Motivic GUT Part I: Grand Unified Theory of Topological Order

TL;DR

This paper introduces a unified, higher-categorical framework for classifying gapped topological phases across dimensions, emphasizing the importance of the ambient higher-categorical space for capturing the full structure of topological order.

Contribution

It proposes a novel definition of gapped topological order using unitary fusion $( abla,d)$-categories up to Morita equivalence, extending known 2D structures to higher dimensions with intrinsically higher-categorical data.

Findings

Recovers unitary modular tensor categories for 2D topological order

Extends classification to higher dimensions with higher-categorical structures

Suggests reinterpreting existing classifications as shadows of $ abla$-categorical objects

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.