On 1D, $$ \mathcal{N} $$ = 4 supersymmetric SYK-type models. Part I

Proposals are made to describe 1D, $$ \mathcal{N} $$ N = 4 supersymmetrical systems that ex- tend SYK models by compactifying from 4D, $$ \mathcal{N} $$ N = 1 supersymmetric Lagrangians involving chiral, vector, and tensor supermultiplets. Quartic fermionic vertices are generated via in- tegrals over the whole superspace, while 2(q − 1)-point fermionic vertices are generated via superpotentials. The coupling constants in the superfield Lagrangians are arbitrary, and can be chosen to be Gaussian random. In that case, these 1D, $$ \mathcal{N} $$ N = 4 supersymmetric SYK models would exhibit Wishart-Laguerre randomness, which share the same feature among other 1D supersymmetric SYK models in literature. One difference with 1D, $$ \mathcal{N} $$ N = 1 and $$ \mathcal{N} $$ N = 2 models though, is our models contain dynamical bosons, but this is consistent with other 1D, $$ \mathcal{N} $$ N = 4 and 2D, $$ \mathcal{N} $$ N = 2 models in literature. Added conjectures on duality and possible mirror symmetry realizations in these models is noted.