A Non-Perturbative Definition of the Standard Models

Abstract

The Standard Models contain chiral fermions coupled to gauge theories. It has been a long-standing problem to give such gauged chiral fermion theories a quantum non-perturbative definition. By classification of quantum anomalies and symmetric invertible topological orders via a mathematical cobordism theorem for differentiable and triangulable manifolds, and the existence of symmetric gapped boundary for the trivial symmetric invertible topological orders, we propose that Spin(10) chiral fermion theories with Weyl fermions in 16-dimensional spinor representations can be defined on a 3+1D lattice, and subsequently dynamically gauged to be a Spin(10) chiral gauge theory. As a result, the Standard Models from the 16n-chiral fermion SO(10) Grand Unification can be defined non-perturbatively via a 3+1D local lattice model of bosons or qubits. Furthermore, we propose that Standard Models from the 15n-chiral fermion SU(5) Grand Unification can also be realized by a 3+1D local lattice model of fermions.

24 pages. Two columns. v3: Refinement with detailed discussions. Appendices include viewpoints from perturbative local anomalies and non-perturbative global anomalies (e.g. arXiv:1810.00844, SU(2) = Spin(3) $\subset$ Spin(10)), and co/bordism theories (e.g. ${\Omega }_D^{(\mathrm{Spin}(D)\times \mathrm{Spin}(10))/{\mathbb{Z}}_2^f}$)

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• Global anomalies in the Standard Model(s) and Beyond
• Symmetric Mass Generation in the 1+1 Dimensional Chiral Fermion 3-4-5-0 Model
• Gapped Chiral Fermions
• Nonperturbative effects in the Standard Model with gauged 1-form symmetry
• Anomalies and symmetric mass generation for Kaehler-Dirac fermions
• Cobordism and Deformation Class of the Standard Model
• Chiral Lattice Fermions From Staggered Fields
• Ultra Unification
• Unwinding Fermionic SPT Phases: Supersymmetry Extension
• Categorical Symmetry of the Standard Model from Gravitational Anomaly
• Gauge Enhanced Quantum Criticality Beyond the Standard Model
• Fermi Surface Symmetric Mass Generation
• Anomaly-free $U(1{)}^m$ extensions of the Standard Model
• Definition and Classification of Fermi Surface Anomalies
• 3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems
• A no-go result for implementing chiral symmetries by locality-preserving unitaries in a 3 dimensional Hamiltonian lattice model of fermions
• CT or P Problem and Symmetric Gapped Fermion Solution
• Hamiltonian models of lattice fermions solvable by the meron-cluster algorithm
• Lattice Regularization of Reduced Kähler-Dirac Fermions and Connections to Chiral Fermions
• The Smith Fiber Sequence and Invertible Field Theories
• Smith homomorphisms and Spin${}^h$ structures
• Flavor Hierarchy from Smooth Confinement