Beyond the standard model with six-dimensional spinors (2002.02391v3)

Abstract: 6D spinors with $Spin(3,3)$ symmetry are utilized to efficiently encode three generations of matter. $E_{8(-24)}$ is shown to contain physically relevant subgroups with representations for GUT groups, spacetime symmetries, three generations of the standard model fermions, and Higgs bosons. Pati-Salam, $SU(5)$, and $Spin(10)$ grand unified theories are found when a single generation is isolated. For spacetime symmetries, $Spin(4,2)$ may be used for conformal symmetry, $AdS_5\rightarrow dS_4$, or simply broken to $Spin(3,1)$ of Minkowski space. Another class of representations finds $Spin(2,2)$ and can give $AdS_3$ with various GUTs. An action for three generations of fermions in the Majorana-Weyl spinor ${\bf 128}$ of $Spin(4,12)$ is found with $Spin(3)$ flavor symmetry inside $E_{8(-24)}$. The ${\bf 128}$ of $Spin(4,12)$ can be regarded as the tangent space to a particular pseudo-Riemannian form of the octo-octonionic Rosenfeld projective plane $E_{8(-24)}/Spin(4,12)= (\mathbb{O}_s\times\mathbb{O})\mathbb{P}^2$.

• The paper uses six-dimensional spinors and the E8(-24) exceptional Lie algebra to compactly encode three generations of standard model matter and house various GUT subgroups.
• By exploring the subalgebraic structure of E8(-24), the framework identifies various grand unified theories including Spin(10), SU(5), and Pati-Salam, along with potential spacetime symmetries.
• The model employs 6D chiral Majorana-Weyl spinors to offer new perspectives on fermionic masses and generations from a higher dimension, suggesting a potential unification of internal gauge and spacetime symmetries.

Beyond the Standard Model with Six-Dimensional Spinors

The paper "Beyond the standard model with six-dimensional spinors" explores a novel approach to representing three generations of matter, grand unified theories (GUTs), and high energy physics by employing six-dimensional (6D) spinors with $Spin(3,3)$ symmetry. This framework provides a new perspective on unifying spacetime with gauge symmetries, offering potential insights into the Grand Unified Theories (GUTs), such as Pati-Salam, $SU(5)$, and $Spin(10)$, as well as spacetime algebras.

Key Findings and Methodologies

1. 6D Spinors and $E_{8(-24)}$: The paper employs 6D spinors to encode the standard model's three generations of matter compactly. These spinors are organized within the symmetry structure of $Spin(3,3)$. The researchers utilize the exceptional Lie algebra $E_{8(-24)}$, a noncompact real form, to house the necessary subgroups representing GUT groups, spacetime symmetries, Higgs bosons, and fermion generations.
2. Decomposition of $E_{8(-24)}$:

By exploring the vast landscape of subalgebraic structures in $E_{8(-24)}$, this work identifies various grand unified theories: - The $Spin(10)$, which includes fermions in chiral spinors ${\bf 16}$ and GUT bosons. - $SU(5)$ and Pati-Salam, found by leveraging the representations $({\bf 10}, {\bf 5})$ in the context of GUT construction. - The researchers also propose that spacetime symmetries like $Spin(4,2)$ could be used for representations pertinent to anti-de Sitter (AdS) spaces or can be broken into profoundly relevant algebras such as $Spin(3,1)$ for Minkowski space.

1. Majorana-Weyl Spinors in 6D: The exploration involves using the chiral Majorana-Weyl spinor ${\bf 128}$ of $Spin(4,12)$, allowing three distinct 4D spacetime configurations with varying mass projections from an overarching 6D framework. These constructs open new avenues for understanding fermionic masses and generations through a higher-dimensional perspective.
2. Implications for Quantum Gravity and GraviGUT Frameworks: The proposed model has implications for the unification of matter and spacetime gauge algebras, which could be interpreted as an internal gravitational theory and an existing lower energy gauge theory like the Standard Model. This duality could potentially lead to insights into unexplored interactions between internal and external spacetime symmetries.
3. Future Directions: The framework establishes a potential for bridging differences between the classical and quantum realms, opening discussions for new connections between string theory, supergravity, and higher-dimensional theories. This discourse encourages further exploration of dualities—and their realization—through $E_{8}$ symmetries, promoting advancements in GUTs and quantum gravity research.

Conclusion and Implications

The paper of $E_{8(-24)}$ through 6D spinors introduces a sophisticated approach to model building beyond the standard model. By efficiently encompassing three generations of fermions, linking spacetime and internal gauge symmetries, and providing a compact framework for exploring the interactions of matter, the paper makes significant strides in theoretical physics. The exploration of intricate Lie algebras, remarkable containment of GUT subalgebras, and associations with different spacetime symmetries portray a promising field for expanding our understanding of high-energy physics and potential future developments in the field.

Potential Developments in AI and Theoretical Physics:

Going forward, advancements in AI and computational models could enable more comprehensive analysis and simulation of the proposed theoretical frameworks. Leveraging AI to explore these high-dimensional algebraic structures could yield novel insights into the unification of physical laws and improve our understanding of fundamental particle interactions within these complex mathematical constructs.