[论文解读] Exceptional $\mathfrak{g}_2$ deformations and gauge symmetries

G. Karapetyan|arXiv (Cornell University)|Jan 10, 2026

Algebraic and Geometric Analysis被引用 0

一句话总结

本文引入 Clifford 代数参数化的八元数乘积变形（X、XY 和新的 u-乘积），以研究 G₂ 导出的变形及在变形的 G₂ 结构中如何出现 SU(3)-样子子代数作为残留对称性。

ABSTRACT

Deformed $\mathfrak{g}_2$ exceptional applications are introduced via the Clifford algebra-parametrized formalism. Using the products between multivectors of $\cl_{0,7}$, the Clifford algebra over the metric vector space $\RR^{0,7}$, and octonions, resulting in an octonion, we generalize the exceptional Lie algebra $\mathfrak{g}_2$ applications, also associated with the transformation rules for bosonic and fermionic fields on the 7-sphere $S^7$. The emergence of $SU(3)$-like subalgebras within the exceptional Lie algebra $\mathfrak{g}_2$ provides an algebraic framework reminiscent of the $SU(3)$ gauge symmetry of QCD.

研究动机与目标

• Motivate the study of octonionic and G₂ structures in relation to SU(3)-like gauge symmetries.
• Introduce a Clifford-algebra based framework to deform octonionic multiplication via X-, XY-, and u-products.
• Analyze how deformations induce residual SU(3)-like subalgebras within the Lie algebra 𝔤₂.
• Show how twisted algebras 𝕆_u and 𝕆_{1,u} have automorphisms and derivations connected to 𝔤₂ and its deformations.

提出的方法

• Define the octonion product inside the Clifford algebra 𝒞ℓ_{0,7} using the element ψ to obtain the octonionic product A∘B.
• Generalize the X-product and XY-product to U-produced variants ∘_u, ∘_{1,u}, and ∘_{u,v} acting on Clifford multivectors and octonions.
• Introduce the u-product A∘_uB=(A∘u)∘(u^{-1}∘B) and its variants to deform multiplication.
• Demonstrate that automorphisms/derivations of the deformed algebras 𝕆_u and 𝕆_{1,u} are obtained via conjugation by powers of u, yielding twisted G₂ structures.
• Show that maps like f_u(A)=u^{-1/3}∘A∘u^{1/3} realize G₂_u automorphisms, connecting deformations to 𝔤₂ derivations.
• Provide Lemmas 1 and 2 establishing that certain conjugations produce derivations of 𝕆_u and 𝕆_{1,u}.

实验结果

研究问题

• RQ1 How do Clifford-algebra parametrizations deform the octonionic product and affect associated derivation algebras 𝔤₂?
• RQ2 Can SU(3)-like subalgebras be realized as residual symmetries within deformed G₂ structures via A∘_u and related products?
• RQ3 What is the relationship between automorphism groups G₂_u and the original G₂ via conjugation by u?
• RQ4 Do the generalized products introduce a controlled interpolation between inequivalent SU(3) embeddings inside 𝔤₂?
• RQ5 How can these deformations inform geometric/topological interpretations of gauge symmetries in higher-dimensional contexts?

主要发现

• The X-, XY-, and new u-products extend octonionic multiplication to interact with full Clifford multivectors beyond paravectors.
• Clifford-parametrized deformations interpolate between inequivalent embeddings of SU(3) inside 𝔤₂, yielding SU(3)-like residual symmetries.
• The twisted algebras 𝕆_u and 𝕆_{1,u} have automorphism groups G₂_u conjugate to G₂ via a specific u-dependent map f_u.
• Derivations of the deformed algebras 𝕆_u and 𝕆_{1,u} are explicitly constructed, showing closure under a Leibniz rule with respect to ∘_u and ∘_{1,u} (Lemmas 1 and 2).
• The maps A ↦ f_u(A)=u^{-1/3}∘A∘u^{1/3} realize automorphisms of 𝕆 and connect to the twisted Lie algebra 𝔤₂_u.
• The framework provides a versatile algebraic setting to explore deformations of exceptional structures with potential physical and geometric applications.