I am reading the book Supergravity.(adsbygoogle = window.adsbygoogle || []).push({});

In chapter 4, section 4.3.2

- Duality for gauge fields and complex scalar:

The simplest case of electromagnetic duality in an interacting field theeory occurs with one abelian gauge field ##A_{\mu}(x)## and a complex scalar field ##Z(x)##. The electromagnetic part of the Lagrangian is:

$$L=-\frac{1}{4}(ImZ)F_{\mu\nu}F^{\mu\nu}-\frac{1}{8}(ReZ)\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$$

The author said:

The gauge Bianci identity and E.OM of our theory are:

$$\partial_{\mu}\tilde{F}^{\mu\nu}=0, \hspace{2cm} \partial_{\mu}[(ImZ)F^{\mu\nu} +i(ReZ)\tilde{F}^{\mu\nu}]=0$$

He continued to say:

$$G^{\mu\nu}=\epsilon^{\mu\nu\rho\sigma}\frac{\delta S}{\delta F^{\rho \sigma}}=-i(ImZ)\tilde{F}^{\mu\nu}+(ReZ)F^{\mu\nu}$$

My question is I tried to carry on the calculations as I moved with the reading, but I failed in deriving the ##G^{\mu\nu}## and I am not getting this final result because I am new to tensors and am trying to learn GR and SUGRA simultaneously and want to get a better picture of how to work this out before I move to more advanced levels.. Any sort of help is appreciated.

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Find G_{\mu\nu} from this lagrangian.

Loading...

Similar Threads - Find G_{munu} lagrangian | Date |
---|---|

I Is this the general way to find the center of mass motion? | Jan 19, 2018 |

I How did LIGO know where to find merging black holes? | Oct 18, 2017 |

Riemann tensor and derivatives of ##g_{\mu\nu}## | Dec 23, 2014 |

**Physics Forums - The Fusion of Science and Community**