(Anti) de Sitter Geometry, Complex Conformal Gravity-Maxwell Theory from a Cl(4, C) Gauge Theory of Gravity and Grand Unification

IF 1.1 2区 数学 Q2 MATHEMATICS, APPLIED
Carlos Castro Perelman
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引用次数: 0

Abstract

We present the deep connections among (Anti) de Sitter geometry, and complex conformal gravity-Maxwell theory, stemming directly from a gauge theory of gravity based on the complex Clifford algebra Cl(4, C). This is attained by simply promoting the de (Anti) Sitter algebras so(4, 1), so(3, 2) to the real Clifford algebras Cl(4, 1, R), Cl(3, 2, R), respectively. This interplay between gauge theories of gravity based on Cl(4, 1, R), Cl(3, 2, R) , whose bivector-generators encode the de (Anti) Sitter algebras so(4, 1), so(3, 2), respectively, and 4D conformal gravity based on Cl(3, 1, R) is reminiscent of the \(AdS_{ D+1}/CFT_D\) correspondence between \(D+1\)-dim gravity in the bulk and conformal field theory in the D-dim boundary. Although a plausible cancellation mechanism of the cosmological constant terms appearing in the real-valued curvature components associated with complex conformal gravity is possible, it does not occur simultaneously in the imaginary curvature components. Nevertheless, by including a Lagrange multiplier term in the action, it is still plausible that one might be able to find a restricted set of on-shell field configurations leading to a cancellation of the cosmological constant in curvature-squared actions due to the coupling among the real and imaginary components of the vierbein. We finalize with a brief discussion related to \(U(4) \times U(4)\) grand-unification models with gravity based on \( Cl (5, C) = Cl(4,C) \oplus Cl(4,C)\). It is plausible that these grand-unification models could also be traded for models based on \( GL (4, C) \times GL(4, C) \).

从Cl(4,C)规范理论看(反)de Sitter几何、复共形引力Maxwell理论
我们提出了(反)de Sitter几何和复共形引力Maxwell理论之间的深层联系,它们直接源于基于复Clifford代数Cl(4,C)的引力规范理论。这是通过简单地将de(Anti)Sitter代数so(4,1),so(3,2)分别推广到实Clifford代数Cl(4,2,R),Cl(3,1,R)来实现的。基于Cl(4,1,R)、Cl(3,2,R)的引力规范理论之间的这种相互作用让人想起体中的\(D+1\)-dim引力和D-dim边界中的共形场论之间的\(AdS_{D+1}/CFT_D\)对应关系。尽管宇宙常数项出现在与复共形引力相关的实值曲率分量中的一种看似合理的抵消机制是可能的,但它不会同时出现在虚曲率分量中。然而,通过在作用中包含拉格朗日乘子项,仍然有可能找到一组有限的壳上场配置,由于vierbein的实分量和虚分量之间的耦合,导致曲率平方作用中的宇宙学常数被抵消。最后,我们简要讨论了基于(Cl(5,C)=Cl(4,C)\oplus Cl(4,C)\)的重力大统一模型。这些大统一模型也可以交换为基于\(GL(4,C)\乘以GL(4、C)\的模型。
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来源期刊
Advances in Applied Clifford Algebras
Advances in Applied Clifford Algebras 数学-物理:数学物理
CiteScore
2.20
自引率
13.30%
发文量
56
审稿时长
3 months
期刊介绍: Advances in Applied Clifford Algebras (AACA) publishes high-quality peer-reviewed research papers as well as expository and survey articles in the area of Clifford algebras and their applications to other branches of mathematics, physics, engineering, and related fields. The journal ensures rapid publication and is organized in six sections: Analysis, Differential Geometry and Dirac Operators, Mathematical Structures, Theoretical and Mathematical Physics, Applications, and Book Reviews.
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