Hypertoric varieties, $W$-Hilbert schemes, and Coulomb branches

TL;DR

This paper explores the geometric and metric structures of hypertoric varieties and Coulomb branches, linking Hilbert schemes, hyperk"ahler metrics, and Nahm's equations, revealing new connections and descriptions.

Contribution

It demonstrates that Coulomb branches can be realized as transverse equivariant Hilbert schemes or Hamiltonian reductions of hypertoric varieties, introducing a new hyperspherical variety.

Findings

Coulomb branches are obtainable as Hilbert schemes or Hamiltonian reductions.

Hyperk"ahler metrics are described via twistor spaces and solutions to Nahm's equations.

A new hyperspherical variety is introduced, linked to hypertoric varieties.

Abstract

We study transverse equivariant Hilbert schemes of affine hypertoric varieties equipped with a symplectic action of a Weyl group. In particular, we show that the Coulomb branches of Braverman, Finkelberg, and Nakajima can be obtained either as such Hilbert schemes or Hamiltonian reductions thereof. Furthermore, we propose that the Coulomb branches for representations of non-cotangent type are also obtained in this way. We also investigate the putative complete hyperk\"ahler metrics on these objects. We describe their twistor spaces and, in the case when the symplectic quotient construction of the hypertoric variety is $W$-equivariant (which includes Coulomb branches of cotangent type), we show that the hyperk\"ahler metric can be described as the natural $L^2$-metric on a moduli space of solutions to modified Nahm's equations on an interval with poles at both ends and a discontinuity in the middle, with the latter described by a new object: a hyperspherical variety canonically associated to a hypertoric variety.