C-P-T Fractionalization

Abstract

Discrete spacetime symmetries of parity P or reflection R, and time-reversal T, act naively as ${\mathbb{Z}}_2$-involutions in the passive transformation on the spacetime coordinates; but together with a charge conjugation C, the total C-P-R-T symmetries have enriched active transformations on fields in representations of the spacetime-internal symmetry groups of quantum field theories (QFTs). In this work, we derive that these symmetries can be further fractionalized, especially in the presence of the fermion parity $(-1{)}^{\mathrm{F}}$. We elaborate on examples including relativistic Lorentz invariant QFTs (e.g., spin-1/2 Dirac or Majorana spinor fermion theories) and nonrelativistic quantum many-body systems (involving Majorana zero modes), and comment on applications to spin-1 Maxwell electromagnetism (QED) or interacting Yang-Mills (QCD) gauge theories. We discover various C-P-R-T-$(-1{)}^{\mathrm{F}}$ group structures, e.g., Dirac spinor is in a projective representation of ${\mathbb{Z}}_2^{\mathrm{C}}\times {\mathbb{Z}}_2^{\mathrm{P}}\times {\mathbb{Z}}_2^{\mathrm{T}}$ but in an (anti)linear representation of an order-16 nonabelian finite group, as the central product between an order-8 dihedral (generated by C and P) or quaternion group and an order-4 group generated by T with T${}^2=(-1{)}^{\mathrm{F}}$. The general theme may be coined as C-P-T or C-R-T fractionalization.

arXiv comment8 pages. Heredity of setups: Spinor theories follow any standard QFT textbook. The 0+1d Majorana zero modes analysis follows arXiv:2011.13921, arXiv:2011.12320. Special thanks to Shing-Tung Yau on "Can C-P-T symmetries be fractionalized more than involutions?" v4: refinement

References in corpus (43)

• Non-Abelian Anyons and Topological Quantum Computation
• Unpaired Majorana fermions in quantum wires
• Classification of topological insulators and superconductors in three spatial dimensions
• Novel Symmetry Classes in Mesoscopic Normal-Superconducting Hybrid Structures
• Periodic table for topological insulators and superconductors
• Generalized Global Symmetries
• Topological phases of fermions in one dimension
• The effects of interactions on the topological classification of free fermion systems
• Theta, Time Reversal, and Temperature
• Fermion Path Integrals And Topological Phases
• Classification of Topological Insulators and Superconductors
• Interacting fermionic topological insulators/superconductors in three dimensions
• Reflection positivity and invertible topological phases
• Braiding statistics of loop excitations in three dimensions
• Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory
• Symmetry protected topological phases in non-interacting fermion systems
• Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions
• Symmetric Gapped Interfaces of SPT and SET States: Systematic Constructions
• Cobordism Conjecture, Anomalies, and the String Lamppost Principle
• Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding
• Adjoint QCD${}_4$, Deconfined Critical Phenomena, Symmetry-Enriched Topological Quantum Field Theory, and Higher Symmetry-Extension
• Global Anomalies on the Hilbert Space
• An Adventure in Topological Phase Transitions in 3 + 1-D: Non-abelian Deconfined Quantum Criticalities and a Possible Duality
• Two-flavor adjoint QCD
• Time-Reversal Symmetry, Anomalies, and Dualities in (2+1)$d$
• Time Reversal, SU(N) Yang-Mills and Cobordisms: Interacting Topological Superconductors/Insulators and Quantum Spin Liquids in 3+1D
• Quantum 4d Yang-Mills Theory and Time-Reversal Symmetric 5d Higher-Gauge Topological Field Theory
• Particle-Hole Symmetries in Condensed Matter
• Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism
• CPT theorem and classification of topological insulators and superconductors
• New Higher Anomalies, SU(N) Yang-Mills Gauge Theory and ${\mathbb{CP}}^{\mathrm{N}-1}$ Sigma Model
• Lorentz Symmetry Fractionalization and Dualities in (2+1)d
• Higher Anomalies, Higher Symmetries, and Cobordisms II: Lorentz Symmetry Extension and Enriched Bosonic/Fermionic Quantum Gauge Theory
• Quantum Statistics and Spacetime Surgery
• Supersymmetry in the nonsupersymmetric Sachdev-Ye-Kitaev model
• Unwinding Fermionic SPT Phases: Supersymmetry Extension
• Quantum Statistics and Spacetime Topology: Quantum Surgery Formulas
• Dihedral symmetry in $SU(N)$ Yang-Mills theory
• Boundary supersymmetry of 1+1 d fermionic SPT phases
• The CPT Group of the Dirac Field
• Supersymmetric boundaries of one-dimensional phases of fermions beyond symmetry-protected topological states
• Fractionalized time reversal, parity and charge conjugation symmetry in topological superconductor: a possible origin of three generations of neutrinos and mass mixing
• On the group generated by $\mathbf{C}$, $\mathbf{P}$ and $\mathbf{T}$: ${\mathbf{I}}^2={\mathbf{T}}^2={\mathbf{P}}^2=\mathbf{ITP}=-1$, with applications to pseudo-scalar mesons

Cited by in corpus (4)

• CT or P Problem and Symmetric Gapped Fermion Solution
• Discrete symmetry transformations on non-abelian gauge fields
• Three point interaction of Dirac fermions with higher spin particles and discrete symmetries
• Operator Dimension Parity Fractionalization