$K(2)$-local splittings of finite Galois extensions of $MU\langle6\rangle$ and $MString$

TL;DR

This paper demonstrates that certain spectra related to complex cobordism and string theory split into simpler components after localization at the prime 2, revealing new structural insights and refinements in their Galois extensions and orientations.

Contribution

It provides a novel $K(2)$-local splitting of $MU extless 6 extgreater$ and $MString$ spectra into Morava $E$-theories, with detailed refinements and connections to TMF.

Findings

Spectra split into Morava $E$-theories after Galois extension.

Refined splitting of $MString$ using spin characteristic classes.

Existence of unital sections for orientations after extension.

Abstract

Using a Milnor-Moore argument we show that, $K(2)$-locally at the prime $2$, the spectra $MU\langle 6\rangle$ and $MString$ split as direct sums of Morava $E$-theories after tensoring with a finite Galois extension of the sphere called $E^{hF_{3/2}}$. In the case of $MString$ we are able to refine this splitting in several ways: we show that the projection maps are determined by spin characteristic classes, that the Ando-Hopkins-Rezk orientation admits a unital section after tensoring with $E^{hF_{3/2}}$, and that the splitting can be improved to one of $E^{hH}\otimes MString$ into a direct sum of shifts of $TMF_0(3)$ where $H$ is an open subgroup of the Morava stabilizer group of index $4$.