The slice spectral sequence for the C4$C_{4}$ analog of real K-theory

We describe the slice spectral sequence of a 32-periodic $C_{4}$-spectrum $K_{[2]}$ related to the $C_{4}$ norm ${\mathrm{MU}^{((C_{4}))}=N_{C_{2}}^{C_{4}}\mathrm{MU}_{\mathbb{R}}}$ of the real cobordism spectrum $\mathrm{MU}_{\mathbb{R}}$. We will give it as a spectral sequence of Mackey functors converging to the graded Mackey functor $\underline{\pi}_{*}K_{[2]}$, complete with differentials and exotic extensions in the Mackey functor structure. The slice spectral sequence for the 8-periodic real K-theory spectrum $K_{\mathbb{R}}$ was first analyzed by Dugger. The $C_{8}$ analog of $K_{[2]}$ is 256-periodic and detects the Kervaire invariant classes $\theta_{j}$. A partial analysis of its slice spectral sequence led to the solution to the Kervaire invariant problem, namely the theorem that $\theta_{j}$ does not exist for ${j\geq 7}$.