Riemann-Roch for real varieties

If E is a C^\infty complex vector bundle on an oriented C^\infty manifold \Sigma, diffeomorphic to a circle, then the space of sections of E has a canonical polarization in the sense of Pressley and Segal and so one has its determinantal gerbe with lien C^*, the group of nonzero complex numbers. If q:\Sigma-->B is a smooth family of circles as above and E is a vector bundle on \Sigma, then the smooth direct image q_*(E) is an infinite-dimensional bundle with fibers as above and so we have its determinantal gerbe on B with lien being the sheaf of invertible complex valued C^\infty functions, it gives a class in H^3(B, Z). In this paper we consider a family q:\Sigma-->B as above but with fibers being compact oriented C^\infty manifolds of dimension d. For a bundle E on \Sigma one expects q_*(E) to possess a determinantal d-gerbe and hence to give a class in H^{d+2}(B, Z). We construct directly, by means of a version of the Chern-Weil theory, the real version of this would be class. We further prove a real version of the Grothendieck-Riemann-Roch theorem describing this class as a direct image of a certain characteristic class of E.