A Towards formulating an invariant Lagrangian

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Assuming a Lagrangian proportional to the following terms:

##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!
 
Maybe I am misunderstanding your notation, but just ##G_{a\bar{b}}G^{a\bar{b}}## written out is not the term you are replacing in the lagrangian (even ignoring the different number of terms that would be summed), so how are you supposing this should work?
 

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