Motivic GUT Part I: Grand Unified Theory of Topological Order
Masahiko G. Yamada

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 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…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Topological and Geometric Data Analysis · Constraint Satisfaction and Optimization
