The Atiyah–Sutcliffe conjecture and En algebras

We show that a certain conjecture by Atiyah and Sutcliffe implies the existence of an E3-algebra (respectively E2-algebra) structure on the disjoint union of all complex (respectively real) full flag manifolds modulo symmetric groups. Moreover, we show that these structures are liftings of exotic E3 (respectively E2) structures on the free E∞-algebras on BU(1)+ (respectively BO(1)+), that do not extend to E4 (respectively E3) structures. We also provide some (co)homological calculations supporting the conjecture. This article is part of the theme issue 'Derived Lie algebras in geometry and topology'.