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Gibbs manifolds

Dmitrii Pavlov¹, Bernd Sturmfels^1,2, Simon Telen^1,3

¹MPI-MiS, Leipzig, Germany.

Information Geometry

|November 24, 2025

Abstract

None:

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.

Keywords:

Gibbs variety Quantum optimal transport Semidefinite programming Toric geometry

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