$${{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2$$-graded mechanics: the classical theory

<jats:title>Abstract</jats:title><jats:p><jats:inline-formula><jats:alternatives><jats:tex-math>$${{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>-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 <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\mathbb {Z}}}_2\times {{\mathbb {Z}}}_2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>-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, <jats:inline-formula><jats:alternatives><jats:tex-math>$$(1,2,1)_{[00]}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mn>00</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:msub></mml:math></jats:alternatives></jats:inline-formula> (the ordinary boson is propagating, while the exotic boson is an auxiliary field), <jats:inline-formula><jats:alternatives><jats:tex-math>$$(1,2,1)_{[11]}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mn>11</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:msub></mml:math></jats:alternatives></jats:inline-formula> (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 <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\mathbb {Z}}}_2\times {\mathbb {Z}}_2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>-graded superalgebra are constructed for both single multiplets and interacting multiplets. Furthermore, scale-invariant actions can possess a full <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\mathbb {Z}}}_2\times {\mathbb {Z}}_2$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mi>Z</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>-graded conformal invariance spanned by 10 generators and containing an <jats:italic>sl</jats:italic>(2) subalgebra.</jats:p>