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

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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 \( GL (4, C) \times GL(4, C) \).

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Notes

  1. Due to \(\epsilon ^{ \mu _1 \mu _2 \mu _3 \mu _4} = \epsilon ^{ \mu _3 \mu _4 \mu _1 \mu _2} \), the Lagrangian \( {{\mathcal {L}}}_2 \) is not zero.

  2. This can be simply verified, for example, in the case of \( \Gamma _{mn} = i \epsilon _{mnpqr} \Gamma ^{pqr}\) when \(m = 1\) is a temporal-like index, and the rest of the indices are spatial-like, because \(\Gamma _{1n}\) is Hermitian but \(\Gamma ^{pqr}\) is anti-Hermitian, when \( n,p,q,r = 2,3,4,5\). Without the i factor there would be an inconsistency. Similar findings apply to the other combinations, for example, \( \Gamma _{mnpqr} = i \epsilon _{mnpqr} \textbf{1}\).

  3. \(\Gamma ^A = \textbf{1}, \gamma ^a, \gamma ^{ab}, \gamma ^{abc}, \gamma ^{abcd}\) leading to \(2^4 = 16\) generators.

  4. We should notice that the \( K_a, P_a \) generators in (2.31) are both comprised of Hermitian \( \Gamma _i\), and anti-Hermitian \( \Gamma _i \Gamma _5 \) matrices, when \( i = 2,3,4\). Whereas, one has an anti-Hermitian \( \Gamma _1 \) and a Hermitian \(\Gamma _1 \Gamma _5\) matrix. As a result, \( P_a, K_a\) are represented by \( 4 \times 4 \) nilpotent matrices \(P_a^2 = K_a^2 = 0\) (no sum over a).

  5. The length parameter l is the same as the de Sitter throat size \( \rho \).

  6. Gravity involves invariance under diffeomorphisms (coordinate transformations) and gravitons have spin 2, not 1. What occurs is that the torsion constraint \( F^a_{\mu \nu } = 0 \) allows to convert a combination of translations, Lorentz and dilation transformations of the vierbein \(e^a_\mu \) into general coordinate transformations of the vierbein, see [16, 22] for further details.

  7. No boundaries of the bulk spacetime have been invoked. From the isomorphism displayed in Eq. (2.25a) one learns that the bivectors of Cl(2, 3, R) generate the so(2, 3) algebra which is not the same as the Anti de Sitter algebra so(3, 2).

  8. Note that one has not complexified spacetime as in Twistor theory.

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Acknowledgements

We thank M. Bowers for assistance, and to the reviewers for their many suggestions to improve this work.

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  1. Ronin Institute, 127 Haddon Place, Montclair, NJ, 07043, USA

    Carlos Castro Perelman

  2. Plaksha University, Mohali, Punjab, India

    Carlos Castro Perelman

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Perelman, C.C. (Anti) de Sitter Geometry, Complex Conformal Gravity-Maxwell Theory from a Cl(4, C) Gauge Theory of Gravity and Grand Unification. Adv. Appl. Clifford Algebras 33, 54 (2023). https://doi.org/10.1007/s00006-023-01299-3

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