Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fixed points and holomorphic structures on line bundles over the quantum projective line $\mathbb{C}\mathrm{P}_q^1$

Indranil Biswas, Satyajit Guin, Pradip Kumar

TL;DR

This work develops a fixed-point framework for analyzing holomorphic structures on line bundles over the quantum projective line $\mathbb{C}\mathrm{P}_q^1$ by studying flat $\overline{\partial}$-connections and their gauge transformations. It embeds the problem in a Banach space via a bounded left-inverse of $\overline{\partial}$ and derives a concrete necessary-and-sufficient criterion for gauge equivalence in terms of fixed points of nonlinear maps, enabling existence results for small-norm gauge transformations and a fixed-point formulation of the mixed gauge equation. The authors introduce abstract perturbative criteria and the notion of admissible pairs to argue for the potential of uncountably many pairwise non-gauge-equivalent holomorphic structures, hinting at a quantum Picard-type moduli space for $\mathbb{C}\mathrm{P}_q^1$. The approach connects noncommutative differential geometry on the Podleś sphere with fixed-point analysis to illuminate quantum complex geometry and lays groundwork for extensions to higher-dimensional quantum projective spaces.

Abstract

It has recently been observed that, in contrast to the classical case, holomorphic structures on line bundles over the quantum projective line $\mathbb{C}\mathrm{P}_q^1$ are not uniquely determined by degree. In this work, we develop a fixed-point-theoretic framework for the analysis of flat $\overline\partial$-connections that define holomorphic structures on line bundles over the quantum projective line. Within this framework, we establish sufficient conditions ensuring the gauge equivalence of holomorphic connections. Furthermore, we obtain a necessary and sufficient criterion characterising when two such holomorphic connections are gauge equivalent. This criterion is formulated in terms of the existence of fixed points, lying in the open unit ball, of certain nonlinear maps acting on an appropriate Banach space.

Fixed points and holomorphic structures on line bundles over the quantum projective line $\mathbb{C}\mathrm{P}_q^1$

TL;DR

This work develops a fixed-point framework for analyzing holomorphic structures on line bundles over the quantum projective line by studying flat -connections and their gauge transformations. It embeds the problem in a Banach space via a bounded left-inverse of and derives a concrete necessary-and-sufficient criterion for gauge equivalence in terms of fixed points of nonlinear maps, enabling existence results for small-norm gauge transformations and a fixed-point formulation of the mixed gauge equation. The authors introduce abstract perturbative criteria and the notion of admissible pairs to argue for the potential of uncountably many pairwise non-gauge-equivalent holomorphic structures, hinting at a quantum Picard-type moduli space for . The approach connects noncommutative differential geometry on the Podleś sphere with fixed-point analysis to illuminate quantum complex geometry and lays groundwork for extensions to higher-dimensional quantum projective spaces.

Abstract

It has recently been observed that, in contrast to the classical case, holomorphic structures on line bundles over the quantum projective line are not uniquely determined by degree. In this work, we develop a fixed-point-theoretic framework for the analysis of flat -connections that define holomorphic structures on line bundles over the quantum projective line. Within this framework, we establish sufficient conditions ensuring the gauge equivalence of holomorphic connections. Furthermore, we obtain a necessary and sufficient criterion characterising when two such holomorphic connections are gauge equivalent. This criterion is formulated in terms of the existence of fixed points, lying in the open unit ball, of certain nonlinear maps acting on an appropriate Banach space.
Paper Structure (20 sections, 30 theorems, 177 equations)

This paper contains 20 sections, 30 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. The Hopf algebras $\mathcal{U}_q(\mathfrak{su}(2)),\,\mathcal{O}(\mathrm{SU}_q(2))$, and $\mathbb{C}\mathrm{P}_q^1$
  3. The Hopf algebras $\mathcal{U}_q(\mathfrak{su}(2))\hbox{and}\mathcal{O}(\mathrm{SU}_q(2))$
  4. The dual pairing
  5. The homogeneous space $\mathcal{O}(S_q^2)$
  6. The $C^*$-algebras $C(\mathrm{SU}_q(2))\hbox{and}C(S_q^2)$
  7. $\mathbb{C}\mathrm{P}_q^1$ and holomorphic line bundles
  8. The Dirac operator for $\mathbb{C}\mathrm{P}_q^1$ and the Gauge equation
  9. The Dirac operator and holomorphic differential
  10. Gauge equivalence and the noncommutative exponential equation
  11. Fixed Points and the Gauge equation
  12. A bounded left-inverse for $\overline\partial$
  13. Fixed-point formulation for $\overline{\partial}u = -\overline{\partial}f\cdot u$, where $f\in\mathbb{C}\mathrm{P}_q^1$
  14. Gauge equivalence of connections
  15. Two connections and the mixed gauge equation
  16. ...and 5 more sections

Key Result

Lemma 2.1

The following holds: for each $n\,\in\,\mathbb{Z}$, where $S$ is the antipode of $\,\mathcal{O}(\mathrm{SU}_q(2))$.

Theorems & Definitions (63)

  • Lemma 2.1
  • proof
  • Definition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2: GravemanLaRueMacArthurPesinWei2025NonStandardHolomorphic
  • proof
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3: GravemanLaRueMacArthurPesinWei2025NonStandardHolomorphic
  • ...and 53 more