A1-representability of hermitian K-theory and Witt groups

We show that hermitian K-theory and Witt groups are representable both in the unstable and in the stable A1-homotopy category of Morel and Voevodsky. In particular, Balmer Witt groups can be nicely expressed as homotopy groups of a topological space. The proof includes a motivic version of real Bott periodicity. Consequences include other new results related to projective spaces, blow ups and homotopy purity. The results became part of the proof of Morel's conjecture on certain A1-homotopy groups of spheres.