Gibbs Manifolds

TL;DR

Gibbs manifolds are geometric structures derived from affine spaces of symmetric matrices via the exponential map, with applications across optimization, statistics, and quantum physics, and their algebraic properties are characterized in this paper.

Contribution

The paper computes the defining polynomials of Gibbs varieties and demonstrates their low-dimensionality, extending the understanding of Gibbs manifolds in various scientific fields.

Findings

Gibbs varieties are low-dimensional algebraic sets.

Explicit polynomials defining Gibbs varieties are computed.

Applications include matrix pencils and quantum optimal transport.

Abstract

Gibbs manifolds are images of affine spaces of symmetric matrices under the exponential map. They arise in applications such as optimization, statistics and quantum~physics, where they extend the ubiquitous role of toric geometry. The Gibbs variety is the zero locus of all polynomials that vanish on the Gibbs manifold. We compute these polynomials and show that the Gibbs variety is low-dimensional. Our theory is applied to a wide range of scenarios, including matrix pencils and quantum optimal transport.