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Anton Sutulin

Publications and source records attributed to Anton Sutulin.

18 recordsLinked to original sources
arXiv2022

Integrability of supersymmetric Calogero-Moser models

We analyze the integrability of the ${\cal N}$-extended supersymmetric Calogero-Moser model. We explicitly construct the Lax pair $\{L,A\}$ for this system, which properly reproduces all equations of motion. After adding a supersymmetric oscillator potential we reduce the latter to solving $\dot{U}\,{=}\,A\,U$ for the time evolution operator $U(t)$. The bosonic variables, however, evolve independently of $U$ on closed trajectories, as is required for superintegrability. To visualize the structure of the conserved currents we derive the complete set of Liouville charges up to the fifth power in the momenta, for the ${\cal N}{=}\,2$ supersymmetric model. The additional, non-involutive, conserved charges needed for a maximal superintegrability of this model are also found.

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arXiv2020

New ${\cal N}{=}\,2$ superspace Calogero models

Starting from the Hamiltonian formulation of ${\cal N}{=}\,2$ supersymmetric Calogero models associated with the classical $A_n, B_n, C_n$ and $D_n$ series and their hyperbolic/trigonometric cousins, we provide their superspace description. The key ingredients include $n$ bosonic and $2n(n{-}1)$ fermionic ${\cal N}{=}\,2$ 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 ${\cal N}{=}\,2$ supersymmetrization for bosonic potentials with arbitrary repulsive two-body interactions.

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arXiv2019

Extended supersymmetric Calogero model

We present a surprising redefinition of matrix fermions which brings the supercharges of the $\cal N$-extended supersymmetric $A_{n-1}$ Calogero model introduced in [1] to the standard form maximally cubic in the fermions. The complexity of the model is transferred to a non-canonical and nonlinear conjugation property of the fermions. Employing the new cubic supercharges, we apply a supersymmetric generalization of a "folding" procedure for $A_{2n-1}\oplus A_1$ to explicitly construct the supercharges and Hamiltonian for arbitrary even-$\cal N$ supersymmetric extensions of the $B_n$, $C_n$ and $D_n$ rational Calogero models. We demonstrate that all considered models possess a dynamical $osp({\cal N}|2)$ superconformal symmetry.

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arXiv2019

Supersymmetric many-body Euler-Calogero-Moser model

We explicitly construct a supersymmetric $so(n)$ spin-Calogero model with an arbitrary even number $\cal N$ of supersymmetries. It features $\frac{1}{2}{\cal N}n(n{+}1)$ rather than ${\cal N}n$ fermionic coordinates and a very simple structure of the supercharges and the Hamiltonian. The latter, together with additional conserved currents, form an $osp({\cal N}|2)$ superalgebra. We provide a superspace description for the simplest case, namely ${\cal N}{=}2$ supersymmetry. The reduction to an $\cal N$-extended supersymmetric goldfish model is also discussed.

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arXiv2018

${\cal N}$-extended supersymmetric Calogero models

We propose a new ${\cal N}$-extended supersymmetric $su(n)$ spin-Calogero model. Employing a generalized Hamiltonian reduction adopted to the supersymmetric case, we explicitly construct a novel rational $n$-particle Calogero model with an arbitrary even number of supersymmetries. It features ${\cal N}n^2$ rather than ${\cal N}n$ fermionic coordinates and increasingly high fermionic powers in the supercharges and the Hamiltonian.

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arXiv2018

${\cal N}{=}\,4$ supersymmetric mechanics on curved spaces

We present ${\cal N}{=}\,4$ supersymmetric mechanics on $n$-dimensional Riemannian manifolds constructed within the Hamiltonian approach. The structure functions entering the supercharges and the Hamiltonian obey modified covariant constancy equations as well as modified Witten-Dijkgraaf-Verlinde-Verlinde equations specified by the presence of the manifold's curvature tensor. Solutions of original Witten-Dijkgraaf-Verlinde-Verlinde equations and related prepotentials defining ${\cal N}{=}\,4$ superconformal mechanics in flat space can be lifted to $so(n)$-invariant Riemannian manifolds. For the Hamiltonian this lift generates an additional potential term which, on spheres and (two-sheeted) hyperboloids, becomes a Higgs-oscillator potential. In particular, the sum of $n$ copies of one-dimensional conformal mechanics results in a specific superintegrable deformation of the Higgs oscillator.

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arXiv2017

SU(2$|$1) supersymmetric mechanics on curved spaces

We present SU$(2|1)$ supersymmetric mechanics on $n$-dimensional Riemannian manifolds within the Hamiltonian approach. The structure functions including prepotentials entering the supercharges and the Hamiltonian obey extended curved WDVV equations specified by the manifold's metric and curvature tensor. We consider the most general $u(2)$-valued prepotential, which contains both types (with and without spin variables), previously considered only separately. For the case of real Kähler manifolds we construct all possible interactions. For isotropic ($so(n)$-invariant) spaces we provide admissible prepotentials for any solution to the curved WDVV equations. All known one-dimensional SU$(2|1)$ supersymmetric models are reproduced.

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arXiv2017

Curved Witten-Dijkgraaf-Verlinde-Verlinde equation and ${\cal N}{=}\,4$ mechanics

We propose a generalization of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation from ${\mathbb R}^n$ to an arbitrary Riemannian manifold. Its form is obtained by extending the relation of the WDVV equation with ${\cal N}{=}\,4$ supersymmetric $n$-dimensional mechanics from flat to curved space. The resulting `curved WDVV equation' is written in terms of a third-rank Codazzi tensor. For every flat-space WDVV solution subject to a simple constraint we provide a curved-space solution on any isotropic space, in terms of the rotationally invariant conformal factor of the metric.

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arXiv2011

N=4 chiral supermultiplet interacting with a magnetic field

We couple N=4 chiral supermultiplet with an auxiliary N=4 fermionic supermutiplet containing on-shell four physical fermions and four auxiliary bosons. The latter ones play the role of isospin variables. We choose the very specific coupling which results in a component action containing only time derivatives of fermionic components presented in the auxiliary supermultiplet, which therefore may be dualized into auxiliary ones. The resulting component action describes the interaction of the chiral supermultiplet with a magnetic field constant on the pseudo-sphere $SU(1,1)/U(1)$. Then we specify the prepotential of our theory to get, in the bosonic sector, the action for the particle moving over the pseudo-sphere -- Lobachevsky space. We provided also the Hamiltonian formulation of this system and show that the full symmetry group of our system is $SU(1,1)\times U(1)$. The currents forming the $su(1,1)$ algebra are modified, as compared to the bosonic case, by the fermionic and isospin terms, while the additional u(1) current contains only isospin variables. One of the most important features of our construction is the presence in the Hamiltonian and supercharges of all currents of the isospin group SU(2). Despite the fact that two of the su(2) currents ${T,{\bar{T}}}$ enter the Hamiltonian only through the Casimir operator of the SU(2) group, they cannot be dropped out, even after fixing the total isospin of the system, because these currents themselves enter into the supercharges. We also present the Hamiltonian and supercharges describing the motion of a particle over the sphere $S^2$ in the background of constant magnetic field. In this case the additional isospin currents form the $su(1,1)$ algebra.

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arXiv2011

SU(2) reductions in N=4 multidimensional supersymmetric mechanics

We perform an su(2) Hamiltonian reduction in the bosonic sector of the su(2)-invariant action for two free (4, 4, 0) supermultiplets. As a result, we get the five dimensional N=4 supersymmetric mechanics describing the motion of an isospin carrying particle interacting with a Yang monopole. We provide the Lagrangian and Hamiltonian descriptions of this system. Some possible generalizations of the action to the cases of systems with a more general bosonic action, a four-dimensional system which still includes eight fermionic components, and a variant of five-dimensional N=4 mechanics constructed with the help of the ordinary and twisted N=4 hypermultiplets were also considered.

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arXiv2010

N=4, d=1 Supersymmetric Hyper-Kaehler Sigma Models and Non-Abelian Monopole Background

We construct a Lagrangian formulation of \Nf supersymmetric mechanics with hyper-Kähler sigma models in a bosonic sector in the non-Abelian background gauge field. The resulting action includes a wide class of \Nf supersymmetric mechanics describing the motion of an isospin-carrying particle over spaces with non-trivial geometry. In two examples we discuss in details, the background fields are identified with the field of BPST instantons in the flat and Taub-NUT spaces.

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arXiv2010

Five-dimensional N=4 supersymmetric mechanics

We perform an $su(2)$ Hamiltonian reduction in the bosonic sector of the $su(2)$-invariant action for two free $(4,4,0)$ supermultiplets. As a result, we get the five dimensional \Nf supersymmetric mechanics describing the motion of an isospin carrying particle interacting with a Yang monopole. Some possible generalizations of the action to the cases of systems with a more general bosonic action constructed with the help of the ordinary and twisted \Nf hypermultiplets are considered.

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arXiv2010

N=4, d=1 Supersymmetric Hyper-Kaehler Sigma Models with Isospin Variables

We provide a Lagrangian formulation of \Nf supersymmetric mechanics describing the motion of an isospin carrying particle on conformal to hyper-Kähler spaces in a non-Abelian background gauge field. In two examples we discuss in details, this background field is identified with the field of BPST instantons in the flat and Taub-NUT spaces.

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arXiv2010

Dual multiplets in N=4 superconformal mechanics

We propose Lagrangian formulations of a three-particles translation invariant $N=4$ superconformal mechanics based on the standard and twisted $(1,4,3)$ supermultiplets. We show that in the appropriate set of coordinates, in which the bosonic kinetic terms for each of the two cases take conformally-flat forms, the corresponding models produce just similar physical systems with the flipped angular part of bosonic potential $U(x)\rightarrow 1/U(x)$. This flipping looks so simple only being written in the "angle" variable, while in the standard variables it looks more complicated to preserve the superconformal symmetry. We demonstrate at both the superfield and the component level, how familiar $N=4$ supersymmetric 3-particles models, including $3-Calogero$, ${\it BC_2}$ and ${\it G_2}$ ones, can be constructed with twisted supermultiplets. We also present some new explicit examples of 3-dimensional superconformal mechanics.

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arXiv2010

N=4 Supersymmetry and the BPST Instanton

In this paper we construct the Lagrangian and Hamiltonian formulations of N=4 supersymmetric systems describing the motion of an isospin particle on a conformally flat four-manifold with SO(4) isometry carrying the non-Abelian field of a BPST instanton. The conformal factor can be specified to yield various particular systems, such as superconformally invariant mechanics as well as a particle on the four-sphere, the pseudosphere or on R x S^3. The isospin degrees of freedom arise as bosonic components of an additional fermionic N=4 supermultiplet, whose other components are rendered auxiliary by a nonlocal redefinition. Our on-shell component action coincides with the one recently proposed in arXiv:0912.3289.

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arXiv2009

Three dimensional N=4 supersymmetric mechanics with Wu-Yang monopole

We propose Lagrangian and Hamiltonian formulations of a N=4 supersymmetric three-dimensional isospin-carrying particle moving in the non-Abelian field of a Wu-Yang monopole and in some specific scalar potential. This additional potential is completely fixed by N=4 supersymmetry and in the simplest case of flat metrics it coincides with that which provides the existence of the Runge-Lenz vector for the bosonic subsector. The isospin degrees of freedom are described on the Lagrangian level by bosonic auxiliary variables forming N=4 supermultiplet with additional, also auxiliary fermions. Being quite general, the constructed systems include such interesting cases as N=4 superconformally invariant systems with Wu-Yang monopole, the particles living in the flat R^3 and in the R x S^2 spaces and interacting with the monopole, and also the particles moving on three-dimensional sphere and pseudo-sphere with Wu-Yang monopole sitting in the center. The superfield Lagrangian description of these systems is so simple that one could wonder to see how all couplings and the proper coefficients arise while passing to the component action.

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arXiv2007

Hierarchy of N=8 Mechanics Models

Using the N=4 superspace approach in one dimension (time), we construct general N=8 supersymmetric mechanics actions for the multiplets (b,8,8-b) classified in hep-th/0406015, with the main focus on the previously unexplored cases of (8,8,0), (7,8,1) and (6,8,2), as well as on (5,8,3) for completeness. N=8 supersymmetry of the action amounts to a harmonicity condition for the Lagrangian with respect to its superfield arguments. We derive the generic off-shell component action for the ``root'' multiplet (8,8,0), prove that the actions for all other multiplets follow from it through automorphic dualities and argue that this hierarchical structure is universal. The bosonic target geometry in all cases is conformally flat, with a unique scalar potential (except for the root multiplet). We show that the N=4 superfield constraints respect the full R-symmetry and find the explicit realization of its quotient over the manifest R-symmetry on superfields and component fields. Several R-symmetric N=4 superfield Lagrangians with N=8 supersymmetry are either newly found or reproduced by a simple universal method.

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