Exceptional field theory: SL(5)

In this work the exceptional field theory formulation of supergravity with SL(5) gauge group is considered. This group appears as a U-duality group of $D=7$ maximal supergravity. In the formalism presented the hidden global duality group is promoted into a gauge group of a theory in dimensions 7+number of extended directions. This work is a continuation of the series of works for $E_{8,7,6}, SO(5,5)$ and $SL(3)\times SL(2)$ duality groups.