Fully developed isotropic turbulence: symmetries and exact identities

We consider the regime of fully developed isotropic and homogeneous turbulence of the Navier-Stokes equation with a stochastic forcing. We present two gauge symmetries of the corresponding Navier-Stokes field theory, and derive the associated general Ward identities. Furthermore, by introducing a local source bilinear in the velocity field, we show that these symmetries entail an infinite set of exact and local relations between correlation functions. They include in particular the K\'arm\'an-Howarth relation and another exact relation for a pressure-velocity correlation function recently derived in Ref. [G. Falkovich, I. Fouxon, Y. Oz,J. Fluid Mech. 644, 465 (2010)], that we further generalize.