Nonlinear supersymmetry: from classical to quantum mechanics

Quantization of the nonlinear supersymmetry faces a problem of a quantum anomaly. For some classes of superpotentials, the integrals of motion admit the corrections guaranteeing the preservation of the nonlinear supersymmetry at the quantum level. With an example of the system realizing the nonlinear superconformal symmetry, we discuss the nature of such corrections and speculate on their possible general origin.Comment: 11 page

On manifolds admitting the consistent Lagrangian formulation for higher spin fields

We study a possibility of Lagrangian formulation for free higher spin bosonic totally symmetric tensor field on the background manifold characterizing by the arbitrary metric, vector and third rank tensor fields in framework of BRST approach. Assuming existence of massless and flat limits in the Lagrangian and using the most general form of the operators of constraints we show that the algebra generated by these operators will be closed only for constant curvature space with no nontrivial coupling to the third rank tensor and the strength of the vector fields. This result finally proves that the consistent Lagrangian formulation at the conditions under consideration is possible only in constant curvature Riemann space.Comment: 11 pages; v2: minor typos corrected, a reference adde

Dolan-Grady Relations and Noncommutative Quasi-Exactly Solvable Systems

We investigate a U(1) gauge invariant quantum mechanical system on a 2D noncommutative space with coordinates generating a generalized deformed oscillator algebra. The Hamiltonian is taken as a quadratic form in gauge covariant derivatives obeying the nonlinear Dolan-Grady relations. This restricts the structure function of the deformed oscillator algebra to a quadratic polynomial. The cases when the coordinates form the su(2) and sl(2,R) algebras are investigated in detail. Reducing the Hamiltonian to 1D finite-difference quasi-exactly solvable operators, we demonstrate partial algebraization of the spectrum of the corresponding systems on the fuzzy sphere and noncommutative hyperbolic plane. A completely covariant method based on the notion of intrinsic algebra is proposed to deal with the spectral problem of such systems.Comment: 25 pages; ref added; to appear in J. Phys.

On the existence of the second Dirac operator in Riemannian space

We describe a Riemannian space class where the second Dirac operator arises and prove that the operator is always equivalent to a standard Dirac one. The particle state in this gravitational field is degenerate to some extent and we introduce an additional value in order to describe a particle state completely. Some supersymmetry constructions are also discussed. As an example we study all Riemannian spaces with a five-dimentional motion group and find all metrics for which the second Dirac operator exists. On the basis of our discussed examples we hypothesize about the number of second Dirac operators in Riemannian space.Comment: LaTex, 10 pages, no figure

Supersymmetry and discrete transformations of the Dirac operators in Taub-NUT geometry

It is shown that the N=4 superalgebra of the Dirac theory in Taub-NUT space has different unitary representations related among themselves through unitary U(2) transformations. In particular the SU(2) transformations are generated by the spin-like operators constructed with the help of the same covariantly constant Killing-Yano tensors which generate Dirac-type operators. A parity operator is defined and some explicit transformations which connect the Dirac-type operators among themselves are given. These transformations form a discrete group which is a realization of the quaternion discrete group. The fifth Dirac operator constructed using the non-covariant Killing-Yano tensor of the Taub-NUT space is quite special. This non-standard Dirac operator is connected with the hidden symmetry and is not equivalent to the Dirac-type operators of the standard N=4 supersymmetry.Comment: 14 pages, latex, no figure

Subjective Assessment of the Effective Contract Impact in Universities on Improving the Indicators of Lectures` Research Work

В предлагаемой исследовательской статье освещены результаты социологического опроса, направленного на изучение мнения профессорско-преподавательского состава о влиянии системы эффективного контракта на повышение показателей научно-исследовательской работы в вузах России и Белоруссии. Авторами анализируются зарубежные и российские оценки эффективности деятельности научно-педагогических работников. Дизайном выполненной работы явилось поперечное (одномоментное) исследование. В качестве объектов исследования были выбраны представители профессорско-преподавательского состава вуза, перешедшего на эффективный контракт (Северный государственный медицинский университет, Россия), и вузов, не перешедших на него (Тобольский пединститут им. Д. И. Менделеева – филиал ТюмГУ, Россия); Республиканский институт высшей школы и Белорусский государственный педагогический университет им. Максима Танка (Республика Беларусь). Общая численность респондентов – 395 человек. В качестве инструментария социологического исследования использовалась полуструктурированная анкета, разработанная Национальным исследовательским университетом «Высшая школа экономики». Исследование показало, что представители обеих групп были осведомлены о назначении эффективного контракта и частоте оценки критериев деятельности преподавателей для назначения им стимулирующих выплат. Респонденты из вузов, не перешедших на эффективный контракт, получают информацию о последнем из сети Интернет и имеют завышенный уровень ожиданий относительно своих будущих доходов. Респонденты из обеих групп подчеркнули важность определения наибольшего веса при расчете стимулирующих выплат за публикационную и грантовую активность и за участие в научных конференциях. Также установлено, что не все преподаватели готовы к выполнению изложенных в положениях эффективного контракта требований к научно-исследовательской работе, но отмечают позитивное влияние его внедрения на показатели научной деятельности. В группе респондентов из вузов, не перешедших на эффективный контракт, одна треть высказала сомнения в его существенном влиянии на повышение показателей научно-исследовательской работы. Данная статья будет полезна руководству вузов при расчете весовых коэффициентов по показателям и критериям оценки деятельности профессорско-преподавательского состава для назначения стимулирующих выплат. Интересным исследованием в дальнейшем может быть сравнение субъективных оценок, полученных в вузах, не использующих эффективный контракт, и после его внедрения в этих же учреждениях.Article shows the results of a sociological survey aimed at studying the opinion of the teaching staff on the impact of an effective contract on improving the research work indicators in universities in Russia and Belarus. The survey analyzes foreign and Russian evaluation of academic staff performance. A cross-sectional (single-point) study was the design. The focus of research was the academic staff of educational institutions that went over to an effective contract (Northern State Medical University, Russia) and without the implementation of an effective contract (Tobolsk Pedagogical Institute named after D. I. Mendeleev (branch) the University of Tyumen, Russia; National Institute for Higher Education and Belarusian State Pedagogical University named after Maxim Tank, Belarus), the overall number of respondents is 395. A semi-structured questionnaire created by the National Research University Higher School of Economics was used as a tool for sociological research. The study showed that both study groups are aware of an effective contract purpose and criteria evaluation frequency. Respondents from universities without an effective contract receive information from the Internet and have overestimated expectations of the future income level. Respondents of both groups underlined the importance of determining the greatest weight in calculating incentive payments for publication and grant activity, participation in scientific conferences. When implementing an effective contract, not all lecturers are ready to fulfill the criteria for research work, but they note the positive effect of its implementation on the indicators of scientific activity. In the group of respondents without an effective contract, one third expressed doubts about the significant impact of the implementation of an effective contract on improving the performance of research work. This article will be useful to the management of universities when calculating weighting coefficients according to indicators and criteria for evaluating the activities of the teaching staff for the appointment of incentive payments. An interesting study in the future may be a comparison of subjective assessments obtained in universities without an effective contract, and after its implementation in the same universities

PT symmetric models in more dimensions and solvable square-well versions of their angular Schroedinger equations

For any central potential V in D dimensions, the angular Schroedinger equation remains the same and defines the so called hyperspherical harmonics. For non-central models, the situation is more complicated. We contemplate two examples in the plane: (1) the partial differential Calogero's three-body model (without centre of mass and with an impenetrable core in the two-body interaction), and (2) the Smorodinsky-Winternitz' superintegrable harmonic oscillator (with one or two impenetrable barriers). These examples are solvable due to the presence of the barriers. We contemplate a small complex shift of the angle. This creates a problem: the barriers become "translucent" and the angular potentials cease to be solvable, having the sextuple-well form for Calogero model and the quadruple or double well form otherwise. We mimic the effect of these potentials on the spectrum by the multiple, purely imaginary square wells and tabulate and discuss the result in the first nontrivial double-well case.Comment: 21 pages, 5 figures (see version 1), amendment (a single comment added on p. 7

On the cubic interactions of massive and partially-massless higher spins in (A)dS

Cubic interactions of massive and partially-massless totally-symmetric higher-spin fields in any constant-curvature background of dimension greater than three are investigated. Making use of the ambient-space formalism, the consistency condition for the traceless and transverse parts of the parity-invariant interactions is recast into a system of partial differential equations. The latter can be explicitly solved for given s_1-s_2-s_3 couplings and the 2-2-2 and 3-3-2 examples are provided in detail for general choices of the masses. On the other hand, the general solutions for the interactions involving massive and massless fields are expressed in a compact form as generating functions of all the consistent couplings. The St\"uckelberg formulation of the cubic interactions as well as their massless limits are also analyzed.Comment: 42 pages, 2 tables, LaTex. Comments on two-derivative couplings involving partially-massless spin-2 fields added, typos corrected, references added. v2: final version to appear in JHEP. v3: formulae (3.4) and (3.9) correcte