The Atiyah-Sutcliffe conjecture and $E_n$-algebras

TL;DR

This paper links a conjecture by Atiyah and Sutcliffe to the existence of specific $E_n$-algebra structures on flag manifolds, providing algebraic and homological insights into these geometric objects.

Contribution

It demonstrates that the Atiyah-Sutcliffe conjecture implies $E_3$ and $E_2$-algebra structures on flag manifolds and explores their liftings from free $E_$-algebras, with supporting (co)homological calculations.

Findings

Existence of $E_3$-algebra on complex flag manifolds

Existence of $E_2$-algebra on real flag manifolds

Homological calculations supporting the conjecture

Abstract

We show that a certain conjecture by Atiyah and Sutcliffe implies the existence of an $ E_3 $-algebra (respectively $ E_2 $-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 $ E_3 $ (respectively $ E_2 $) structures on the free $ E_\infty $-algebras on $ BU(1)+ $ (respectively $ BO(1)+ $), that do not extend to $ E_4 $ (respectively $ E_3 $) structures. We also provide some (co)homological calculations supporting the conjecture.