KK Compactification of Supergravity Models

  • Horaţiu NăstaseEmail author
Part of the Fundamental Theories of Physics book series (FTPH, volume 197)


In this chapter, we will study the KK compactification of supergravity models in higher dimensions. Supergravity models in higher dimensions have: -a graviton, described by the metric \(g_{\mu \nu }\) (which can be written as \(e^a_\mu \eta _{ab}e^b_\nu \)), except for the coupling to fermions, which is written in terms of \(\omega _\mu ^{ab}(e)\). -gravitino(s) \(\psi _{\mu \alpha }^i\) , one for each supersymmetry. Indeed, each supersymmetry \(\epsilon ^i\) takes us from the graviton to a different gravitino, \(\delta e_\mu ^a=\bar{\epsilon }^i\gamma ^a\psi _\mu ^i\). -other fields: scalars \(\phi ^I\), vectors \(A_\mu ^I\), spinors \(\lambda _\alpha ^I\), and also antisymmetric tensors \(A_{\mu _1...\mu _r}\). The antisymmetric tensors are generalizations of the gauge fields (Maxwell fields) \(A_\mu \), that have also a field strength,
$$\begin{aligned} F_{\mu _1...\mu _{r+1}}=(n+1)\partial _{[\mu _1}A_{\mu _2...\mu _{r+1}]}\;, \end{aligned}$$
and so satisfy a gauge invariance
$$\begin{aligned} \delta A_{\mu _1...\mu _r}=\partial _{[\mu _1}\Lambda _{\mu _2...\mu _r]}. \end{aligned}$$
The gauge invariant action for the antisymmetric tensor field is
$$\begin{aligned} S=-\frac{1}{2(r+1)!}\int d^dx\sqrt{-\det g}F^2_{\mu _1...\mu _{r+1}}. \end{aligned}$$

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Instituto de Física Teórica, UNESPSão PauloBrazil

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