# Going from a complex rep to a real rep

1. Jan 17, 2007

### AlphaNumeric

Not sure if this is appropriate here or the differential geometry forum, but anyway....

I've got a Kahler potential and from that can compute it's metric and then move onto the dynamics of the fields which the potential is a function of. One of the terms I consider is $$K_{a\bar{b}}\dot{\Phi}^{a}\dot{\Phi}^{\bar{b}}$$ where $$\Phi$$ is a complex field.

Now instead of considering the complex metric $$K_{a\bar{b}}$$ I want to go a point of view involving just real fields, though obviously twice as many. Papers I've read and books I've flicked through just seem to say "And $$G_{ij}$$ is the metric associated with the real representation of the Kahler potential" or words to that effect. Problem is, I'm not sure how to compute it, but have a guess by the following logic :

Let K be a function of n complex fields $$\Phi$$ and so by definition of metric :

$$ds^{2} = K_{a\bar{b}}d\Phi^{a} d\Phi^{\bar{b}} = G_{ij}d\phi^{i}d\phi^{j}$$

where $$\Phi^{a} = \phi^{a} + i\phi^{a+n}$$. Putting that into the Kahler form gives

$$K_{a\bar{b}}(d\phi^{a} + id\phi^{a+n})(d\phi^{a} - id\phi^{a+n}) = K_{ab}d\phi^{a}d\phi^{b}+K_{ab}d\phi^{a+n}d\phi^{b+n} = G_{ij}d\phi^{i}d\phi^{j}$$

Matching efficents of $$d\phi^{i}d\phi^{j}$$ gives $$G_{ab} = K_{ab} = G_{a+n \, b+n}$$

This then gives the general form

$$G_{ij} = \left( \begin{array}{cc}K_{ab} & 0 \\ 0 & K_{ab} \end{array} \right)$$

This seems to make sense, is it correct?

Last edited: Jan 17, 2007