[Paper Review] Solitonic symmetry as non-invertible symmetry: cohomology theories with TQFT coefficients

Shi Chen, Yuya Tanizaki|arXiv (Cornell University)|Jul 3, 2023

Algebraic structures and combinatorial models12 citations

TL;DR

The paper reframes solitonic symmetry as non-invertible generalized symmetry generated by partition functions of auxiliary TQFTs coupled to the path-integral target space Y, forming solitonic cohomology theories with TQFT coefficients.

ABSTRACT

Originating from the topology of the path-integral target space $Y$, solitonic symmetry describes the conservation law of topological solitons and the selection rule of defect operators. As Ref.~\cite{Chen:2022cyw} exemplifies, the conventional treatment of solitonic symmetry as an invertible symmetry based on homotopy groups is inappropriate. In this paper, we develop a systematic framework to treat solitonic symmetries as non-invertible generalized symmetries. We propose that the non-invertible solitonic symmetries are generated by the partition functions of auxiliary topological quantum field theories (TQFTs) coupled with the target space $Y$. We then understand solitonic symmetries as non-invertible cohomology theories on $Y$ with TQFT coefficients. This perspective enables us to identify the invertible solitonic subsymmetries and also clarifies the topological origin of the non-invertibility in solitonic symmetry. We finally discuss how solitonic symmetry relies on and goes beyond the conventional wisdom of homotopy groups. This paper is aimed at a tentative general framework for solitonic symmetry, serving as a starting point for future developments.

Motivation & Objective

• Motivate a new foundation for solitonic symmetry beyond homotopy group descriptions.
• Propose that solitonic symmetries are generated by partition functions of auxiliary fully-extended TQFTs coupled to the target space Y.
• Clarify the algebraic structure of solitonic symmetry and its relation to invertible subsymmetries.
• Explain how solitonic symmetry can go beyond conventional homotopy-based classifications.

Proposed method

• Define solitonic symmetry as the symmetry generated by topological functionals depending on the target space Y.
• Argue that locality forces topological functionals to arise from partition functions of auxiliary fully-extended TQFTs (TQFT coefficients).
• Describe the resulting symmetric fusion higher-categories Rep•(Y) and sRep•(Y) as the algebraic framework.
• Show how invertible subsymmetries relate to ordinary cohomology theories with TQFT coefficients.
• Discuss how non-invertibility emerges from decomposing Rep•(Y) into condensations and sphere-based topological data.
• Provide the coherence and locality principles guiding the construction of topological functionals.

Experimental results

Research questions

• RQ1 What is the appropriate foundation for solitonic symmetry beyond the homotopy group description?
• RQ2 How can topological functionals be consistently defined to generate solitonic symmetry while satisfying locality and coherence?
• RQ3 What is the algebraic structure governing solitonic symmetry in higher dimensions?
• RQ4 How do invertible and non-invertible solitonic subsymmetries relate to conventional cohomology theories with TQFT coefficients?
• RQ5 In what sense do solitonic symmetries go beyond homotopy theory and encode non-invertible fusion data?

Key findings

• Solitonic symmetry is naturally generated by partition functions of auxiliary fully-extended TQFTs coupled to the target space Y.
• Topological functionals form a commutative, non-invertible fusion structure captured by symmetric fusion higher-categories Rep•(Y) and sRep•(Y).
• Solitonic symmetry can be viewed as non-invertible cohomology theories on Y with TQFT coefficients, generalizing invertible cohomology theories.
• Invertible solitonic subsymmetries arise from orthodox cohomology theories, while non-invertibility is tied to the topological data of Y and sphere nontrivialities.
• The framework distinguishes charges carried by solitonic defects and shows how locality constrains acceptable functionals via generalized orientations.