A Givental like decomposition formula for $T^4$ tensor model

In this talk I will review recent results in the study of the simplest interacting tensor model using its multi-matrix representation. This representation is obtained through a Hubbard-Stratanovitch transformation applied to the quartic melonic tensor model. I first use this transformation to give a description á la Givental of this tensor model, i.e. as a differential operator acting on a product of Hermitian matrix model. However the reader should be warned that the differential is not a Givental operator although the decomposition looks like the same in spirit. This decomposition allows to understand how the Hirota's equations of the Hermitian one matrix model transform to give bilinear identities on the partition function of the quartic melonic tensor model. In a second part I will present results obtained in collaboration with Viet Anh Nguyen and Bertrand Eynard. In this works we rederive known results about this tensor model (2-point function, NLO computations) using this multi-matrix representation.