The Riemannian metrics on a manifold are the real points of a complex domain of complex-valued metrics which has the Lorentzian metrics on its boundary. In quantum field theory the positivity of energy is encoded in the assertion that the theory extends holomorphically from Lorentzian manifolds to this complex domain. (The expresssion ``Wick rotation" is often used for this extension.) The domain has interseting geometric properties, suggestive for both mathematics and physics, which are especially vivid in the two-dimwnsional case. I shall discuss some of these properties in the talk, which is based on work in progress with Maxim Kontsevich. G.Segal, All Souls College, Oxford.