Solitonic symmetry as non-invertible symmetry: 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.

43 pages, 0 figures

Cited by in corpus (8)

• Gapped Phases with Non-Invertible Symmetries: (1+1)d
• Anomalies of Non-Invertible Symmetries in (3+1)d
• The Club Sandwich: Gapless Phases and Phase Transitions with Non-Invertible Symmetries
• Emergent generalized symmetries in ordered phases and applications to quantum disordering
• Hasse Diagrams for Gapless SPT and SSB Phases with Non-Invertible Symmetries
• Fermionic Non-Invertible Symmetries in (1+1)d: Gapped and Gapless Phases, Transitions, and Symmetry TFTs
• Topological aspects of brane fields: solitons and higher-form symmetries
• Non-invertible symmetries of two-dimensional Non-Linear Sigma Models