Back

Indexes:
$\alpha-\omega$ = 4-curved
$a-h,o-z$ = 3-curved
$A-Z$ = 4-flat
$i-n$ = 3-flat

dual of a dual of a degree-2 tensor:
$ \star \star T^{IJ} = {1\over4} {\epsilon^{IJ}}_{KL} {\epsilon^{KL}}_{MN} T^{MN} $
$ = -\delta^I_{[M} \delta^J_{N]} T^{MN}$
$ = -T^{[IJ]}$

dual of dual of antisymmetric tensor:
$ \star \star T^{IJ} = -T^{[IJ]} = -T^{IJ}$

A tensor $ T^{IJ} $ is self-dual if it fulfills:
$ \star T^{IJ} = i T^{IJ} $
for $ \star $ is the dual of the Lorentz coordinates

self-dual projection opertor:
$ P^\pm = {1\over 2}(1\mp i\star) $

self-dual projection tensor:
$ {P^{\pm IJ}}_{KL} = {1 \over 2} ( \delta^I_K \delta^J_L \mp {i\over 2} {\epsilon^{IJ}}_{KL} ) $

any antisymmetric tensor can be decomposed into self-dual and anti-self-dual components:
self-dual: $ {^+T}^{IJ} = P^+ T^{IJ} = {1\over 2} (T^{IJ} - i {\star T}^{IJ}) = {1\over 2} ( T^{IJ} - {i\over 2} {\epsilon^{IJ}}_{KL} T^{KL} ) $
anti-self-dual: $ {^-T}^{IJ} = P^- T^{IJ} = {1\over 2} (T^{IJ} + i {\star T}^{IJ}) = {1\over 2} ( T^{IJ} + {i\over 2} {\epsilon^{IJ}}_{KL} T^{KL} ) $
such that: $ {^+T}^{IJ} + {^-T}^{IJ} = T^{IJ} $

verify self-duality:
$ \star {^+T}^{IJ} $
$ = {1\over 2} {\epsilon^{IJ}}_{KL} {^+T}^{KL} $
$ = {1\over 2} {\epsilon^{IJ}}_{KL} {1\over 2} (T^{KL} - {i\over 2} {\epsilon^{KL}}_{MN} T^{MN}) $
$ = {1\over4} ({\epsilon^{IJ}}_{KL} T^{KL} - {i\over 2} {\epsilon^{IJ}}_{KL} {\epsilon^{KL}}_{MN} T^{MN}) $
$ = {i\over 2} (-{1\over4} {\epsilon^{IJ}}_{KL} {\epsilon^{KL}}_{MN} T^{MN} - {i\over 2} {\epsilon^{IJ}}_{KL} T^{KL}) $
$ = {i\over 2} (\delta^{IJ}_{MN} T^{MN} - {i\over 2} {\epsilon^{IJ}}_{KL} T^{KL}) $
$ = {i\over 2} ({1\over 2} (\delta^I_M \delta^J_N - \delta^I_N \delta^J_M) T^{MN} - {i\over 2} {\epsilon^{IJ}}_{KL} T^{KL}) $
$ = {i\over 2} ({1\over 2} (T^{IJ} - T^{JI}) - {i\over 2} {\epsilon^{IJ}}_{KL} T^{KL}) $
using $ T^{IJ} = -T^{JI} $
$ = i {1\over 2} (T^{IJ} - {i\over 2} {\epsilon^{IJ}}_{KL} T^{KL}) $
$ = i \cdot {^+T}^{IJ} $

self-dual of spin connection:
$ {A_\alpha}^{IJ} = {{^+\omega}_\alpha}^{IJ} $
$ = {1\over 2} {\omega_\alpha}^{IJ} - {i\over 4} {\epsilon^{IJ}}_{KL} {\omega_\alpha}^{KL} $

self-dual covariant derivative on Lorentz indexes:
$ \mathcal{D}_\alpha {T_\beta}^I = \partial_\alpha {T_\beta}^I + {{A_\alpha}^I}_J {T_\beta}^J $
(I know I was using $\mathcal{D}$ in the Palatini variation part of the last page. Just forget about that. I'm not sure what to do with it just yet.)

Make the self-dual covariant derivative on Lorentz indexes compatible with Lorentz metric:
follows the same process as the similar proof for $ D_a $
$ 0 = \mathcal{D}_\alpha \eta_{IJ} = \partial_\alpha \eta_{IJ} - {{A_\alpha}^K}_I \eta_{KJ} - {{A_\alpha}^K}_J \eta_{IK} $
$ 0 = 0 - A_{\alpha JI} - A_{\alpha IJ} $
$ A_{aIJ} = -A_{\alpha JI} $

curvature of self-dual spin connection on Lorentz indexes:
$ {{F_{\alpha\beta}}^I}_J V^J = \mathcal{D}_\alpha \mathcal{D}_\beta V^I - \mathcal{D}_\beta \mathcal{D}_\alpha V^I $
$ = \mathcal{D}_\alpha (\partial_\beta V^I + {{A_\beta}^I}_J V^J) - \mathcal{D}_\beta (\partial_\alpha V^I + {{A_\alpha}^I}_J V^J) $
$ = \partial_\alpha (\partial_\beta V^I + {{A_\beta}^I}_J V^J) + {{A_\alpha}^I}_K (\partial_\beta V^K + {{A_\beta}^K}_J V^J) - \partial_\beta (\partial_\alpha V^I + {{A_\alpha}^I}_J V^J) - {{A_\beta}^I}_K (\partial_\alpha V^K + {{A_\alpha}^K}_J V^J) $
$ = \partial_\alpha \partial_\beta V^I + \partial_\alpha ({{A_\beta}^I}_J V^J) + {{A_\alpha}^I}_K \partial_\beta V^K + {{A_\alpha}^I}_K {{A_\beta}^K}_J V^J - \partial_\beta \partial_\alpha V^I - \partial_\beta ({{A_\alpha}^I}_J V^J) - {{A_\beta}^I}_K \partial_\alpha V^K - {{A_\beta}^I}_K {{A_\alpha}^K}_J V^J $
$ = V^J \partial_\alpha {{A_\beta}^I}_J + {{A_\beta}^I}_J \partial_\alpha V^J - V^J \partial_\beta {{A_\alpha}^I}_J - {{A_\alpha}^I}_J \partial_\beta V^J + {{A_\alpha}^I}_K \partial_\beta V^K - {{A_\beta}^I}_K \partial_\alpha V^K + {{A_\alpha}^I}_K {{A_\beta}^K}_J V^J - {{A_\alpha}^K}_J {{A_\beta}^I}_K V^J $
$ {{F_{\alpha\beta}}^I}_J = 2 \partial_{[\alpha} {{A_{\beta]}}^I}_J + 2 {{A_{[\alpha}}^I}_{|K|} {{A_{\beta]}}^K}_J $
$ {F_{\alpha\beta}}^{IJ} = 2 (\partial_{[\alpha} {A_{\beta]}}^{IJ} + {A_{[\alpha}}^{IK} {A_{\beta]K}}^J) $

useful identity:
${\epsilon^{IK}}_{MN} {\omega_\alpha}^{MN} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_\beta}^{QR}$
$= {\epsilon^I}_{MNP} {{\epsilon^J}_{QR}}^P {\omega_\alpha}^{MN} {\omega_\beta}^{QR}$
$= -\epsilon^{IKLP} \epsilon_{SQRP} \eta_{KM} \eta_{LN} \eta^{JS} {\omega_\alpha}^{MN} {\omega_\beta}^{QR}$
$= \delta^{IKL}_{SQR} \eta_{KM} \eta_{LN} \eta^{JS} {\omega_\alpha}^{MN} {\omega_\beta}^{QR}$
$= (\delta^I_S \delta^K_Q \delta^L_R + \delta^I_Q \delta^K_R \delta^L_S + \delta^I_R \delta^K_S \delta^L_Q - \delta^I_R \delta^K_Q \delta^L_S - \delta^I_Q \delta^K_S \delta^L_R - \delta^I_S \delta^K_R \delta^L_Q) \eta^{JS}{\omega_\alpha}^{MN} {\omega_\beta}^{QR}$
$= \omega_{\alpha QR} {\omega_\beta}^{QR} \eta^{IJ} + {\omega_{\alpha R}}^J {\omega_\beta}^{IR} + {{\omega_\alpha}^J}_Q {\omega_\beta}^{QI} - {\omega_{\alpha Q}}^J {{\omega_\beta}^QI} - {{\omega_\alpha}^J}_R {\omega_\beta}^{IR} - \omega_{\alpha RQ} {\omega_\beta}^{QR} \eta^{IJ} $
$= 2 \eta^{IJ} \omega_{\alpha KL} {\omega_\beta}^{KL} + 2 {\omega_{\alpha K}}^J {\omega_\beta}^{IK} + 2 {{\omega_\alpha}^J}_K {\omega_\beta}^{KI}$
$= 4 {{\omega_\beta}^I}_K {\omega_\alpha}^{KJ} + 2 \eta^{IJ} \omega_{\alpha KL} {\omega_\beta}^{KL}$
Therefore
${\epsilon^{IK}}_{MN} {\omega_{[\alpha}}^{MN} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_{\beta]}}^{QR}$
$= 4 {\omega_{[\alpha}}^{KJ} {{\omega_{\beta]}}^I}_K + 2 \eta^{IJ} \omega_{{[\alpha} KL} {\omega_{\beta]}}^{KL}$
$= 4 {\omega_{[\alpha}}^{KJ} {{\omega_{\beta]}}^I}_K$

antisymmetric operator applied to curved coordinates of square of spin connection is antisymmetric in Lorentz coordinates:
${\omega_{[\alpha}}^{IK} {\omega_{{\beta]}K}}^J$
apply antisymmetric relation on curved coordinate indexes:
$= -{\omega_{[\beta}}^{IK} {\omega_{{\alpha]}K}}^J$
reverse order of terms
$= -{\omega_{{[\alpha}K}}^J {\omega_{\beta]}}^{IK}$
apply antisymmetric property of Lorentz indexes of spin connection
$= -{{\omega_{[\alpha}}^J}_K {\omega_{\beta]}}^{KI}$
index gymnastics:
$= -{\omega_{[\alpha}}^{JK} {\omega_{\beta]K}}^I$

TODO show the antisym of spin times dual spin and antisym of dual spin times spin are both antisym themselves
then, because antisym of spin squared and antisym of dual-spin spin and spin dual-spin are all antisym,
use the property that $\star \star = -1$ for antisym only to show the next identity:
also useful:
$\star({\omega_{[\alpha}}^{IK} {\omega_{{\beta]}K}}^J) = {1\over2} ({\omega_{[\alpha}}^{IK} {(\star\omega)_{{\beta]}K}}^J + {(\star\omega)_{[\alpha}}^{IK} {\omega_{{\beta]}K}}^J)$
${1\over2} {\epsilon^{IJ}}_{KL} ({\omega_\alpha}^{KM} {\omega_{\beta M}}^L - {\omega_\beta}^{KM} {\omega_{\alpha M}}^L) = {1\over4} ({\omega_\alpha}^{IK} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_\beta}^{QR} + {\epsilon^{IK}}_{MN} {\omega_\alpha}^{MN} \eta_{KP} {\omega_\beta}^{PJ} - {\omega_\beta}^{IK} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_\alpha}^{QR} - {\epsilon^{IK}}_{MN} {\omega_\beta}^{MN} \eta_{KP} {\omega_\alpha}^{PJ}) $

The curvature of the self-dual of the spin connection is the self-dual of the curvature of the spin-connection (... which itself is the curvature of the affine connetion with indexes transformed)
$ {F_{\alpha\beta}}^{IJ} = \partial_\alpha {A_\beta}^{IJ} - \partial_\beta {A_\alpha}^{IJ} + {A_\alpha}^{IK} {A_{\beta K}}^J - {A_\beta}^{IK} {A_{\alpha K}}^J $
$= \partial_\alpha ({1\over2} {\omega_\beta}^{IJ} - {i\over4} {\epsilon^{IJ}}_{MN} {\omega_\beta}^{MN}) - \partial_\beta ({1\over2} {\omega_\alpha}^{IJ} - {i\over4} {\epsilon^{IJ}}_{MN} {\omega_\alpha}^{MN}) + ({1\over2} {\omega_\alpha}^{IK} - {i\over4} {\epsilon^{IK}}_{MN} {\omega_\alpha}^{MN} ) \eta_{KP} ({1\over2} {\omega_\beta}^{PJ} - {i\over4} {\epsilon^{PJ}}_{QR} {\omega_\beta}^{QR} ) - ({1\over2} {\omega_\beta}^{IK} - {i\over4} {\epsilon^{IK}}_{MN} {\omega_\beta}^{MN} ) \eta_{KP} ({1\over2} {\omega_\alpha}^{PJ} - {i\over4} {\epsilon^{PJ}}_{QR} {\omega_\alpha}^{QR}) $
$= {1\over2} \partial_\alpha {\omega_\beta}^{IJ} - {1\over2} \partial_\beta {\omega_\alpha}^{IJ} + {1\over4} {\omega_\alpha}^{IK} {\omega_{\beta K}}^J - {1\over4} {\omega_\beta}^{IK} {\omega_{\alpha K}}^J - {i\over4} {\epsilon^{IJ}}_{KL} (\partial_\alpha {\omega_\beta}^{KL} - \partial_\beta {\omega_\alpha}^{KL}) - {1\over16} {\epsilon^{IK}}_{MN} {\omega_\alpha}^{MN} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_\beta}^{QR} + {1\over16} {\epsilon^{IK}}_{MN} {\omega_\beta}^{MN} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_\alpha}^{QR} - {i\over8} ({\omega_\alpha}^{IK} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_\beta}^{QR} + {\epsilon^{IK}}_{MN} {\omega_\alpha}^{MN} \eta_{KP} {\omega_\beta}^{PJ}) + {i\over8} ({\omega_\beta}^{IK} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_\alpha}^{QR} + {\epsilon^{IK}}_{MN} {\omega_\beta}^{MN} \eta_{KP} {\omega_\alpha}^{PJ}) $
using the identity above, ${\epsilon^{IK}}_{MN} {\omega_{[\alpha}}^{MN} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_{\beta]}}^{QR} = 4 {\omega_{[\alpha}}^{KJ} {{\omega_{\beta]}}^I}_K$
$= {1\over2} \partial_\alpha {\omega_\beta}^{IJ} - {1\over2} \partial_\beta {\omega_\alpha}^{IJ} + {1\over4} {\omega_\alpha}^{IK} {\omega_{\beta K}}^J - {1\over4} {\omega_\beta}^{IK} {\omega_{\alpha K}}^J - {i\over4} {\epsilon^{IJ}}_{KL} (\partial_\alpha {\omega_\beta}^{KL} - \partial_\beta {\omega_\alpha}^{KL}) - {1\over4} {\omega_\alpha}^{KJ} {{\omega_\beta}^I}_K + {1\over4} {\omega_\beta}^{KJ} {{\omega_\alpha}^I}_K - {i\over8} ({\omega_\alpha}^{IK} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_\beta}^{QR} + {\epsilon^{IK}}_{MN} {\omega_\alpha}^{MN} \eta_{KP} {\omega_\beta}^{PJ}) + {i\over8} ({\omega_\beta}^{IK} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_\alpha}^{QR} + {\epsilon^{IK}}_{MN} {\omega_\beta}^{MN} \eta_{KP} {\omega_\alpha}^{PJ}) $
$= {1\over2} \partial_\alpha {\omega_\beta}^{IJ} - {1\over2} \partial_\beta {\omega_\alpha}^{IJ} + {1\over2} {\omega_\alpha}^{IK} {\omega_{\beta K}}^J - {1\over2} {\omega_\beta}^{IK} {\omega_{\alpha K}}^J - {i\over4} {\epsilon^{IJ}}_{KL} (\partial_\alpha {\omega_\beta}^{KL} - \partial_\beta {\omega_\alpha}^{KL}) - {i\over8} ({\omega_\alpha}^{IK} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_\beta}^{QR} + {\epsilon^{IK}}_{MN} {\omega_\alpha}^{MN} \eta_{KP} {\omega_\beta}^{PJ}) + {i\over8} ({\omega_\beta}^{IK} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_\alpha}^{QR} + {\epsilon^{IK}}_{MN} {\omega_\beta}^{MN} \eta_{KP} {\omega_\alpha}^{PJ}) $
$= {1\over2} {\Omega_{\alpha\beta}}^{IJ} - {i\over4} {\epsilon^{IJ}}_{KL} (\partial_\alpha {\omega_\beta}^{KL} - \partial_\beta {\omega_\alpha}^{KL}) - {i\over8} ({\omega_\alpha}^{IK} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_\beta}^{QR} + {\epsilon^{IK}}_{MN} {\omega_\alpha}^{MN} \eta_{KP} {\omega_\beta}^{PJ}) + {i\over8} ({\omega_\beta}^{IK} \eta_{KP} {\epsilon^{PJ}}_{QR} {\omega_\alpha}^{QR} + {\epsilon^{IK}}_{MN} {\omega_\beta}^{MN} \eta_{KP} {\omega_\alpha}^{PJ}) $
using the identity above that $\star \star [a,b] = \star [\star a,b] = [a,\star b]$
$= {1\over2} {\Omega_{\alpha\beta}}^{IJ} - {i\over4} {\epsilon^{IJ}}_{KL} (\partial_\alpha {\omega_\beta}^{KL} - \partial_\beta {\omega_\alpha}^{KL}) - {i\over4} {\epsilon^{IJ}}_{KL} ({\omega_\alpha}^{KM} {\omega_{\beta M}}^L - {\omega_\beta}^{KM} {\omega_{\alpha M}}^L) $
$= {1\over2} {\Omega_{\alpha\beta}}^{IJ} - {i\over4} {\epsilon^{IJ}}_{KL} (\partial_\alpha {\omega_\beta}^{KL} - \partial_\beta {\omega_\alpha}^{KL} + {\omega_\alpha}^{KM} {\omega_{\beta M}}^L - {\omega_\beta}^{KM} {\omega_{\alpha M}}^L) $
$ = {1\over2} {\Omega_{\alpha\beta}}^{IJ} - {i\over4} {\epsilon^{IJ}}_{KL} {\Omega_{\alpha\beta}}^{KL} $
$ {F_{\alpha\beta}}^{IJ} = {^+\Omega_{\alpha\beta}}^{IJ} $

Self-dual Palatini action:
$S = \int e {e^\alpha}_I {e^\beta}_J {(^+\Omega)_{\alpha\beta}}^{IJ} dx^4 = \int e {e^\alpha}_I {e^\beta}_J {F_{\alpha\beta}}^{IJ} dx^4 $

Back