[Paper Review] Higher condensation theory

Liang Kong, Zhi-Hao Zhang+2 more|arXiv (Cornell University)|Mar 12, 2024

Machine Learning in Materials ScienceMaterials Science3 citations

TL;DR

This paper develops a unified higher categorical framework for condensing k-codimensional topological defects in (n+1)D topological orders using higher algebras and higher representations. It introduces a k-step condensation process via Ek-algebras, characterizing resulting defects as Ei-modules over condensed algebras, and establishes connections to gauging and anomaly cancellation, generalizing anyon condensation to higher dimensions.

ABSTRACT

We develop a unified mathematical theory of defect condensations for topological orders in all dimensions based on higher categories, higher algebras and higher representations. A k-codimensional topological defect $A$ in an n+1D (potentially anomalous) topological order $C^{n+1}$ is condensable if it is equipped with the structure of a condensable $E_k$-algebra. Condensing such a defect $A$ amounts to a k-step process. In the first step, we condense the defect $A$ along one of its transversal directions, thus obtaining a (k-1)-codimensional defect $ΣA$, which is naturally equipped with the structure of a condensable $E_{k-1}$-algebra. In the second step, we condense the defect $ΣA$ in one of the remaining transversal directions, thus obtaining a (k-2)-codimensional defect $Σ^2 A$, so on and so forth. In the k-th step, we condense the 1-codimensional defect $Σ^{k-1}A$ along the only transversal direction, thus defining a phase transition from $C^{n+1}$ to a new n+1D topological order $D^{n+1}$. We give precise mathematical descriptions of each step in above process, including the precise mathematical characterization of the condensed phase $D^{n+1}$. When $C^{n+1}$ is anomaly-free, the same phase transition can be alternatively defined by replacing the last two steps by a single step of condensing the $E_2$-algebra $Σ^{k-2}A$ directly along the remaining two transversal directions. When n=2, this modified last step is precisely a usual anyon condensation in a 2+1D topological order. We derive many new mathematical results physically along the way. We also establish the connections among various notions of "gauging" symmetries. We also briefly discuss questions, generalizations and applications that naturally arise from our theory, including higher Morita theory, a theory of integrals and the condensations of liquid-like gapless defects in topological orders.

Motivation & Objective

• To formulate a unified mathematical theory of defect condensation in topological orders across all dimensions using higher categories and higher algebras.
• To generalize 2+1D anyon condensation to higher-dimensional topological orders via a systematic k-step condensation process.
• To characterize the resulting topological phases and defects after condensation using module categories over condensed higher algebras.
• To establish connections between condensation, gauging symmetries, and the non-surjectivity/non-fully-faithfulness of center functors.
• To develop tools for constructing and classifying condensable E1-algebras and Lagrangian E2-algebras from categorical obstructions.

Proposed method

• Formalizes k-codimensional defects in (n+1)D topological orders as condensable Ek-algebras within a higher categorical framework.
• Introduces a k-step condensation process: successively condensing along transversal directions, reducing codimension step-by-step.
• Characterizes the i-codimensional defects in the final phase Dn+1 as Ei-modules over the condensed Ek−i-algebra Σk−iA.
• Uses the center functor and its failure to be surjective or fully faithful as a key tool to derive obstructions and classify condensable algebras.
• Applies generalized center functors and internal hom constructions in higher categories to describe the condensation process categorically.
• Leverages higher Morita theory and E1/E2-algebra structures to unify algebraic and geometric descriptions of condensation.

Experimental results

Research questions

• RQ1 How can condensation of k-codimensional defects in (n+1)D topological orders be systematically described using higher algebraic structures?
• RQ2 What is the precise mathematical characterization of the resulting topological order and its defects after a k-step condensation process?
• RQ3 How does the condensation of a k-codimensional defect relate to the gauging of symmetries in topological orders?
• RQ4 What role do the non-surjectivity and non-fully-faithfulness of the center functor play in classifying condensable higher algebras?
• RQ5 Can the standard 2+1D anyon condensation be recovered as a special case of this higher condensation theory?

Key findings

• The k-step condensation process is fully characterized: condensing a k-codimensional defect A via successive condensations along transversal directions yields a new (n+1)D topological order Dn+1.
• The i-codimensional defects in Dn+1 are precisely Ei-modules over the condensed Ek−i-algebra Σk−iA, providing a complete classification of defects in the condensed phase.
• When the original order Cn+1 is anomaly-free, the last two steps of the k-step process can be unified into a single step condensing an E2-algebra, generalizing 2+1D anyon condensation.
• The theory provides a geometric description of defect condensation, linking algebraic structures to physical intuition via precise categorical constructions.
• The failure of the center functor to be surjective or fully faithful is used to derive new mathematical results and classify condensable E1-algebras and Lagrangian E2-algebras.
• The framework establishes a deep connection between condensation, gauging, and higher Morita theory, showing that condensation can be interpreted as a form of generalized gauging.