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Miura transform for W-algebras of exceptional type

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Miura transform for W-algebras of classical types can be found in e.g. Sec. 6.3.3 of Bouwknegt-Schoutens. Is there a similar explicit Miura transform for W-algebras of exceptional types, say, E6? It's 20 years since the review by B-S, so I'd hope somebody worked this out ...


This post has been migrated from (A51.SE)

asked Oct 25, 2011 in Theoretical Physics by Yuji (1,390 points) [ revision history ]
retagged Apr 19, 2014 by dimension10

1 Answer

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Yes, for the "quasi-classical" case(i.e. for the case when the $W$-algebra is commutative, which occurs when the level is either infinite or critical) it was defined by Drinfeld and Sokolov long time ago; you can look at Section 4 of http://arxiv.org/PS_cache/math/pdf/0305/0305216v1.pdf for a good review.

For the "quantum" case (i.e. for arbitrary level) it was studied by Feigin and Frenkel, but I am not sure what the right reference is; you can look for example at Section 4 of http://arxiv.org/PS_cache/hep-th/pdf/9408/9408109v1.pdf, but there should be more modern references. In fact, the main tool in the work of Feigin and Frenkel is the screening operators, which describe the $W$-algebra explicitly as a subalgebra of (the vertex operator algebra associated to) the Heisenberg algebra (where the embedding to the Heisenberg algebra is the Miura transformation).

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answered Oct 26, 2011 by Alexander Braverman (570 points) [ no revision ]
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Thank you, but my main problem is to explicitly write down the subalgebra commuting with the screening operators. For A and D, it's done by Fateev-Zamolodchikov and Fateev-Lukyanov. Their forms are quite useful because it can be readily implemented in a computer algebra system. I just want to perform a few stupid calculation inside W-algebra of type E6, but I first need to realize it inside computer.

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Since I don't believe in explicit formulas, I won't be able to say anything intelligent here:) One remark, though: you can describe the image of the W-algebra without the screening operators. It is just equal to the intersection over all simple roots of things like Virasoro$\otimes$Heisenberg of smaller rank (I hope it is clear what I mean)

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Yes you're right. Physicists cover their lack of deep thinking by lots of explicit calculation:p I've been using that approach to find generators of W(E6), but that's still quite messy. That's why I asked the question here.

This post has been migrated from (A51.SE)
Do you want just generators, or generators and relations?

This post has been migrated from (A51.SE)
I think he wants the fields for each exponent of $E6$ together with their OPE. I don't think you'll find those Yuji, at least at the principal nilpotent. In the case of the minimal nilpotent, Kac and Wakimoto have explicit formulas in [this paper](http://arxiv.org/abs/math-ph/0304011)

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Thanks, I managed to get the generators. The degree-9 one was not so bad; but the degree-12 one, when dumped to a file, has ~ 100MB as an expression. Oh Buddha.

This post has been migrated from (A51.SE)
By improving the program now the expression is about ~0.9MB :)

This post has been migrated from (A51.SE)

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