I'm attempting to reproduce the results in http://arxiv.org/PS_cache/hep-th/pdf/0611/0611332.pdf [Broken], notably the V_F(T) potential involved with the Kahler potential(adsbygoogle = window.adsbygoogle || []).push({});

[tex]K = -2 \ln \left( \hat{\mathcal{V}} + \frac{\hat{\xi}}{2} \right)[/tex]

[tex]W = W_{0} + Ae^{-aT}[/tex]

[tex]\hat{\mathcal{V}} = \gamma (T+\bar{T})^{\frac{3}{2}}[/tex]

[tex]\hat{\xi} = \xi (S+\bar{S})^{\frac{3}{2}}[/tex]

Firsly I compute K_ab' using

[tex]K_{a\bar{b}} = \frac{\partial^{2}K}{\partial \phi^{a} \partial \bar{\phi}^{b}}[/tex]

This gives me

[tex]K_{a\bar{b}} = \left(\frac{1}{\hat{\mathcal{V}} + \frac{\hat{\xi}}{2}}\right)^{2} \left( \begin{array}{cc} -\frac{3}{4} \gamma^{\frac{4}{3}} \hat{\mathcal{V}}^{-\frac{1}{3}}(\hat{\xi}-4\hat{\mathcal{V}}) & \frac{9}{4}(\xi \gamma)^{\frac{2}{3}} (\hat{\mathcal{V}}\hat{\xi})^{\frac{1}{3}} \\ \frac{9}{4}(\xi \gamma)^{\frac{2}{3}} (\hat{\mathcal{V}}\hat{\xi})^{\frac{1}{3}} & \frac{3}{4}\xi^{\frac{4}{3}}\hat{\xi}^{-\frac{1}{3}}(\hat{\xi}-\hat{\mathcal{V}}) \end{array} \right)[/tex]

(god that's a pain to type!)

The problem is, when computing the inverse of this metric [tex]K^{a\bar{b}}[/tex], I don't get the quoted form of the paper (which is also used in several other papers by Quevado), which gives each entry of the metric being proportional to

[tex]\frac{1}{\hat{\xi}-\hat{\mathcal{V}}}[/tex]

This gives an important result for Westphal, since the potential V_F turns out to have this same singularity and it gives a vacuum uplifting. If only I could get that far :uhh: I'm guessing it's the det of K_ab' which gives that factor, but that's not what I get when I compute |K_ab'| to get K^ab'.

Where am I going wrong? Am I neglecting a term I shouldn't in section 2 (equations 2.5 through to 2.7) or is it just I've slipped up on the algebra. I've done it about 10 times and still get the same. Given the symmetry between [tex]\hat{\xi}[/tex] and [tex]\hat{\mathcal{V}}[/tex] the Kahler metric has the right form, taking account of the factor of 2 here and there due to the 1/2 in the Kahler potential for xi-hat, so I'm at a lose to see anything obvious I've done wrong unless there's some fundamental result in the theory which I've missed or I've misunderstood the paper itself (both pretty likely knowing me!).

Thanks for any help :)

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# Kahler geometry and potentials

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