Real topological Hochschild and cyclic homology are refinements of the classical topological Hochschild and cyclic homology which take into account the reflection of the circle. The talk will introduce these theories, with a focus on their geometric fixed points under the action of a reflection. We will give a formula for these geometric fixed-points in terms of tensor products of Hill-Hopkins-Ravenel norms, in line with a similar formula for normal L-theory of Harpaz-Nikolaus-Shah. We will then use this formula to carry out calculations for perfect fields and for the ring of integers. The first part of the talk will introduce some background on equivariant homotopy theory and THH, and the second part will focus on the real theories and the calculations.
This is joint work with Kristian Moi and Irakli Patchkoria.