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Conformal transformations and equivariance in unbounded KK-theory

Ada Masters, Adam Rennie

TL;DR

This work extends unbounded Kasparov theory to incorporate conformal, group, and quantum group equivariance by introducing conformally generated cycles and a robust multiplicative perturbation theory. Central ideas include the logarithmic transform that converts multiplicative to additive perturbations and the demonstration that bounded transforms preserve KK-classes under conformal changes, enabling descent and Green–Julg-type constructions in new settings. The paper provides explicit unbounded γ-elements for real and complex Lorentz groups and exhibits conformal equivariance in noncommutative examples such as the Heisenberg group and the Podleś sphere, using both classical and quantum group frameworks. These developments yield a unified approach to equivariant KK-theory that accommodates twisted spectral triples and paves the way for broader applications to quantum groups and noncommutative geometry, with a coherent pathway from unbounded to bounded KK-theory via conformal perturbations and logarithmic dampening.

Abstract

We extend unbounded Kasparov theory to encompass conformal group and quantum group equivariance. This new framework allows us to treat conformal actions on both manifolds and noncommutative spaces. As examples, we present unbounded representatives of Kasparov's $γ$-element for the real and complex Lorentz groups and display the conformal $SL_q(2)$-equivariance of the standard spectral triple of the Podleś sphere. In pursuing descent for conformally equivariant cycles, we are led to a new framework for representing Kasparov classes. Our new representatives are unbounded, possess a dynamical quality, and also include known twisted spectral triples. We define an equivalence relation on these new representatives whose classes form an abelian group surjecting onto KK. The technical innovation which underpins these results is a novel multiplicative perturbation theory. By these means, we obtain Kasparov classes from the bounded transform with minimal side conditions.

Conformal transformations and equivariance in unbounded KK-theory

TL;DR

This work extends unbounded Kasparov theory to incorporate conformal, group, and quantum group equivariance by introducing conformally generated cycles and a robust multiplicative perturbation theory. Central ideas include the logarithmic transform that converts multiplicative to additive perturbations and the demonstration that bounded transforms preserve KK-classes under conformal changes, enabling descent and Green–Julg-type constructions in new settings. The paper provides explicit unbounded γ-elements for real and complex Lorentz groups and exhibits conformal equivariance in noncommutative examples such as the Heisenberg group and the Podleś sphere, using both classical and quantum group frameworks. These developments yield a unified approach to equivariant KK-theory that accommodates twisted spectral triples and paves the way for broader applications to quantum groups and noncommutative geometry, with a coherent pathway from unbounded to bounded KK-theory via conformal perturbations and logarithmic dampening.

Abstract

We extend unbounded Kasparov theory to encompass conformal group and quantum group equivariance. This new framework allows us to treat conformal actions on both manifolds and noncommutative spaces. As examples, we present unbounded representatives of Kasparov's -element for the real and complex Lorentz groups and display the conformal -equivariance of the standard spectral triple of the Podleś sphere. In pursuing descent for conformally equivariant cycles, we are led to a new framework for representing Kasparov classes. Our new representatives are unbounded, possess a dynamical quality, and also include known twisted spectral triples. We define an equivalence relation on these new representatives whose classes form an abelian group surjecting onto KK. The technical innovation which underpins these results is a novel multiplicative perturbation theory. By these means, we obtain Kasparov classes from the bounded transform with minimal side conditions.

Paper Structure

This paper contains 31 sections, 108 theorems, 470 equations.

Table of Contents

  1. Introduction
  2. Conformal transformations
  3. Motivating examples
  4. Technical preliminaries and additive perturbation theory
  5. A multiplicative perturbation theory
  6. A partial converse
  7. The logarithmic transform: multiplicative to additive
  8. The singular case
  9. Group-equivariant KK-theory
  10. Uniformly equivariant unbounded KK-theory
  11. Conformally equivariant unbounded KK-theory
  12. The γ-element for the real and complex Lorentz groups
  13. The case of $SO(2n+1, 1)$
  14. The case of $SO(2n, 1)$
  15. The case of $SU(n, 1)$
  16. ...and 16 more sections

Key Result

Theorem 1

Let $(U, μ)$ be a conformal transformation from the order-$\frac{1}{1 - α}$ cycle $(A, E_B, D_1)$ to the order-$\frac{1}{1 - α}$ cycle $(A, E'_B, D_2)$. Then the bounded transforms $(A, E_B, F_{D_1})$ and $(A, E'_B, F_{D_2})$ are unitarily equivalent up to locally compact perturbation via the unitar for all $a ∈ A$. Hence $[(A, E_B, F_{D_1})]=[(A, E'_B, F_{D_2})]\in KK(A,B).$

Theorems & Definitions (255)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • ...and 245 more