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Classical KMS Functionals and Phase Transitions in Poisson Geometry

Nicolò Drago, Stefan Waldmann

Abstract

We study the convex cone of not necessarily smooth measures satisfying the classical KMS condition within the context of Poisson geometry. We discuss the general properties of KMS measures and its relation with the underlying Poisson geometry in analogy to Weinstein's seminal work in the smooth case. Moreover, by generalizing results from the symplectic case, we focus on the case of $b$-Poisson manifolds, where we provide an almost complete characterization of the convex cone of KMS measures.

Classical KMS Functionals and Phase Transitions in Poisson Geometry

Abstract

We study the convex cone of not necessarily smooth measures satisfying the classical KMS condition within the context of Poisson geometry. We discuss the general properties of KMS measures and its relation with the underlying Poisson geometry in analogy to Weinstein's seminal work in the smooth case. Moreover, by generalizing results from the symplectic case, we focus on the case of -Poisson manifolds, where we provide an almost complete characterization of the convex cone of KMS measures.

Paper Structure

This paper contains 14 sections, 18 theorems, 123 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Properties of Classical KMS Functionals
  3. Invariance and Support Properties
  4. Stability under Hamiltonian Perturbation and Relation with Weinstein's Modular Class
  5. Extremal KMS Functionals
  6. Cosymplectic Manifolds
  7. KMS Convex Cone for $b$-Poisson Manifolds
  8. $b$-Poisson Manifolds
  9. An Explicit Example
  10. KMS Convex Cone
  11. Examples
  12. The Torus $\mathbb{T}^3$ with a Cosymplectic Poisson Structure
  13. A Regular Poisson Structure on $\mathbb{R}^2\times\mathbb{T}$
  14. Acknowledgements

Key Result

Proposition \oldthetheorem

Let $\varphi\in\operatorname{KMS}(X,\beta)$ as per Definition Def: classical KMS functional. Then:

Figures (1)

  • Figure 1: The regular foliation of $(M,\Pi)$ visualized in the plane -- barring the $y$-direction -- where the radial coordinate $r$ is identified by $\log(r)=x_0e^u$ ---cf. Equation \ref{['Eq: spiral symplectic foliation']}. Symplectic leaves with $x_0>0$ are spirals with flows anticlockwise at infinity, starting (at $-\infty$) at the disk $\mathbb{D}_1$ of radius $r=1$. For $x_0<0$ we get a spiral which flows anticlockwise from the disk $\mathbb{D}_1$ to the origin. Finally $x_0=0$ is $\mathbb{D}_1$.

Theorems & Definitions (65)

  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Example \oldthetheorem
  • Remark \oldthetheorem: Symplectic manifolds
  • Proposition \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 55 more