A Non-Perturbative Definition of the Standard Models (1809.11171v3)

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.

• The paper introduces a lattice-based framework that non-perturbatively defines chiral fermion theories in Standard Models.
• It employs cobordism theory to classify quantum anomalies and ensure consistent, anomaly-free gauge theories.
• Numerical results using Thom-Madsen-Tillmann spectra and Adams spectral sequences validate the anomaly-free construction of Spin(10) models.

A Non-Perturbative Definition of the Standard Models

In their paper, Juven Wang and Xiao-Gang Wen tackle a critical issue concerning the non-perturbative definition of Standard Models, particularly those involving chiral fermions coupled to gauge theories. The primary obstacle has been defining these theories in a quantum non-perturbative manner due to challenges such as the absence of convergence in perturbative expansions and issues related to fermion doubling. Their approach uses a combination of quantum anomaly classifications and symmetric invertible topological orders guided by the advancements in cobordism theory. They propose a lattice model-based solution, challenging a problem with deep implications for both theoretical physics and practical computations in quantum field theories.

Theoretical Framework

The authors focus on defining Spin(10) chiral fermion theories on a 3+1D lattice. These theories initially comprise Weyl fermions in 16-dimensional spinor representations. The classification of quantum anomalies and topological orders via cobordism theory forms the basis of their approach. Utilizing cobordism, which provides a robust mathematical form for enduring challenges in defining chiral gauge theories non-perturbatively, they identify conditions under which Spin(10) theories can be realized without encountering perturbative anomalies, thus ensuring a consistent and well-defined quantum theory.

Strong Numerical Results

The authors present convincing theoretical evidence supported by calculations involving Thom-Madsen-TiLLMann spectra and Adams spectral sequences leading to the classification of cobordism groups. Their rigorous mathematical approach reveals that both the Spin(10) chiral fermion theories and their crossover into the Spin(10) chiral gauge theories (after dynamically gauging) are anomaly-free. Numerically, these cobordism groups simplify to $Z_2$ or similar finite classifications, enabling pathways to construct lattice models without the usual impediments related to gauge anomalies.

Implications and Future Directions

Practically, this work implies a shift towards using lattice models of bosons or fermions for defining Standard Models, offering a stable quantum setup resistant to issues like fermion doubling. The theoretically grounded use of cobordism and lattice models extends beyond Spin(10) theories, potentially impacting the definition of other gauge theories, including SU(5) grand unifications. Moreover, the application of the cobordism theory and mathematical reasoning in defining gauge theories could have substantial implications for quantum simulation and computational applications in other fields.

Future work could leverage these findings by exploring extensions of lattice models into higher dimensions or investigating the inclusion of further interactions that respect symmetric gapped boundary conditions. These efforts would further refine the approach and bring us closer to practical implementations that leverage quantum lattice models, particularly as quantum computing infrastructures become increasingly sophisticated.

In summary, Wang and Wen’s paper presents a compelling argument for a quantum non-perturbative framework within the quantum lattice paradigm, emphasizing the power of cobordism in classifying anomalies and defining consistent, anomaly-free quantum theories. Their work signifies a major stride in understanding how lattice frameworks can offer robust, computable explanations for complex quantum phenomena in the Standard Models.