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A reconstruction theorem for Connes-Landi deformations of commutative spectral triples

Branimir Ćaćić

TL;DR

This work extends Connes's reconstruction theorem to Connes-Landi deformations of commutative spectral triples along a compact Abelian group $G$, introducing the notion of $\theta$-commutative spectral triples parameterised by $H^2(\widehat{G},\mathbb{T})$ and proving that a deformation by $\theta'$ yields a $(\theta+\theta')$-commutative triple.Using strict deformation quantisation, the authors develop a functorial deformation theory for $G$-equivariant Fréchet algebras and modules, and they establish that Connes-Landi deformations preserve regularity properties required for spectral geometry.They prove an extended reconstruction theorem guaranteeing that such deformed triples correspond to $G$-equivariant commutative geometry up to isomorphism, and they refine the splitting homomorphism to handle rational deformations, showing these are almost-commutative when the deformation parameter has finite order.Overall, the framework supports quantisation and dequantisation between $G$-equivariant commutative spectral triples and their Connes-Landi deformations, with significant implications for the spectral action on rational toric noncommutative manifolds and for Morita-type classifications in the rational regime.

Abstract

We formulate and prove an extension of Connes's reconstruction theorem for commutative spectral triples to so-called Connes-Landi or isospectral deformations of commutative spectral triples along the action of a compact Abelian Lie group $G$, also known as toric noncommutative manifolds. In particular, we propose an abstract definition for such spectral triples, where noncommutativity is entirely governed by a deformation parameter sitting in the second group cohomology of the Pontrjagin dual of $G$, and then show that such spectral triples are well-behaved under further Connes-Landi deformation, thereby allowing for both quantisation from and dequantisation to $G$-equivariant abstract commutative spectral triples. We then use a refinement of the Connes-Dubois-Violette splitting homomorphism to conclude that suitable Connes-Landi deformations of commutative spectral triples by a rational deformation parameter are almost-commutative in the general, topologically non-trivial sense.

A reconstruction theorem for Connes-Landi deformations of commutative spectral triples

TL;DR

This work extends Connes's reconstruction theorem to Connes-Landi deformations of commutative spectral triples along a compact Abelian group $G$, introducing the notion of $\theta$-commutative spectral triples parameterised by $H^2(\widehat{G},\mathbb{T})$ and proving that a deformation by $\theta'$ yields a $(\theta+\theta')$-commutative triple.Using strict deformation quantisation, the authors develop a functorial deformation theory for $G$-equivariant Fréchet algebras and modules, and they establish that Connes-Landi deformations preserve regularity properties required for spectral geometry.They prove an extended reconstruction theorem guaranteeing that such deformed triples correspond to $G$-equivariant commutative geometry up to isomorphism, and they refine the splitting homomorphism to handle rational deformations, showing these are almost-commutative when the deformation parameter has finite order.Overall, the framework supports quantisation and dequantisation between $G$-equivariant commutative spectral triples and their Connes-Landi deformations, with significant implications for the spectral action on rational toric noncommutative manifolds and for Morita-type classifications in the rational regime.

Abstract

We formulate and prove an extension of Connes's reconstruction theorem for commutative spectral triples to so-called Connes-Landi or isospectral deformations of commutative spectral triples along the action of a compact Abelian Lie group , also known as toric noncommutative manifolds. In particular, we propose an abstract definition for such spectral triples, where noncommutativity is entirely governed by a deformation parameter sitting in the second group cohomology of the Pontrjagin dual of , and then show that such spectral triples are well-behaved under further Connes-Landi deformation, thereby allowing for both quantisation from and dequantisation to -equivariant abstract commutative spectral triples. We then use a refinement of the Connes-Dubois-Violette splitting homomorphism to conclude that suitable Connes-Landi deformations of commutative spectral triples by a rational deformation parameter are almost-commutative in the general, topologically non-trivial sense.

Paper Structure

This paper contains 20 sections, 35 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Strict deformation quantisation
  3. Group-theoretic preliminaries
  4. Peter--Weyl theory for G-equivariant Fréchet *-algebras and modules
  5. Deformation of Fréchet G-*-algebras
  6. Deformation of G-equivariant Fréchet modules
  7. Connes--Landi deformation
  8. Deformation of G-equivariant spectral triples
  9. Group cohomology and symmetries of Connes--Landi deformation
  10. Connes--Landi deformations of commutative spectral triples
  11. A reconstruction theorem
  12. The proof of Theorem \ref{['deform']}
  13. Order zero; order one
  14. Orientability
  15. Finiteness and absolute continuity
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $(\mathcal{A},H,D)$ be a $p$-dimensional $\theta$-commutative spectral triple and let $\theta^\prime \in H^2(\widehat{G},\mathbb{T})$. Then the Connes--Landi deformation $(\mathcal{A}_\theta,H,D)$ of $(\mathcal{A},H,D)$ by $\theta^\prime$ defines a $p$-dimensional $(\theta+\theta^\prime)$-commut

Theorems & Definitions (115)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Kleppner Klepp
  • Remark 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 105 more