpc and a heliocentric radial velocity +79 km/s, and after subtraction of the solar motion, our measurement indicates a total orbital motion 266.1+/-128.7 km/s in the Galactocentric reference frame, with a radial component +21.5+/-4.3 km/s and tangential component 265.2+/-129.4 km/s. The small radial component indicates that Leo II either has a low-eccentricity orbit, or is currently close to perigalacticon or apogalacticon distance. We see evidence for systematic errors in the astrometry of the extragalactic sources which, while close to being point sources, are slightly resolved in the HST images. We argue that more extensive observations at later epochs will be necessary to better constrain the proper motion of Leo II. We provide a detailed catalog of the stellar and extragalactic sources identified in the HST data which should provide a solid early-epoch reference for future astrometric measurements.Comment: 10 pages, accepted for publication in the Astrophysical Journa

Classical conformal blocks, Coulomb gas integrals and Richardson-Gaudin models

Virasoro conformal blocks are universal ingredients of correlation functions of two-dimensional conformal field theories (2d CFTs) with Virasoro symmetry. It is acknowledged that in the (classical) limit of large central charge of the Virasoro algebra and large external, and intermediate conformal weights with fixed ratios of these parameters Virasoro blocks exponentiate to functions known as Zamolodchikovs' classical blocks. The latter are special functions which have awesome mathematical and physical applications. Uniformization, monodromy problems, black holes physics, quantum gravity, entanglement, quantum chaos, holography, N=2 gauge theory and quantum integrable systems (QIS) are just some of contexts, where classical Virasoro blocks are in use. In this paper, exploiting known connections between power series and integral representations of (quantum) Virasoro blocks, we propose new finite closed formulae for certain multi-point classical Virasoro blocks on the sphere. Indeed, combining classical limit of Virasoro blocks expansions with a saddle point asymptotics of Dotsenko-Fateev (DF) integrals one can relate classical Virasoro blocks with a critical value of the "Dotsenko-Fateev matrix model action". The latter is the "DF action" evaluated on a solution of saddle point equations which take the form of Bethe equations for certain QIS (Gaudin spin models). A link with integrable models is our main motivation for this research line. ... .Comment: 49 pages, several diagrams and tables; conformal to version accepted for publication in JHE

Core Formation by a Population of Massive Remnants

Core radii of globular clusters in the Large and Small Magellanic Clouds show an increasing trend with age. We propose that this trend is a dynamical effect resulting from the accumulation of massive stars and stellar-mass black holes at the cluster centers. The black holes are remnants of stars with initial masses exceeding 20-25 solar masses; as their orbits decay by dynamical friction, they heat the stellar background and create a core. Using analytical estimates and N-body experiments, we show that the sizes of the cores so produced and their growth rates are consistent with what is observed. We propose that this mechanism is responsible for the formation of cores in all globular clusters and possibly in other systems as well.Comment: 5 page

Clustering and Sharing Incentives in BitTorrent Systems

Peer-to-peer protocols play an increasingly instrumental role in Internet content distribution. Consequently, it is important to gain a full understanding of how these protocols behave in practice and how their parameters impact overall performance. We present the first experimental investigation of the peer selection strategy of the popular BitTorrent protocol in an instrumented private torrent. By observing the decisions of more than 40 nodes, we validate three BitTorrent properties that, though widely believed to hold, have not been demonstrated experimentally. These include the clustering of similar-bandwidth peers, the effectiveness of BitTorrent's sharing incentives, and the peers' high average upload utilization. In addition, our results show that BitTorrent's new choking algorithm in seed state provides uniform service to all peers, and that an underprovisioned initial seed leads to the absence of peer clustering and less effective sharing incentives. Based on our observations, we provide guidelines for seed provisioning by content providers, and discuss a tracker protocol extension that addresses an identified limitation of the protocol