N=4 Multi-Particle Mechanics, WDVV Equation and Roots

Lechtenfeld, Olaf

doi:10.3842/sigma.2011.023

Cited by 14 publications

(41 citation statements)

References 32 publications

(139 reference statements)

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“…The leftmost equation in the first line in (7) is a variant of the WDVV equation. With regard to the SU(1, 1|2) mechanics it has been extensively studied in [4,6,7,27,28]. In particular, each solution of the WDVV equation satisfying (10) will qualify to describe some SU(1, 1|n) superconformal mechanics.…”

Section: Prepotentials F Related To Root Systemsmentioning

confidence: 99%

“…The explicit construction of the m-particle SU(1, 1|2) superconformal Calogero model reduces to solving a variant of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation [1,4]. Although plenty of interesting solutions to the WDVV equation were found in terms of root systems and their deformations [1,6,7,26,27,28,29], the construction of interacting models seems unfeasible beyond m = 3. Since, in the context of [25], it is the structure of the superconformal group which matters, any multi-particle SU(1, 1|2) mechanics appears to be a good candidate.…”

Section: Introductionmentioning

confidence: 99%

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Superconformal SU(1, 1|n) mechanics

Galajinsky

Lechtenfeld

2016

J. High Energ. Phys.

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Recent years have seen an upsurge of interest in dynamical realizations of the superconformal group SU(1, 1|2) in mechanics. Remarking that SU(1, 1|2) is a particular member of a chain of supergroups SU(1, 1|n) parametrized by an integer n, here we begin a systematic study of SU(1, 1|n) multi-particle mechanics. A representation of the superconformal algebra su(1, 1|n) is constructed on the phase space spanned by m copies of the (1, 2n, 2n−1) supermultiplet. We show that the dynamics is governed by two prepotentials V and F , and the Witten-Dijkgraaf-Verlinde-Verlinde equation for F shows up as a consequence of a more general fourth-order equation. All solutions to the latter in terms of root systems reveal decoupled models only. An extension of the dynamical content of the (1, 2n, 2n−1) supermultiplet by angular variables in a way similar to the SU(1, 1|2) case is problematic.

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Section: Prepotentials F Related To Root Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Superconformal SU(1, 1|n) mechanics

Galajinsky

Lechtenfeld

2016

J. High Energ. Phys.

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“…The choice of OSp(4|2) is the unique one consistent with free kinetic terms for the bosonic fields, as long as one insists on the supercharges (4.30) being linear in fermionic variables [41]. Allowing for supercharge terms cubic in the fermionic operators will constrain their coefficient functions by the so-called WDVV equations [38,39,40,42,43,41]. It will be interesting to develop a superspace variant of this more general situation.…”

Section: Concluding Remarks and Outlookmentioning

confidence: 99%

Quantum SU(2|1) supersymmetric Calogero-Moser spinning systems

Fedoruk¹,

Ivanov²,

Lechtenfeld

et al. 2018

J. High Energ. Phys.

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SU(2|1) supersymmetric multi-particle quantum mechanics with additional semi-dynamical spin degrees of freedom is considered. In particular, we provide an N = 4 supersymmetrization of the quantum U(2) spin Calogero-Moser model, with an intrinsic mass parameter coming from the centrally-extended superalgebra su(2|1). The full system admits an SU(2|1) covariant separation into the center-of-mass sector and the quotient. We derive explicit expressions for the classical and quantum SU(2|1) generators in both sectors as well as for the total system, and we determine the relevant energy spectra, degeneracies, and the sets of physical states.2 It was shown in [17], that the centrally extended superalgebra su(2|1) can be represented as a semi-direct sum of su(2|1) and an extra R-symmetry generator: su(2|1) ≃ su(2|1)+ ⊃ u(1). The central charge is a combination of the R-symmetry generator and the internal U(1) generator of su(2|1). In the models under consideration the central charge operator is identified with the canonical Hamiltonian.3 The kinetic term of the variables Z k a in the action (1.5) is of the first-order in the time derivatives, in contrast to the dynamical variable X a b with the second-order kinetic term. Just for this reason we call Z k a ,Z a k semi-dynamical variables. In the Hamiltonian (see below), they appear only in the interaction terms and enter through the SU(2) current S (ik) .

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“…The examples of ∨-systems include all two-dimensional systems, Coxeter systems and the so-called deformed root systems [33,49,53], but the full classification is still an open problem (see the latest results in [15,16,29,47]). The combinatorial (or matroidal) structure of all known ∨-systems is quite special, but there are no general results known so far.…”

Section: Introductionmentioning

confidence: 99%

$\vee$ -Systems, Holonomy Lie Algebras, and Logarithmic Vector Fields

Feigin

Веселов

2017

International Mathematics Research Notices

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It is shown that the description of certain class of representations of the holonomy Lie algebra g ∆ associated to hyperplane arrangement ∆ is essentially equivalent to the classification of ∨-systems associated to ∆. The flat sections of the corresponding ∨-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any ∨-system is free in Saito's sense and show this for all known ∨-systems and for a special class of ∨-systems called harmonic, which includes all Coxeter systems. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations. We give formulas for these deformations as well as for the potentials of the classical families of harmonic ∨-systems.

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N=4 Multi-Particle Mechanics, WDVV Equation and Roots

Cited by 14 publications

References 32 publications

Superconformal SU(1, 1|n) mechanics

Superconformal SU(1, 1|n) mechanics

Quantum SU(2|1) supersymmetric Calogero-Moser spinning systems

$\vee$ -Systems, Holonomy Lie Algebras, and Logarithmic Vector Fields

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