The real Betti realization of motivic Thom spectra and of very effective Hermitian K-theory

Abstract: Real Betti realization is a symmetric monoidal functor from the category of motivic spectra to that of topological spectra, extending the functor that associates to a scheme over $\mathbb{R}$ the space of its real points. In this article, we prove some results about the real Betti realizations of certain motivic $\mathcal{E}1$- and $\mathcal{E}\infty$-rings. We show that the motivic Thom spectrum functor and the topological one correspond to each other, as symmetric monoidal functors, under real realization. In particular, we obtain equivalences of $\mathcal{E}\infty$-rings between the real realizations of the variants $\mathsf{MGL}$, $\mathsf{MSL}$, and $\mathsf{MSp}$ of algebraic cobordism, and the variants $\mathsf{MO}$, $\mathsf{MSO}$, and $\mathsf{MU}$ of topological cobordism, respectively. Using this, we identify the $\mathcal{E}_1$-ring structure on the real realization of $\mathsf{ko}$, the very effective cover of Hermitian K-theory, by an explicit 2-local fracture square, as being equivalent to $\mathsf{L}(\mathbb{R}){\geq 0}$, the connective L-theory spectrum of $\mathbb{R}$.