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This physics (bear with me for a while) paper seems to say something about Gal \bar Q/Q:

Let's begin with my limited understanding of what they say. They start with some SUSY QFT ("pure U(N) gauge theory - the most well studied case"). There's a big moduli space of those (you're classifying the vacua of theories here), but then, given some numbers (g_i), you consider a deformation problem and because of some algebra on page 3 you have just isolated points as vacua.

Now if you also fine-tune your parameters (g_i), the connected trees (dessins d'enfants) appear in the moduli space, while, on the other hand, we always had a curve called Seiberg-Witten curve, for a given (g_i) (I think this curve relates to dessin by Belyi map but not sure).

Interestingly, they find a physical counterpart to the statement that each tree parametrizes exactly one Galois orbit (page 16). It gets hard for me to read further, and probably won't make much sense to continue my imperfect retelling, but I'd like to hear (especially from people who read about this topic), the following

Question: is there a new mathematical content described in the article? Is there a mathematical understanding of what they're doing? Or is this direction not interesting?

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I'm testing a new tag for questions that have mathematicians as their target auditorium, but contain references to high-energy physics papers, the hep-th. Feel free to comment on that idea, retag this or pick up a tag! – Ilya Nikokoshev Nov 2 '09 at 20:19
This question in fact was part of an inspiration for my posting of – Ilya Nikokoshev Nov 2 '09 at 20:21
I like the hep-th tag! – Kevin H. Lin Nov 2 '09 at 22:21
Then just dissipate it :) – Ilya Nikokoshev Nov 3 '09 at 8:24
This sounds quite interesting - did you ever get an answer? – Dr Shello Dec 15 '10 at 9:46

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