Taming the zoo of supersymmetric quantum mechanical models

Smilga, Andrei

doi:10.1007/jhep05(2013)119

Cited by 8 publications

(16 citation statements)

References 59 publications

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“…The expressions (2.15) are very much similar to the expressions of the supercharges in the standard Kähler -de Rham model (the SQM version of the Kähler sigma model in two spacetime dimensions introduced in [26]). The latter have the form [22,21],…”

Section: Propositionmentioning

confidence: 99%

“…Indeed, the corresponding canonical momenta commute in this case with the Hamiltonian, and one can perform the Hamiltonian reduction. Many different schemes of the Hamiltonian reduction are possible [22], but in this paper we only consider the reduction killing a half of bosonic dynamical variables.…”

Section: Propositionmentioning

confidence: 99%

“…Note the following. As was shown in [22], in many cases a set of quantum complex supercharges in a nontrivial SQM model, can be obtained from the set of complex supercharges of a free model by two operations: i) a similarity transformation (that needs not to be a unitary transformation) and ii) Hamiltonian reduction. This applies in particular to the quantum counterparts of the supercharges (2.15) of the HKT model, one can represent them aŝ…”

Section: Superchargesmentioning

confidence: 99%

“…= −iδ AB , P A = −i∂ A are the flat canonical momenta that commute with ψ B andψ C ; ω BC is an arbitrary complex matrix. Let us explain it in some more details (in [22], this statement was made only for the simplest case of a 4-dimensional conformally flat manifold). Let U = exp ω BC ψ BψC .…”

Section: Superchargesmentioning

confidence: 99%

“…It was shown in [22] that different SQM sigma models are related to each other by Hamiltonian reduction and/or similarity transformation of the holomorphic supercharges. For example, a Hamiltonian reduction of the Dolbeault model gives a so-called quasicomplex de Rham model [23].…”

Section: Introductionmentioning

confidence: 99%

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Bi-HKT and bi-Kähler supersymmetric sigma models

Fedoruk¹,

Smilga

2016

Journal of Mathematical Physics

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We study CKT (or bi-HKT) N = 4 supersymmetric quantum mechanical sigma models. They are characterized by the usual and the mirror sectors displaying each HKT geometry. When the metric involves isometries, a Hamiltonian reduction is possible. The most natural such reduction with respect to a half of bosonic target space coordinates produces an N = 4 model, related to the twisted Kähler model due to Gates, Hull and Rocek, but including certain extra F -terms in the superfield action.2. The manifolds where the bi-HKT models live have an even complex dimension, their real dimension being an integral multiple of 4. In the case when the metric and the torsions do not depend on a half of real coordinates (on a half of complex coordinates or on the imaginary parts of all complex coordinates), a Hamiltonian reduction eliminating these degrees of freedom is possible. One can describe the model thus obtained in terms of the ordinary and mirror chiral N = 4 (2, 4, 2) multiplets. In this way, one obtains the twisted Kähler (we prefer to call it bi-Kähler) model of Ref.[29] generalized by the inclusion of certain holomorphic terms in the superfield Lagrangian.Alternatively, one can describe it in terms of the real N = 2 (1, 2, 1) superfields. The reduced model has three distinguishable features: i) it belongs to the class of quasicomplex de Rham models with Hermitian (rather than just real) superfield metric, ii) it involves the holomorphic torsions [27,28,25], iii) it involves two different commuting complex structures whose Bismut torsions have an opposite sign. They take their origin in the two hypercomplex structures of the parent bi-HKT model.The plan of the paper is the following. In the next section, we give a detailed self-contained review of the relevant facts of the theory of HKT manifolds. In Section 3, we discuss generic CKT ≡ bi-HKT models, describe them in the superfield and also in the component language. We derive the compact component expressions for the bi-HKT supercharges which reveal the geometric structure of these models. In Section 4, we present and discuss quasicomplex bi-Kähler models, derive their component Lagrangian and the supercharges. We also describe in detail how the Hamiltonian reduction of a generic reducible HKT model and a generic reducible

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Section: Propositionmentioning

confidence: 99%

Section: Propositionmentioning

confidence: 99%

Section: Superchargesmentioning

confidence: 99%

Section: Superchargesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Bi-HKT and bi-Kähler supersymmetric sigma models

Fedoruk¹,

Smilga

2016

Journal of Mathematical Physics

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The geometry of gauged (super)conformal mechanics

Mirfendereski

Raeymaekers

Canberk

et al. 2022

J. High Energ. Phys.

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Motivated by recently explored examples, we undertake a systematic study of conformal invariance in one-dimensional sigma models where an isometry group has been gauged. Perhaps surprisingly, we uncover classes of sigma models which are only scale invariant in their ungauged form and become fully conformally invariant only after gauging. In these cases the target space of the gauged sigma model satisfies a deformation of the well-known conformal geometry constraints. We consider bosonic models as well as their $$ \mathcal{N} $$ N = 1, 2, 4 supersymmetric extensions. We solve the quantum ordering ambiguities in implementing (super-) conformal symmetry on the physical Hilbert space. Examples of our general results are furnished by the D(2, 1; 0)-invariant Coulomb branch quiver models relevant for black hole physics.

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$\!\!\!{\cal N}{=}\,4$ N = 4 mechanics with diverse (4, 4, 0) multiplets: Explicit examples of hyper-Kähler with torsion, Clifford Kähler with torsion, and octonionic Kähler with torsion geometries

Fedoruk¹,

Ivanov²,

Smilga

2014

Journal of Mathematical Physics

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We present simple models of N = 4 supersymmetric mechanics with ordinary and mirror linear (4, 4, 0) multiplets that give a transparent description of HKT, CKT, and OKT geometries. These models are treated in the N = 4 and N =2 superfield approaches, as well as in the component approach. Our study makes manifest that the CKT and OKT supersymmetric sigma models are distinguished from the more simple HKT models by the presence of extra holomorphic torsions in the supercharges. ⋆ On leave of absence from V.N. Karazin Kharkov National University, Ukraine 1 We follow the notation of [7] such that the numerals count the numbers of the physical bosonic, physical fermionic and auxiliary bosonic fields.2 N counts the number of real supercharges.1 [14,15,8] that complex sigma models admit an extended N = 4 supersymmetry for a geometry more general than HKT. This geometry with certain relaxed (compared to HKT) conditions for complex structures has been termed CKT in [8]. For some special CKT metrics (the so called OKT metrics), the models enjoying extended N = 8 supersymmetry can be written. Before explaining (we will do it shortly) what exactly HKT, CKT and OKT mean, let us say a few words about terminology. The abbreviation HKT stands for the Hyper-Kähler with Torsions geometry. It is worth mentioning that, generically, the HKT manifolds are not hyper-Kähler and not even Kähler. Their characteristic feature is the presence of three complex structures which form a quaternion algebra and are covariantly constant with respect to the appropriate connection with torsion. Likewise, CKT means Clifford Kähler with Torsions. This geometry is characterized by three complex structures which form the Clifford algebra but, in general, not the quaternion one. Finally, OKT (Octonionic Kähler with Torsions) manifolds are the manifolds of a dimension which is an integer multiple of 8, such that their geometry involves seven different complex structures forming the 7D Clifford algebra. These structures reveal some relation to the octonion algebra, though do not satisfy it. Neither CKT nor OKT manifolds are Kähler. Despite this mismatch (and the similar one for the generic HKT manifolds), we will follow the established literature tradition and use the names HKT, CKT and OKT for the considered type of geometries.The HKT geometry introduced in [14,15] is by now well understood by mathematicians [17]. This could not be really said, however, about the CKT and OKT geometries. One of the aims of our paper is to treat in detail some simple particular examples of the CKT and OKT manifolds to make more transparent their mathematical structure.

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Taming the zoo of supersymmetric quantum mechanical models

Cited by 8 publications

References 59 publications

Bi-HKT and bi-Kähler supersymmetric sigma models

Bi-HKT and bi-Kähler supersymmetric sigma models

The geometry of gauged (super)conformal mechanics

$\!\!\!{\cal N}{=}\,4$ N = 4 mechanics with diverse (4, 4, 0) multiplets: Explicit examples of hyper-Kähler with torsion, Clifford Kähler with torsion, and octonionic Kähler with torsion geometries

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