Off-diagonal coupling of supersymmetric SYK model

In this work, we consider the off-diagonal coupling between two supersymmetric SYK models, which preserves both supersymmetry and solvability. We found that the interaction terms of the $N=2$ supersymmetric SYK have a holographic interpretation as a possible supersymmetric traversable wormhole. First we introduce the coupling in the trivial Homologic $N=1$ SYK model as a simplified example. Similar couplings can be implied in $N=2$ chiral SYK model with BPS states. We propose a special form of $N=4$ SYK by introducing supermultiplets, and which also naturally include the coupling terms. The holographic picture of $N=4$ SYK does not have an eternal solution in the low energy limit. And the effective actions are studied in both thermal limit and low energy limit. We also investigate the SYK-like thermal field double states of the supersymmetric SYK and the transmission amplitude between single-side $N=2$ models in Lorentz time. Additionally, the multi-side $N=2,4$ OTOCs are also studied.