A Non-Perturbative Definition of the Standard Models

TL;DR

The paper addresses the long-standing problem of a non-perturbative, finite-dimensional lattice definition of gauged chiral fermion theories that underlie the Standard Model and Grand Unified Theories. It develops a cobordism- and iTQFT-based framework to classify symmetric invertible topological orders and uses the existence of symmetric gapped boundaries for trivial cobordism classes to realize mirror sectors separately on higher-dimensional lattices, enabling the construction of chiral gauge theories on the same-dimensional lattice. Concrete lattice realizations are provided for Spin(10) with 16 Weyl fermions and for SU(5) chiral fermions, showing that the corresponding 4+1D bulk invertible phases can be trivial and hence admit $3+1$D lattice regularizations which can then be gauged. An enriching perspective is offered by the “it from qubit” viewpoint, arguing that the resulting Standard Model sectors can emerge from qubit-based bosonic lattice models with appropriate gauge extensions. Overall, the work proposes a non-perturbative lattice definition of SM-like theories via cobordism classifications, with potential implications for UV completions and the gauge structure of lattice-regularized chiral theories.

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.

Figures (3)

• Figure 1: (a) A lattice construction of a single Weyl fermion is given in Section \ref{['so10']}, the subfigure shows the gapless energy spectrum $E(\vec{k})$ of Brillouin zone in the schematic 3-dimensional momentum $\vec{k}=({k}_x,{k}_y,{k}_z)$-space with a linear dispersion $|E(\vec{k})| \propto c |\vec{k}|$ for some effective speed of light $c$. (b) The 16 copies of the same lattice model (\ref{['H4d']}) give rise to the 3+1D Weyl fermions at the low energy in the 16-dimensional spinor representation of the Spin(10) on the lattice boundary shown in Section \ref{['so10']}. The 16 gapless Weyl points (schematically the 16 dots $\bullet$) may be separated but can be tuned to the same point on the $\vec{k}$-space Brillouin zone. We show this Spin(10) chiral Weyl fermion theory is free from all 't Hooft anomalies via a cobordism theory in Sec. \ref{['sec:Cobordism']}. (c) There are two ways to obtain the symmetric gapped boundary for the bulk of the 16 copies of the lattice model: First, via \ref{['propoi']}, there exists a symmetric gapped boundary for the corresponding $d+2$D bulk regularized lattice model (without the need to access from gapping out the gapless theories from interactions. Second, via \ref{['propoiii']}, there exist non-perturbative symmetric interactions to fully gap this well-defined all anomaly-free Spin(10) chiral fermion theory with 16 Weyl fermions in 16-dimensional spinor representation of the Spin(10). (Schematic interactions are drawn in the shaded blue region.) In this work, we only prove \ref{['propoi']}, but we suggest some supportive evidence for \ref{['propoiii']} but without proving \ref{['propoiii']}. However, applying only \ref{['propoi']} (but without requiring \ref{['propoiii']}) is sufficient enough for us to construct the Spin(10) chiral fermion theory on the lattice via \ref{['propoI']}.
• Figure 2: Adams chart for $\Omega_D^{{\rm Spin} \times {\rm SU}(5)}$.
• Figure 3: Adams chart for $\Omega_D^{\frac{{\rm Spin} \times {\rm Spin}(10)}{\mathbb{Z}_2^F}}$, also for $\Omega_D^{\frac{{\rm Spin} \times {\rm Spin}(18)}{\mathbb{Z}_2^F}}$.

Theorems & Definitions (3)

• Proposition 1
• Proposition 2
• Proposition 3