\({{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2\)-graded mechanics admits four types of particles: ordinary bosons, two classes of fermions (fermions belonging to different classes commute among each other) and exotic bosons. In this paper we construct the basic \({{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2\)-graded worldline multiplets (extending the cases of one-dimensional supersymmetry) and compute, based on a general scheme, their invariant classical actions and worldline sigma-models. The four basic multiplets contain two bosons and two fermions. They are (2, 2, 0), with two propagating bosons and two propagating fermions, \((1,2,1)_{[00]}\) (the ordinary boson is propagating, while the exotic boson is an auxiliary field), \((1,2,1)_{[11]}\) (the converse case, the exotic boson is propagating, while the ordinary boson is an auxiliary field) and, finally, (0, 2, 2) with two bosonic auxiliary fields. Classical actions invariant under the \({{\mathbb {Z}}}_2\times {\mathbb {Z}}_2\)-graded superalgebra are constructed for both single multiplets and interacting multiplets. Furthermore, scale-invariant actions can possess a full \({{\mathbb {Z}}}_2\times {\mathbb {Z}}_2\)-graded conformal invariance spanned by 10 generators and containing an sl(2) subalgebra.A preprint version of the article is available at ArXiv.

中文翻译：

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$$ {{\ mathbb {Z}}} _ 2 \ times {{\ mathbb {Z}}} _ 2 $$ Z2×Z2级力学：经典理论

\（{{\ mathbb {Z}}} _ 2 \ times {{\ mathbb {Z}}} _ 2 \）-等级力学允许四种类型的粒子：普通玻色子，两类费米子（属于不同类的费米子在彼此）和异国玻色子。在本文中，我们构造了基本的\（{{\ mathbb {Z}}} _ 2 \ times {{\ mathbb {Z}}} _ 2 \）级的世界线多重峰（扩展了一维超对称的情况）并进行计算，基于通用方案，它们的不变经典动作和世界线sigma模型。四个基本多重峰包含两个玻色子和两个费米子。它们是（2，2，0），具有两个正在传播的玻色子和两个正在传播的费米子\（（1,2,1）_ {[00]} \）（普通玻色子正在传播，而奇异玻色子是辅助字段），\（（1,2,1）_ {[11]} \）（相反的情况是，外来玻色子正在传播，而普通玻色子是一个辅助场），最后（0，2，2）具有两个玻色子辅助场。在\（{{\ mathbb {Z}}} _ 2 \ times {\ mathbb {Z}} _ 2 \）分级的超代数下构造的经典动作是不变的，它同时针对单个多重峰和交互多重峰构造。此外，尺度不变动作可以具有完整的\（{{{mathbb {Z}}} _ 2 \ times {\ mathbb {Z}} _ 2 \）级的共形不变性，该共形不变性由10个生成器构成，并包含sl（2）子代数。该文章的预印本可从ArXiv获得。