Equivariant Twisted R-algebras via Thom Spectra

For a $C_2$-commutative ring spectrum $R$, a twisted $R$-algebra is an $R$-module with a multiplication whose order is switched by the $C_2$-action. In this talk, we shall construct various quotients of $R$ as twisted $R$-algebras, when $R$ is an even real commutative ring spectrum. These are constructed as Thom spectra of maps out of suitable $C_2$ -actions on $S^1$ and $U(n)$. One such example is given by $KR/2$ which is endowed with a twisted $KR$-algebra structure.  In the context of twisted $R$-algebras, one may consider the real topological Hochschild homology, and for Thom spectra, one has a nice formula again as a Thom spectrum. We use this to obtain computations for the real topological Hochschild homology of $KR/2$ as a twisted $KR$-algebra. The computation also involves a splitting of the units spectrum, which is an analogue of the classical non-equivariant splitting.