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A proof of Bott periodicity via Quot schemes and bulk-edge correspondence

Masaki Natori

Abstract

In this paper, we give alternative proofs of Bott periodicity of $ K $-theory and the bulk-edge correspondence of integer quantum Hall effect. Regarding Bott periodicity, we connect its proof with configuration spaces and use Quot schemes in algebraic geometry in our proof. Regarding the bulk-edge correspondence, we formulate edge indices based on the consideration of Graf--Porta and give a more elementary and self-contained proof.

A proof of Bott periodicity via Quot schemes and bulk-edge correspondence

Abstract

In this paper, we give alternative proofs of Bott periodicity of -theory and the bulk-edge correspondence of integer quantum Hall effect. Regarding Bott periodicity, we connect its proof with configuration spaces and use Quot schemes in algebraic geometry in our proof. Regarding the bulk-edge correspondence, we formulate edge indices based on the consideration of Graf--Porta and give a more elementary and self-contained proof.
Paper Structure (23 sections, 25 theorems, 98 equations, 4 figures)

This paper contains 23 sections, 25 theorems, 98 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. On the Bott periodicity theorem
  4. On the bulk-edge correspondence
  5. An alternative proof of the Bott periodicity theorem
  6. The statement and strategy of the proof
  7. Construction of configuration spaces
  8. Construction of configuration spaces and global section bundles
  9. Restriction of configuration spaces
  10. Comparison with Quot schemes
  11. Construction and continuity of configuration maps
  12. Definition of configuration maps
  13. Continuity of configuration maps
  14. Proof of the Bott periodicity theorem
  15. Comparison with other proofs of the Bott periodicity theorem
  16. ...and 8 more sections

Key Result

Theorem 1.1

The configuration map is continuous.

Figures (4)

  • Figure 1: The graph of eigenvalues of $\{A_t\}_t$ in \ref{['ex:spectral-flow']}.
  • Figure 2: An example of spectrum of $H$.
  • Figure 3: An example of spectrum of $H_1$ with $\mathop{\mathrm{ind}}\nolimits_1(H) = -1$ and an example of spectrum of $H_2$ with $\mathop{\mathrm{ind}}\nolimits_2(H) = -1$.
  • Figure 4: An example of spectrum of $\overline{H}$.

Theorems & Definitions (62)

  • Theorem 1.1: see \ref{['thm:coker-conti2']}
  • Definition 2.1
  • Theorem 2.2: MR0178470, MR0228000
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 52 more