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Super-Galilean conformal algebras of higher n, their properties and applications

Abstract

Conformal symmetry and supersymmetry play very important roles in modern physics and mathematics. The combined notion, superconformal symmetry, is also very important today. In most cases such symmetries are considered in the framework of relativistic theories. Supersymmetry is also applied to various problems in non-relativistic theories. On the other hand the role of conformal symmetry in a non-relativistic setting was considered limited. It has been observed recently that conformal algebras (mathematical tools for conformal symmetry) play important roles in a wide class of non-relativistic problems ranging from mechanics and electrodynamics to gravity. It is then natural to investigate possible roles of superconformal symmetry in non-relativistic theories. This problem has not been studied extensively. The purpose of our project is to clarify the role and the implications of superconformal symmetry in non-relativistic physics. This will be done in two steps. First step: we develop mathematical structures describing non-relativistic superconformal symmetry. One of these mathematical structure is referred to as the super-Galilean conformal algebras (SGCA). SGCA is not a unique object. There can be many varieties of them, especially linked to the number N of supersymmetries. At present we are very far from a complete understanding of all possible SGCA. Thus, a deeper understanding of the SGCA structures is mandatory before their physical applications. Furthermore, we need the representation theory of SGCA if we want to relate SGCA to physical problems. This is also a huge field of research and much has to be done. Therefore we start our project by studying SGCAs and their representations. Second step: Basing on SGCA representations we build physical models having non-relativistic superconformal symmetry. These models may include superconformal mechanics, sigma models, Chern-Simons matter, quantum many-body systems such as Calogero-type models, physical systems with higher order time derivatives and so on. We then study their properties. This will enable us to relate our models with physical phenomena in real world or other theories in physics. In this way our project will produce new mathematical structures (SGCA) and new physical models with superconformal symmetry. Our project opens new ways bothin physics and in mathematics. (AU)

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Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
AIZAWA, N.; KUZNETSOVA, Z.; TOPPAN, F.. l-oscillators from second-order invariant PDEs of the centrally extended conformal Galilei algebras. Journal of Mathematical Physics, v. 56, n. 3, . (14/03560-0)

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