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##L \sim ( \partial_\mu \sigma) (\partial^\mu \sigma) - g^{m\bar{n}} g^{r\bar{p}} (\partial_\mu g_{mr} ) ( \partial^\mu g_{\bar{n}\bar{p}} ) ~~~~~ \to (1) ##

Where ##\mu =0,1,2,3,4## and m, n,r, p and ##\bar{n}, \bar{p}, \bar{m}## and ##\bar{r}## = 1,2,3 ( complex coordinates )

Now if I have a complex matrix

##G_{a \bar{b}} \sim g^{m\bar{n}} g^{r\bar{p}} (\partial_a g_{mr} ) ( \partial_{\bar{b}} g_{\bar{n}\bar{p}} ) ~~~~~~~~ \to (2) ##

where ##a, \bar{b}= 1, ..., h_{2,1}##and ##h_{2,1}## is an arbitrary large number.

The question now, can I sub. by ##G_{a \bar{b}}##from equ. (2) into equ. ( 1) ? with the summation on the derivatives has different degrees of freedom? Literally, can I write the Lagrangian as:

##L \sim ( \partial_\mu \sigma) (\partial^\mu \sigma) - G_{a \bar{b}} G^{a \bar{b}} ##?

Any help appreciated!