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 A 1 -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 Morelʹs conjecture on certain A 1 -homotopy groups of spheres.