This should probably go somewhere else, but does anyone have any experience with mathematica? I want it to minimze a potential, and this is what I get: \(V[\text{s},\text{t}]\text{:=}\frac{1}{4s t}\left((2 s\text{ }\text{FS}[s,t])^2 + (2 t \text{FT}[s,t])^2 - 3 (W[s,t])^2\right)\) \(\text{Minimize}[V[s,t],\{s,t\}]\) \(\text{Minimize}\left[\frac{-3 \left(e^{-\frac{24 \pi ^2 s}{5}-t}-2 e^{-4 \pi ^2 s-t}\right)^2+4 \left(\frac{-\frac{24}{5} e^{-\frac{24 \pi ^2 s}{5}-t} \pi ^2+8 e^{-4 \pi ^2 s-t} \pi ^2}{e^{-\frac{24 \pi ^2 s}{5}-t}-2 e^{-4 \pi ^2 s-t}}-\frac{1}{2 s}\right)^2 s^2+4 \left(\frac{-e^{-\frac{24 \pi ^2 s}{5}-t}+2 e^{-4 \pi ^2 s-t}}{e^{-\frac{24 \pi ^2 s}{5}-t}-2 e^{-4 \pi ^2 s-t}}-\frac{1}{2 t}\right)^2 t^2}{4 s t},\{s,t\}\right]\) God I love "Copy as LaTeX"...
And before you ask, yes, this IS gaugino condensation, that IS a dilaton, and that IS a Kahler structure modulus, and I AM doing obscene things with supersymmetry...
It's wonderful stuff isn't it Please Register or Log in to view the hidden image! I've had to do exactly the kind of thing. It's been my experience that the huge differences of scales between the minima and the rest of the landscape is so much that you will not be able to use the minimisation routine of Mathematica. After all, if it worked, we'd not have much trouble with vacua, would we? There's a couple of ways to approach this. You can get knee deep in the StringVACUA program James Gray et al wrote, which is going to be WAY more than you want do deal with because you've also got gaugino condensates and my eyes glazed over once I got to the Sturm Queries section of their paper and James didn't really blame me for that. Another way is to do what I did and use numerical methods and actually solve the equations of motion for the moduli for various starting initial conditions and then see how they evolve. It was that method which told me that you need to start Mathematica's Solve[ ] routine VERY close to the minimum or else you aren't going to get it to find it. Plus you can then compute all the great stuff like dark energy of your space-time, e-fold inflation levels and plot your moduli in the landscape! I can provide you with the code if you want, since I originally got it from an ArXiv'd paper which provided it's Mathematica code and I've since given it to a collegue of mine whose doing similar stuff. It's a bit more hit and miss in the sense you don't actually just put in an expression and find the minima but it gives you a much better understanding of what the potential looks like and what properties it has. Here is some of my work from almost 2 years ago typed up into a paper-like format. Never worth trying to submit but I used it for practice in presenting my work. I spent a few months messing around with precisely this kind of thing so if I can be of any specific help, I'm happy to help.
Thanks for the help. I'm just trying to figure out how to fool with this toy problem first, before looking at a more difficult Kahler potential. I may take you up on your offer, though. I have fooled around with the stringVACUA thing, but found that it's only of limited utility. StringVACUA can't deal with (insofar as I can tell) canonical Kahler potentials, much less the types of Kahler potentials that I want to look at.
By canonical I'm assuming you mean \(-\sum_{a} \ln ( \Phi^{a} + \bar{\Phi}^{a} )\) ? It can, it's not something you really want to be doing with it. When doing non-supersymmetric equations, I generally have to mess around with particular constraints by hand until the ideal of constraints are in a very nice form. You'd be amazed the difference something as inoccuous as putting Expand[ ] into your expressions can do when using Singular. The complicated expression I had in that pdf was just something I was messing with at the time. The point is that you can find the minimum pretty straight forwardly for some systems by doing numerical processes. The method Westphal uses in his paper I talk about in that pdf is a pretty good method to find the minimum to within a factor of 2 or so in your moduli. I'd recommend trying that and then putting that approximation into the Mathematica Solve[ ] program and seeing what happens. You'll likely be close enough to the minimum that Mathematica can do it for you from there.
Well, no. I thought canonical means \(C\bar{C} + \ldots\), i.e. "gives canonically normalized kinetic term", etc. And what method did Westphal use? You're talking about Serone and Westphal, right? Let me look at that paper you linked to above.
You have that \(V_{F} = e^{K} \left( K^{a\bar{b}}D_{a}W \overline{D_{b}W} - 3|W|^{2} \right)\) where \(D_{a}W = W\partial_{a}K + \partial_{a}W\), which is your F_a terms, up to a definition factor of \(e^{\frac{K}{2}}\). It seems to me your W is \(\left(e^{-\frac{24\pi^{2}}{5}s} - 2e^{-24\pi^{2}s}\right) e^{-t}\) For some reason you then have that your potential is \(e^{K}\left(-3|W|^{2}+\left(\frac{\partial_{a}W}{W} - \partial_{a}K\right)\overline{\left(\frac{\partial_{a}W}{W} - \partial_{a}K\right)}\right)\), you seem to have dropped a factor of \(|W|^{2}\) from your F terms. The canonical kinetic terms from the Kahler potential, which is usually taken to be their \(K = \Phi\bar{\Phi} + \ldots\) or \(K = -\ln(\Phi+\bar{\Phi}) + \ldots\). The factor of \(\frac{1}{4st}\) implies that \(e^{K} = \frac{1}{4st}\), where \(S = s+i\sigma\) and \(T = t+i\tau\), so you have that \(e^{K} = \frac{1}{(S+\bar{S})(T+\bar{T})\), so \(K = -\ln(S+\bar{S}) - \ln(T+\bar{T})\), just as I said. You can also see this from your F terms, which have a term involving \(\partial_{a}K\), so \(\partial_{S}K = -\frac{1}{S+\bar{S}} = -\frac{1}{2s}\), which is exactly what you have. Did you generate this V from something or are you just taking it as a formula someone else has derived and are working on it from there?
Hmm let's see here... First of all you're right about the W^2. I fixed this already in the notebook (I copied it wrong from the scratch paper I was working on)---I missed the e^G that was sitting out front, and wrote e^K instead. This is what happens when you juggle formalisms. Second, yes the moduli enter the Kahler potential in the normal way, however, things aren't (going to be) that simple. It is clear that there must be some interplay between non-perturbative parts and perturbative parts of the superpotential---the dilaton/Kahler moduli stabilization sector doesn't decouple in the way it does in the type II vacua that (I think) you normally work in. For example, there are many states in the theory with Kahler potential terms which look like \(\frac{C\bar{C}}{(T+\bar{T})^{some\,\,power}}\). In this problem I was just screwing around with mathematica , trying to figure out how to do some numerical work.
