Integration on minimal ${\mathbb{Z}}_2^2$-superspace and emergence of space

Abstract

We investigate the possibilities of integration on the minimal ${\mathbb{Z}}_2^2$-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 ${\mathbb{R}}^2$ despite the fact that the $(0,0)$ part of the minimal ${\mathbb{Z}}_2^2$-superspace is $\mathbb{R}.$

arXiv comment12 pages, no figure

References in corpus (28)

• Double-Graded Supersymmetric Quantum Mechanics
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded parastatistics in multiparticle quantum Hamiltonians
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded mechanics: the classical theory
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded mechanics: the quantization
• On a ${\mathbb{Z}}_2^n$-Graded Version of Supersymmetry
• $\mathcal{N}$-Extension of duble-graded supersymmetric and superconformal quantum mechanics
• ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded supersymmetry: $2$-d sigma models
• Inequivalent quantizations from gradings and ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ parabosons
• Higher Trace and Berezinian of Matrices over a Clifford Algebra
• Is the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded sine-Gordon equation integrable?
• Splitting theorem for ${\mathbb{Z}}_2^n$-supermanifolds
• Classification of minimal ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded Lie (super)algebras and some applications
• ${\mathbb{Z}}_2^n$-Graded extensions of supersymmetric quantum mechanics via Clifford algebras
• Comments of ${\mathbb{Z}}_2^2$-supersymmetry in superfield formalism
• The Graded Differential Geometry of Mixed Symmetry Tensors
• New aspects of the Z${}_{\text{2}}$ $\times$ Z${}_{\text{2}}$-graded 1D superspace: induced strings and 2D relativistic models
• Functional analytic issues in ${\mathbb{Z}}_2^n$-Geometry
• Experimental realization of para-particle oscillators
• Riemannian structures on ${\mathbb{Z}}_2^n$-manifolds
• Local Forms of Morphisms of Colored Supermanifolds
• ${\mathbb{Z}}_2^3$-Graded Extensions of Lie Superalgebras and Superconformal Quantum Mechanics
• The Schwarz-Voronov Embedding of ${\mathbb{Z}}_2^n$-Manifolds
• Cohomological Approach to the Graded Berezinian
• Odd Connections on Supermanifolds: Existence and relation with Affine Connections
• Symplectic ${\mathbb{Z}}_2^n$-manifolds
• Beyond the $10$-fold way: $13$ associative $Z_2\times Z_2$-graded superdivision algebras
• The Geometry of Supersymmetry / A concise introduction
• Linear ${\mathbb{Z}}_2^n$-Manifolds and Linear Actions

Cited by in corpus (5)

• Orthosymplectic $Z_2\times Z_2$-graded Lie superalgebras and parastatistics
• Integrable ${\mathbb{Z}}_2^2$-graded Extensions of the Liouville and Sinh-Gordon Theories
• ${\mathbb{Z}}_2^2$-graded supersymmetry via superfield on minimal ${\mathbb{Z}}_2^2$-superspace
• $\mathcal{N}=2$ Double graded supersymmetric quantum mechanics via dimensional reduction
• Affine extensions of ${\mathbb{Z}}_2^2$-graded $osp(1|2)$ and Virasoro algebra