Killing vectors for a given Kahler potential

[L0r3n20]

Hi all!
It's already a couple of days that I'm trying to solve this problem. I've been given the following Kahler potential
$$\mathcal{K} = - Log\left[ i \left(s-\bar{s}\right)\left(t-\bar{t}\right)\left(u-\bar{u}\right)\right]$$
and I have to compute the Killing vectors of such a manifold. I've tried to compute them using
$$\nabla_a \bar{\xi_b} + \bar{\nabla}_b \xi_a =0$$
but then I can't solve the PDE system if not trivially.
Any suggestion? =)

 

[L0r3n20]

By the way I forgot to tell you I've tried using Cartan's calculus
$$\mathcal{L}\left(g\right) =\iota_{\xi}\left(\mathrm{d}g\right) + \mathrm{d}\left(\iota_{\xi}g\right)$$
but I wasn't able to write the metric in a "covariant" way. It should be something like (I know this formula is wrong it's just to give you the taste)
$$-\frac{1}{\left(x_i-\bar{x}_j\right)\delta^{ij}}\delta_{ij}$$
since I have a diagonal metric...

 

[Ben Niehoff]

By the way I forgot to tell you I've tried using Cartan's calculus
$$\mathcal{L}\left(g\right) =\iota_{\xi}\left(\mathrm{d}g\right) + \mathrm{d}\left(\iota_{\xi}g\right)$$

This formula is not applicable, because g is not a differential form. However, you can use the product rule:

$$\begin{align*}\mathcal{L}_\xi g &= \mathcal{L}_\xi (g_{ij} \, dx^i \otimes dx^j) \\ &= (\mathcal{L}_\xi g_{ij}) \, dx^i \otimes dx^j + g_{ij} \, (\mathcal{L}_\xi dx^i) \otimes dx^j + g_{ij} \, dx^i \otimes (\mathcal{L}_\xi dx^j) \end{align*}$$

but I wasn't able to write the metric in a "covariant" way.

What do you mean?

It should be something like (I know this formula is wrong it's just to give you the taste)
$$-\frac{1}{\left(x_i-\bar{x}_j\right)\delta^{ij}}\delta_{ij}$$
since I have a diagonal metric...

Might want to work on that. It appears that you don't understand how the index gymnastics works. You have too many i, j indices.

 

[L0r3n20]

First of all thank you for your reply.
I knew the metric was wrong but I cannot write it in a compact form. It should read
$$g_{i \bar{j}} \mathrm{d} z^i \wedge \mathrm{d} \bar{z}^{\bar{j}} = -\frac{1}{\left( s-\bar{s}\right)} \mathrm{d} s \wedge \mathrm{d} \bar{s} -\frac{1}{\left( t-\bar{t}\right)} \mathrm{d} t \wedge \mathrm{d} \bar{t} -\frac{1}{\left( u-\bar{u}\right)} \mathrm{d} u \wedge \mathrm{d} \bar{u}$$ from which I'm not able to extract $$g_{i \bar{j}}$$
However this is not my main problem because I wrote the metric in a matrix form and compute the Killing equation but now I can't solve it... :(

 

[blue_raver22]

Hello, thank you for reaching out with your question. I understand that you are having difficulty solving for the Killing vectors of a given Kahler potential. This can be a complex problem, so I will try to provide some guidance and suggestions.

First, let's review the concept of Killing vectors. In general relativity, Killing vectors are vector fields that preserve the metric of a space-time. In other words, they represent symmetries of the space-time, and their corresponding symmetry transformations leave the metric unchanged. In your case, the Kahler potential represents the metric of a manifold, and the Killing vectors represent the symmetries of that manifold.

Now, let's look at the equation you mentioned, \nabla_a \bar{\xi_b} + \bar{\nabla}_b \xi_a =0. This is known as the Killing equation and is used to solve for the Killing vectors. However, as you have mentioned, it can be difficult to solve this equation in general, and it may not always lead to a solution.

One suggestion I have is to try using a different approach. Instead of directly solving the Killing equation, you can try using the properties of the Kahler potential to simplify the problem. For example, since the Kahler potential is in logarithmic form, you can try using logarithmic derivatives to simplify the equations. Additionally, you can also try using the properties of complex variables to simplify the equations.

Another suggestion is to consult with a colleague or mentor who may have experience with solving similar problems. They may be able to provide insight and guidance on how to approach the problem.

I hope these suggestions are helpful to you in solving the problem. Keep in mind that solving for Killing vectors can be a challenging task, but with persistence and the right approach, it is possible to find a solution. Best of luck!