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Non-standard Holomorphic Structures on Line Bundles over the Quantum Projective Line

Mary Graveman, Landen La Rue, Lillian MacArthur, Hunter Pesin, Zhaoting Wei

TL;DR

The paper addresses whether holomorphic structures on quantum line bundles $\mathcal{L}_n$ over ${\mathbb C}P^1_q$ are unique up to gauge. It develops a refined gauge framework using invertible elements in the $C^*$-algebra ${\mathcal C}(\mathbb{C}P^1_q)$ and its spectral-analytic data to construct flat $\bar{\partial}$-connections not gauge equivalent to the standard one, proving the existence of infinitely many gauge-inequivalent holomorphic structures for $0<q<1$. A key technical device is the analysis of the bar-delta operator $\bar{\delta}$ on $C^*(1,B_0)$ and the notion of defective spots, which determines when an invertible gauge-transform $g$ exists solving $\bar{\partial}g=g\bar{\partial}f$. The results show that not only do non-standard holomorphic structures exist, but the associated spaces of holomorphic sections can have arbitrarily large finite dimension, prompting future work on classifying the holomorphic moduli (Picard group) and extending to higher-dimensional quantum projective spaces ${\mathbb C}P^l_q$.

Abstract

In this paper we study non-standard holomorphic structures on line bundles over the quantum projective line $\mathbb{C} P^1_q$. We show that there exist infinitely many non-gauge equivalent holomorphic structures on those line bundles. This gives a negative answer to a question raised by Khalkhali, Landi, and Van Suijlekom in 2011.

Non-standard Holomorphic Structures on Line Bundles over the Quantum Projective Line

TL;DR

The paper addresses whether holomorphic structures on quantum line bundles over are unique up to gauge. It develops a refined gauge framework using invertible elements in the -algebra and its spectral-analytic data to construct flat -connections not gauge equivalent to the standard one, proving the existence of infinitely many gauge-inequivalent holomorphic structures for . A key technical device is the analysis of the bar-delta operator on and the notion of defective spots, which determines when an invertible gauge-transform exists solving . The results show that not only do non-standard holomorphic structures exist, but the associated spaces of holomorphic sections can have arbitrarily large finite dimension, prompting future work on classifying the holomorphic moduli (Picard group) and extending to higher-dimensional quantum projective spaces .

Abstract

In this paper we study non-standard holomorphic structures on line bundles over the quantum projective line . We show that there exist infinitely many non-gauge equivalent holomorphic structures on those line bundles. This gives a negative answer to a question raised by Khalkhali, Landi, and Van Suijlekom in 2011.
Paper Structure (12 sections, 36 theorems, 142 equations)

This paper contains 12 sections, 36 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Review of the Quantum Projective line and line bundles on it
  3. The quantum projective line ${\mathbb C} P^1_q$
  4. $\bar{\partial}$-connections on line bundles
  5. $C^*$-completions, $L^2$-completions, and the spectral triple
  6. Nontrivial Gauge Equivalency Classes of Holomorphic Structures on line bundles
  7. Generalities on gauge equivalences of holomorphic structures
  8. The $C^*$-subalgebra $C^*(1, B_0)$
  9. Existence of non-standard holomorphic structures
  10. Gauge equivalence between $\bar{\partial}$-connections
  11. Holomorphic Sections of Non-Standard Line bundles
  12. Future Work

Key Result

Theorem 1.1

For $0<q<1$, each quantum line bundle $\mathcal{L}_n$ over ${\mathbb C} P^1_q$ admits a flat $\bar{\partial}$-connection $\overline{\nabla}$ which is not gauge equivalent to the standard $\bar{\partial}$-connection.

Theorems & Definitions (90)

  • Theorem 1.1: See Theorem \ref{['thm: gauge equivalent to trivial criterion']} below
  • Proposition 1.2: See Corollary \ref{['coro: ker_dim_lower_bound']} below
  • Theorem 1.3: See Theorem \ref{['thm: infinitely many gauge class']} below
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • ...and 80 more