# Miura transform for W-algebras of exceptional type

+ 14 like - 0 dislike
1453 views

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)

retagged Apr 19, 2014

+ 10 like - 0 dislike

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).

This post has been migrated from (A51.SE)
answered Oct 26, 2011 by (580 points)
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)

This post has been migrated from (A51.SE)
Thanks everyone; I know got the generators at degree 2 and 5. Now I need those at degree 6, 8, 9 and 12 :p

This post has been migrated from (A51.SE)
Good luck, Yuji!

This post has been migrated from (A51.SE)
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)
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)
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsO$\varnothing$erflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). Please complete the anti-spam verification