Reflection Structures and Spin Statistics in Low Dimensions

Abstract

We give a complete classification of topological field theories with reflection structure and spin-statistics in one and two spacetime dimensions. Our answers can be naturally expressed in terms of an internal fermionic symmetry group $G$ which is different from the spacetime structure group. Fermionic groups encode symmetries of systems with fermions and time reversing symmetries. We show that 1-dimensional topological field theories with reflection structure and spin-statistics are classified by finite dimensional hermitian representations of $G$. In spacetime dimension two we give a classification in terms strongly $G$-graded stellar Frobenius algebras. Our proofs are based on the cobordism hypothesis. Along the way, we develop some useful tools for the computation of homotopy fixed points of 2-group actions on bicategories.

arXiv comment74+56 pages, 6 figures

References in corpus (20)

• Periodic table for topological insulators and superconductors
• Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Protected Topological Order
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• Higher-dimensional Algebra and Topological Quantum Field Theory
• Reflection positivity and invertible topological phases
• Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology
• The Classification of Two-Dimensional Extended Topological Field Theories
• Symmetry Protected Topological phases and Generalized Cohomology
• On the cobordism classification of symmetry protected topological phases
• Fermion condensation and super pivotal categories
• A note on the $(\infty ,n)$-category of cobordisms
• Spin Hurwitz numbers and topological quantum field theory
• Spin, statistics, orientations, unitarity
• A Framework for Geometric Field Theories and their Classification in Dimension One
• Fully extended $\mathbf{r}$-spin TQFTs
• The geometric cobordism hypothesis
• The Cobordism Hypothesis in Dimension 1
• The Serre automorphism via homotopy actions and the Cobordism Hypothesis for oriented manifolds
• Lectures on Factorization Homology, Infinity-Categories, and Topological Field Theories
• Two-Dimensional Extended Homotopy Field Theories

Cited by in corpus (3)

• The Smith Fiber Sequence and Invertible Field Theories
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• A Higher Spin-Statistics Theorem for Invertible Quantum Field Theories