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.
Highlights
AND DEFINITIONSThe standard models [1,2,3], gauge theories with the Lie algebra u(1) × su(2) × su(3) in 3 + 1D, coupled to fermions and bosons, are believed to describe elementary particles.1 In the standard standard model, there are 15 two-component complex Weyl fermions per family
We only focus on the smooth differentiable (DIFF) manifolds and their associated all possible invertible topological quantum field theories (iTQFT)
(b) in (2), we only focus on iTQFTs definable on differentiable and triangulable manifolds, and those iTQFTs may be regularized by the same lattice from the simplicial complex of triangulable manifolds
Summary
The standard models [1,2,3], gauge theories with the Lie algebra u(1) × su(2) × su(3) in 3 + 1D, coupled to fermions and bosons, are believed to describe elementary particles. In the standard standard model, there are 15 two-component complex Weyl fermions per family. [23] claimed that 3 + 1D Spin(10) chiral fermion theory with Weyl fermions in a 16-dimensional spinor representation can be defined via an interacting local lattice model with a Spin(10) on-site symmetry which can be gauged.. We will show that a 3 + 1D Spin(10) chiral fermion theory with Weyl fermions in a 16-dimensional spinor representation can be defined via an interacting local lattice model with a Spin(10) on-site internal symmetry which can be gauged. We will show that a 3 + 1D SU(5) chiral fermion theory with Weyl fermions in five-dimensional and 10-dimensional representations can be defined via an interacting local lattice model with an SU(5) on-site symmetry which can be gauged. In this work, we use the following name: a Spin(10) chiral fermion model [rather than an SO(10) chiral fermion model which was sometimes used by others]
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