To a symplectic manifold M one associates the space of smooth loop in M. Although of infinite dimension, it inheri ts a (quasi-) symplectic structure from that of M. An algebraic analogue to the space of smooth loops has been introduced by Kapranov and Vasserot. In this talk, we will define higher dimensional formal loop spaces generalizing that of K and V. We will then develop an appropriate context in which symplectic forms on those infinite dimensional algebraic objects make sense. This makes use of symplectic shifted derived algebraic geometry as introduced by Pantev, Toën, Vaquié and Vezzosi. During this presentation, we will need to introduce Tate modules. Those are infinite dimensional modules which behave nicely regarding duality.