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Motivic GUT Part I: Grand Unified Theory of Topological Order

Masahiko G. Yamada

Abstract

In the series of papers Motivic GUT Part I: Grand Unified Theory of Topological Order, Motivic GUT Part II: Grand Unified Theory of Symmetry-Protected Topological Order, and Motivic GUT Part III: Grand Unified Theory of Symmetry-Enriched Topological Order, we propose a unified framework for gapped topological phases based on the Grothendieck-Kitaev-Lurie motivic yoga. In the spirit of Grothendieck's rising sea, we argue that the classification problem can only be properly addressed after identifying the correct higher-categorical ambient space in which its full richness appears. In this first part, we propose a unified definition of gapped topological order in spatial dimension $d$ in terms of unitary fusion $(\infty,d)$-categorical data, considered up to Morita equivalence. For $d=2$, this framework recovers unitary modular tensor categories. For $d>2$, it naturally leads to genuinely higher-categorical structures. This suggests a Copernican turn in the theory of topological phases: many existing classification schemes should be reinterpreted as lower-categorical shadow realizations of intrinsically $\infty$-categorical objects.

Motivic GUT Part I: Grand Unified Theory of Topological Order

Abstract

In the series of papers Motivic GUT Part I: Grand Unified Theory of Topological Order, Motivic GUT Part II: Grand Unified Theory of Symmetry-Protected Topological Order, and Motivic GUT Part III: Grand Unified Theory of Symmetry-Enriched Topological Order, we propose a unified framework for gapped topological phases based on the Grothendieck-Kitaev-Lurie motivic yoga. In the spirit of Grothendieck's rising sea, we argue that the classification problem can only be properly addressed after identifying the correct higher-categorical ambient space in which its full richness appears. In this first part, we propose a unified definition of gapped topological order in spatial dimension in terms of unitary fusion -categorical data, considered up to Morita equivalence. For , this framework recovers unitary modular tensor categories. For , it naturally leads to genuinely higher-categorical structures. This suggests a Copernican turn in the theory of topological phases: many existing classification schemes should be reinterpreted as lower-categorical shadow realizations of intrinsically -categorical objects.
Paper Structure (64 sections, 31 theorems, 45 equations, 2 tables)

This paper contains 64 sections, 31 theorems, 45 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Axioms of topological order
  3. Axioms
  4. Axiom 1: Excitation data
  5. Axiom 2: No transparent excitation
  6. On the tangential structure $H$
  7. Modularity--Goodwillie equivalence
  8. Claim
  9. Proof of (a) $\Rightarrow$ (b): Excision implies modularity
  10. Proof of (b) $\Rightarrow$ (a): Modularity implies excision
  11. Unitary modular tensor categories in $d=2$
  12. Motivic GUT's Hodge Conjecture
  13. Motivic GUT's Tate Conjecture
  14. Prime fusion $(\infty,d)$-categories
  15. Statement of the Tate Conjecture
  16. ...and 49 more sections

Key Result

Theorem 3.1

Let $\mathcal{E}$ be a fully dualizable fusion $2$-category DouglasReutter with $\dagger$-structure (i.e., Axiom 1 holds with $d=3$). Then the following are equivalent:

Theorems & Definitions (87)

  • Definition 2.1: Topological order
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 3.1: Modularity--Goodwillie equivalence in $d=3$
  • proof : Proof
  • proof : Proof
  • Remark 3.3
  • Remark 3.4
  • Theorem 4.1: Hodge Conjecture
  • ...and 77 more