Linear Z&lt;sub&gt;2&lt;/sub&gt;&lt;sup&gt;n&lt;/sup&gt;-Manifolds and Linear Actions

Bruce, Andrew James; Ibarguengoytia, Eduardo; Poncin, Norbert

doi:10.3842/sigma.2021.060

Cited by 5 publications

(3 citation statements)

References 40 publications

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“…Geometry of supermanifolds has been understood very well (see e.g. [5,6]), while higher graded supergeometry is currently the subject of active research [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. There are many open questions in higher graded supergeometry.…”

Section: Introductionmentioning

confidence: 99%

Integration on minimal Z22 -superspace and emergence of space

Aizawa,

Ito

2023

J. Phys. A: Math. Theor.

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We investigate the possibilities of integration on the minimal Z2 2-superspace. Two definitions are taken from the works by Poncin and Schouten and we examine their generalizations. It is shown that these definitions impose some restrictions on the integrable functions. We then introduce a new definition of integral, which is inspired by our previous work, and show that the definition does not impose restrictions on the integrable functions. An interesting feature of this definition is the emergence of a spatial coordinate which means that the integral is defined on R<sup>2 despite the fact that the (0,0) part of the minimal Z2 2-superspace is R

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Section: Introductionmentioning

confidence: 99%

Integration on minimal Z22 -superspace and emergence of space

Aizawa,

Ito

2023

J. Phys. A: Math. Theor.

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“…To have better understanding of Z n 2 -graded supersymmetry (Z n 2 -SUSY), consideration of Z n 2graded extensions of supersymmetric classical mechanics (Z n 2 -supermechanics) is quite useful as it provides a place where physics, representation theory, calculus on Z n 2 -graded variables and Z n 2 -graded geometry encounter. We remark that geometry on Z n 2 -graded manifolds is one of the topics under extensive studies in mathematics [22][23][24][25][26][27][28][29][30][31][32][33][34][35] and that the representation theory of higher graded superalgebra also attracts mathematical interest [36][37][38][39][40][41].…”

Section: Introductionmentioning

confidence: 99%

Irreducible representations of $\mathbb{Z}_2^2$-graded ${\cal N} =2$ supersymmetry algebra and $\mathbb{Z}_2^2$-graded supermechanics

Aizawa,

Doi

2022

Preprint

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Irreducible representations (irreps) of Z 2 2 -graded supersymmetry algebra of N = 2 are obtained by the method of induced representation and they are used to derive Z 2 2 -graded supersymmetric classical actions. The irreps are four dimensional for λ = 0 where λ is an eigenvalue of the Casimir element, and eight dimensional for λ = 0. The eight dimensional irreps reduce to four dimensional ones only when λ and an eigenvalue of Hamiltonian satisfy a particular relation. The reduced four dimensional irreps are used to define Z 2 2 -graded supersymmetry transformations and two types of classical actions invariant under the transformations are presented. It is shown that one of the Noether charges vanishes if all the variables of specific Z 2 2 -degree are auxiliary.

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“…The proven theorems offer orientation in an environment with many indeterminacies and show that the standard concepts concerned have a pretty good stability with regard to all the necessary choices. Applications can be expected in homotopic algebraic geometry [22,23,2,3,6] and higher supergeometry [7,8,9,21]. Indeed, these areas make extensive use of the functor of points and are the contexts from which the need arose to examine the subjects of this paper.…”

Section: Introductionmentioning

confidence: 99%

Indeterminacies and models of homotopy limits

Govzmann,

Pištalo,

Poncin

2021

Preprint

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In [12] we studied the indeterminacy of the value of a derived functor at an object using different definitions of a derived functor and different types of fibrant replacement. In the present work we focus on derived or homotopy limits, which of course depend on the model structure of the diagram category under consideration. The latter is not necessarily unique, which is an additional source of indeterminacy. In the case of homotopy pullbacks, we introduce the concept of full homotopy pullback by identifying the homotopy pullbacks associated with three different model structures of the category of cospan diagrams, thus increasing the number of canonical representatives. Finally, we define generalized representatives or models of homotopy limits and full homotopy pullbacks. The concept of model is a unifying approach that includes the homotopy pullback used in [17] and the homotopy fiber square defined in [14] in right proper model categories. Properties of the latter are generalized to models in any model category.

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Linear Z<sub>2</sub><sup>n</sup>-Manifolds and Linear Actions

Cited by 5 publications

References 40 publications

Integration on minimal Z22 -superspace and emergence of space

Integration on minimal Z22 -superspace and emergence of space

Irreducible representations of $\mathbb{Z}_2^2$-graded ${\cal N} =2$ supersymmetry algebra and $\mathbb{Z}_2^2$-graded supermechanics

Indeterminacies and models of homotopy limits

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