Topological interpretation of the big Varchenko–Gelfand ring for the braid arrangement

Establish the existence of a topological space \widehat{\mathrm{Conf}}_n, together with an inclusion \mathrm{Conf}_n(\mathbb{R}^3) \subseteq \widehat{\mathrm{Conf}}_n and a natural action of the symmetric group S_n, such that the cohomology ring H^*(\widehat{\mathrm{Conf}}_n) is isomorphic, as a graded S_n-algebra, to the big graded Varchenko–Gelfand ring \widehat{VG}_{M_n} associated with the type A braid arrangement M_n (equivalently, to the orbit harmonics quotient (Z_{M_n})).

Background

In the braid arrangement case, the small graded Varchenko–Gelfand ring _{M_n} is known to admit a topological interpretation: Moseley proved that _{M_n} is isomorphic to the cohomology ring H^*(\mathrm{Conf}_n(\mathbb{R}^3)) of the configuration space of ordered n-tuples of pairwise distinct points in \mathbb{R}^3, with the natural S_n-action.

This paper constructs and analyzes the big graded Varchenko–Gelfand ring \widehat{VG}{M_n} via the orbit harmonics quotient (Z{M_n}), describes its Hilbert series and equivariant structure, and shows that it refines the small ring by incorporating contributions indexed by flats in the lattice L(M_n).

The authors seek a topological model analogous to the configuration space interpretation for the small ring, specifically an enlargement \widehat{\mathrm{Conf}}n of \mathrm{Conf}_n(\mathbb{R}^3) whose cohomology realizes \widehat{VG}{M_n} as a graded S_n-algebra. They also hint that a T-equivariant approach (with a circle action) could be relevant and that similar questions may extend to general real arrangements and convex open sets.