KK Compactification of Supergravity Models

Abstract

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

$$\begin{aligned} S=-\frac{1}{2(r+1)!}\int d^dx\sqrt{-\det g}F^2_{\mu _1...\mu _{r+1}}. \end{aligned}$$