New N $$ \mathcal{N} $$ = 2 superspace Calogero models
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Springer
Received: December 18, Revised: March 23, Accepted: May 6, Published: May 27,
2019 2020 2020 2020
Sergey Krivonos,a Olaf Lechtenfeldb and Anton Sutulina a
Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Russia b Institut f¨ ur Theoretische Physik and Riemann Center for Geometry and Physics, Leibniz Universit¨ at Hannover, Appelstrasse 2, 30167 Hannover, Germany
E-mail: [email protected], [email protected], [email protected] Abstract: Starting from the Hamiltonian formulation of N = 2 supersymmetric Calogero models associated with the classical An , Bn , Cn and Dn series and their hyperbolic/trigonometric cousins, we provide their superspace description. The key ingredients include n bosonic and 2n(n−1) fermionic N = 2 superfields, the latter being subject to a nonlinear chirality constraint. This constraint has a universal form valid for all Calogero models. With its help we find more general supercharges (and a superspace Lagrangian), which provide the N = 2 supersymmetrization for bosonic potentials with arbitrary repulsive two-body interactions. Keywords: Extended Supersymmetry, Superspaces, Integrable Hierarchies ArXiv ePrint: 1912.05989
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)132
JHEP05(2020)132
New N = 2 superspace Calogero models
Contents 1
2 Hamiltonian description of N = 2 supersymmetric Calogero models
2
3 N = 2 superspace A1 ⊕ An−1 Calogero models
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4 N = 2 superspace Bn , Cn and Dn Calogero models
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5 New N = 2 supersymmetric n-particle models
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6 Conclusions
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A Self-consistency of the nonlinear chirality constraint
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1
Introduction
The rational Calogero model [1, 2] and its various generalizations (see [3] for a review) play a special role in mathematical physics. Being an integrable and solvable multi-particle one-dimensional system it appears in many different areas such as fluid mechanics, spin chains, gauge theory and string theory. The standard N = 2 supersymmetric extension of the n-particle Calogero models is based on n scalar bosonic N = 2 superfields with a component content (1, 2, 1) i.e. one physical boson, 2 fermions and one auxiliary field for each particle [4–6]. Being quite satisfactory for N = 2 supersymmetry, this construction fails to reproduce N = 4 supersymmetric Calogero models beyond four particles [7]. Recent progress in the construction of supersymmetric extensions of Calogero models was achieved by adding to the system more fermions as compared to the standard supersymmetrization [4–6]. It was inspired by supersymmetric extensions of the matrix models which, upon reduction or gauge fixing, give rise to the familiar bosonic systems. The superfield approach developed in [8–10] for the rational spin-Calogero models with N = 2, 4 supersymmetry was recently extended to N = 2, 4 supersymmetric hyperbolic Calogero models [11, 12]. However, it suffers from an unclear structure of the bosonic matrix model one has to start from. In the series of papers [1
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