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Quadratic gravity from BF theory in two and three dimensions

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Abstract

A polynomial action in the gauge fields is proposed for two and three dimensions, reproducing quadratic gravity. Such action is further amenable for quantization and based upon the topological BF theory.

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Notes

  1. For the sake of simplicity, we limit our discussion to positive-definite metrics.

  2. In two dimensions, equations can be simplified, by redefining the generators as \(J_I=\frac{1}{2}{\epsilon _I}^{\textit{JK}}T_{\textit{JK}}\). However, we prefer to maintain the \(T_{\textit{IJ}}\) notation, that is straightforward to generalize to three (or higher) dimensions.

  3. Alternatively, we can rescale the generators \(K_1=T_{31}/\ell \), \(K_2=-T_{23}/\ell \) and maintain \(J=T_{12}\). In the limit \(\ell \rightarrow \infty \) we have a Wigner–Inönü contraction \(\textit{SO}(3)\rightarrow \textit{ISO}(2)\) and therefore \(K_1\) and \(K_2\) can be identified with translations and J with rotation.

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Acknowledgments

R. da Rocha is grateful for the CNPq Grants No. 303027/2012-6, No. 451682/2015-7 and No. 473326/2013-2 and also thanks to FAPESP 10270-0 for partial financial support.

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Authors and Affiliations

  1. Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Santo André, SP, 09210-170, Brazil

    Ricardo Paszko

  2. Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, Santo André, SP, 09210-170, Brazil

    Roldão da Rocha

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  1. Ricardo Paszko
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  2. Roldão da Rocha
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Correspondence to Ricardo Paszko.

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Paszko, R., da Rocha, R. Quadratic gravity from BF theory in two and three dimensions. Gen Relativ Gravit 47, 94 (2015). https://doi.org/10.1007/s10714-015-1937-x

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