Connecting M-theory and type IIB parameterizations in Exceptional Field Theory

In the exceptional field theory, there are two natural parameterizations for the generalized metric; in terms of bosonic fields in the eleven-dimensional supergravity (M-theory parameterization) and the type IIB supergravity (type IIB parameterization). In order to translate various results known in the M-theory to the type IIB theory or vice versa, an explicit map between the two parameterizations will be useful. In this note, we present such a linear map. Comparing the two parameterizations under the linear map, we reproduce the known T-duality transformation rules for the supergravity fields. We also obtain the T-duality rules for non-geometric $\beta$-/$\gamma$-fields appearing in the type II theory on a torus.