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Berry phases and connection matrices defined on homogeneous spaces attached to Siegel-Jacobi groups

Stefan Berceanu

TL;DR

This work develops a geometric framework for the Berry phase on Siegel–Jacobi spaces attached to the real Jacobi group by exploiting invariant metrics, Kähler structures, and coherent-state formalisms. It derives explicit Berry connections $A_B$ and curvatures from Kähler potentials, constructs balanced and extended metrics, and provides detailed expressions for connection matrices and covariant derivatives on ${\mathcal{D}}^J_1$, ${\mathcal{X}}^J_1$, and $\tilde{\mathcal{X}}^J_1$. The paper also introduces a generalized transitive almost cosymplectic (GTACOS) structure on the extended space, proves the relevant closedness and nondegeneracy conditions, and presents an extensive Appendix covering foundational theory and representative examples. Overall, it offers concrete computational tools for geometric phases and holonomy in noncompact homogeneous Kähler manifolds associated with Jacobi groups, with potential applications in coherent-state quantization and geometric phase physics.

Abstract

The relation between the Berry phase and connection matrix on the Siegel-Jacobi disk $\mathcal{D}^J_1$ and Siegel-Jacobi upper half-plane$\mathcal{X}^J_1$ are analyzed. The connection matrix and the covariant derivative of one-forms on the extended Siegel-Jacobi upper half-plane $\tilde{\mathcal{X}}^J_1$ are calculated.

Berry phases and connection matrices defined on homogeneous spaces attached to Siegel-Jacobi groups

TL;DR

This work develops a geometric framework for the Berry phase on Siegel–Jacobi spaces attached to the real Jacobi group by exploiting invariant metrics, Kähler structures, and coherent-state formalisms. It derives explicit Berry connections and curvatures from Kähler potentials, constructs balanced and extended metrics, and provides detailed expressions for connection matrices and covariant derivatives on , , and . The paper also introduces a generalized transitive almost cosymplectic (GTACOS) structure on the extended space, proves the relevant closedness and nondegeneracy conditions, and presents an extensive Appendix covering foundational theory and representative examples. Overall, it offers concrete computational tools for geometric phases and holonomy in noncompact homogeneous Kähler manifolds associated with Jacobi groups, with potential applications in coherent-state quantization and geometric phase physics.

Abstract

The relation between the Berry phase and connection matrix on the Siegel-Jacobi disk and Siegel-Jacobi upper half-plane are analyzed. The connection matrix and the covariant derivative of one-forms on the extended Siegel-Jacobi upper half-plane are calculated.
Paper Structure (20 sections, 8 theorems, 348 equations)

This paper contains 20 sections, 8 theorems, 348 equations.

Table of Contents

  1. Introduction
  2. Preparation
  3. Berry phase on Kähler manifolds
  4. Balanced metric
  5. Berry phase on homogenous Kähler manifolds
  6. Linear Hamiltonian in the generators of the Jacobi group $G^J_1(\mathbb{R})$
  7. Kähler two-forms, Christofell's symbols and connections matrices on ${\mathcal{D}}_1$ and ${\mathcal{X}}_1$
  8. Connection matrices on ${\mathcal{D}}^J_1$ and ${\mathcal{X}}^J_1$
  9. Check of formulae \ref{['522a']}
  10. Berry phase on ${\mathcal{D}}^J_1$ and ${\mathcal{X}}^J_1$
  11. Almost cosymplectic manifolds
  12. Definitions
  13. Connection matrix on $\tilde{{\mathcal{X}}}^J_1$
  14. Covariant derivative of one-forms on ${\mathcal{X}}^J_1$ and $\tilde{{\mathcal{X}}}^J_1$
  15. Appendix
  16. ...and 5 more sections

Key Result

Lemma 1

a) The action $G^J(\mathbb{R})_0\times {\mathcal{X}}^J_1\rightarrow {\mathcal{X}}^J_1$ is given by b) If $u'=p'v'+q'$, $v'=x'+\operatorname{i} y'$ as in TAUZ, then the action is given by the formula and c) The action $G^J_1(\mathbb{R})\times \tilde{{\mathcal{X}}}^J_1\rightarrow \tilde{{\mathcal{X}}}^J_1$ is given by and ALIGNN1, AC12.

Theorems & Definitions (23)

  • Lemma 1
  • Proposition 1
  • Remark 1
  • proof
  • Proposition 2
  • Proposition 3
  • Remark 2
  • Proposition 4
  • proof
  • Remark 3
  • ...and 13 more