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Symmetric spaces, non-formal star products and Drinfel'd twists

Pierre Bieliavsky

TL;DR

This work surveys the development of non-formal deformation quantization on highly symmetric spaces, addressing Weinstein’s three-point kernel problem by linking geometric areas of geodesic triangles to star-product kernels. It introduces the Retract Method to construct and relate star-products on symplectic spaces to Drinfel'd twists, providing a bridge from geometric quantization to non-Abelian universal deformation formulas. By examining admissible phases (Weinstein and Ševera areas) and their application to hyperbolic and ax+b-type spaces, the notes derive explicit formulas for geodesic-triangle fluxes and canonical two-point/three-point functions that drive invariant oscillatory integrals. The framework culminates in a structured approach to oscillatory integrals on exponential Lie groups through symbol spaces, enabling concrete non-formal deformation constructions with broad symmetry.

Abstract

These notes refer to a minicourse I gave at the occasion of the conference meeting ``Applications of Noncommutative Geometry to Gauge Theories, Field Theories, and Quantum Space-Time'' to be held from 7 April to 11 April 2025 at the Centre International de Rencontres Mathématiques in Luminy. They consist in a review of a long standing work of mine and collaborators (see references therein) in the field of non-formal deformation quantization admitting a large group of symmetries. But they also contain new material and results. More precisely, in a first part, I present a method (called the Retract Method) to define quantizations/symbolic calculi and associated operator symbol composition formulae (non-formal deformations/star products) of symplectic symmetric spaces such as the hyperbolic plane (Kahler) or symmetric co-adjoint orbits of the Poincaré group (non-metric). In a second part, I explain how to derive non-formal Drinfel'd twists for actions of non-Abelian solvable Lie groups (non-Abelian Universal Deformation Formulae) on or Fr échet algebras from the non-formal noncommutative symmetric spaces defined in the first part.

Symmetric spaces, non-formal star products and Drinfel'd twists

TL;DR

This work surveys the development of non-formal deformation quantization on highly symmetric spaces, addressing Weinstein’s three-point kernel problem by linking geometric areas of geodesic triangles to star-product kernels. It introduces the Retract Method to construct and relate star-products on symplectic spaces to Drinfel'd twists, providing a bridge from geometric quantization to non-Abelian universal deformation formulas. By examining admissible phases (Weinstein and Ševera areas) and their application to hyperbolic and ax+b-type spaces, the notes derive explicit formulas for geodesic-triangle fluxes and canonical two-point/three-point functions that drive invariant oscillatory integrals. The framework culminates in a structured approach to oscillatory integrals on exponential Lie groups through symbol spaces, enabling concrete non-formal deformation constructions with broad symmetry.

Abstract

These notes refer to a minicourse I gave at the occasion of the conference meeting ``Applications of Noncommutative Geometry to Gauge Theories, Field Theories, and Quantum Space-Time'' to be held from 7 April to 11 April 2025 at the Centre International de Rencontres Mathématiques in Luminy. They consist in a review of a long standing work of mine and collaborators (see references therein) in the field of non-formal deformation quantization admitting a large group of symmetries. But they also contain new material and results. More precisely, in a first part, I present a method (called the Retract Method) to define quantizations/symbolic calculi and associated operator symbol composition formulae (non-formal deformations/star products) of symplectic symmetric spaces such as the hyperbolic plane (Kahler) or symmetric co-adjoint orbits of the Poincaré group (non-metric). In a second part, I explain how to derive non-formal Drinfel'd twists for actions of non-Abelian solvable Lie groups (non-Abelian Universal Deformation Formulae) on or Fr échet algebras from the non-formal noncommutative symmetric spaces defined in the first part.
Paper Structure (32 sections, 82 theorems, 675 equations)

This paper contains 32 sections, 82 theorems, 675 equations.

Table of Contents

  1. Symplectic symmetric spaces
  2. Symmetric spaces
  3. Symplectic symmetric spaces
  4. The retract method in Formal Deformation Quantization
  5. Star-products on symplectic manifolds
  6. Drinfel'd twists and symplectic Lie groups
  7. Intertwining Drinfel'd twists -- the Retract method
  8. Symplectic twists and naturality
  9. The retract method I: describing equivalences as evolution solutions
  10. The retract method II: the general context
  11. Admissible phases, Weinstein's and Ševera's areas
  12. The Weinstein area
  13. The Ševera area
  14. The hyperbolic plane
  15. Elementary group type Hermitean spaces
  16. ...and 17 more sections

Key Result

Proposition 1.1.1

Let $(M,s)$ be a symmetric space. Let $X,Y$ be smooth tangent vector fields on $M$ and $x$ be a point in $M$. Then, (i) the following formula defines an affine connection $\nabla$ on $M$. (ii) $\nabla$ is the unique affine connection on $M$ that is invariant under the symmetries. (iii) It is torsion-free and its Riemann curvature tensor is parallel. (iv) The symmetries extend the local geodesic s

Theorems & Definitions (150)

  • Definition 1.1.1
  • Proposition 1.1.1
  • Remark 1.1.1
  • Definition 1.1.2
  • Proposition 1.1.2
  • Proposition 1.1.3
  • Definition 1.1.3
  • Remark 1.1.2
  • Proposition 1.1.4
  • Proposition 1.1.5
  • ...and 140 more