Fuzzy supersphere and supermonopole

Abstract

It is well-known that coordinates of a charged particle in a monopole background become noncommutative. In this paper, we study the motion of a charged particle moving on a supersphere in the presence of a supermonopole. We construct a supermonopole by using a supersymmetric extension of the first Hopf map. We investigate algebras of angular momentum operators and supersymmetry generators. It is shown that coordinates of the particle are described by fuzzy supersphere in the lowest Landau level. We find that there exist two kinds of degenerate wavefunctions due to the supersymmetry. Ground state wavefunctions are given by the Hopf spinor and we discuss their several properties.

Суперсимметризация Моделей Ландау

“…На N -м УЛ физическая киральная волновая функция имеет специальный вид: она выражается через аналитическую функцию от (z, ζ, ξ) согласно формуле 6) можно показать, что…”

“…минимальные суперрасширения модели Хэлдейна на S 2 , рассмотрены в работах [4], [5] (см. также работу [6]). Для лучшего понимания общих особенностей суперсимметричных моделей Лан-дау оказалось полезным изучить планарные пределы таких суперобобщений.…”

SUSY Quantum Hall Effect on Non-Anti-Commutative Geometry

“…2.2 SUSY Hopf map and supermonopole [32,33,34] As is well known, the mathematical background of the Dirac monopole is given by the (1st) Hopf map (see for instance [31]). Similarly, the supermonopole 2 is introduced as a SUSY extension of the Hopf bundle [32].…”

“…The spherical SUSY quantum Hall ef fect 3.1 The spherical SUSY Landau problem [34] With the above set-up, we discuss one-particle problem on a supersphere in a supermonopole background. Since the particle on a supersphere is concerned, the Hamiltonian does not contain the radial part, and is simply given by the angular part…”

“…4 Emergence of non-anti-commutative geometry [34,35] Originally, x a and θ α were the classical coordinates on supersphere and not operators, while, in the LLL, they are effectively regarded as operators. It is because, in the LLL limit (M → 0) the covariant angular momenta can be neglected (see equations (6.2)), and x a and θ α are reduced to the OSp(1|2) operators as indicated by equation (3.5):…”