The literature has definitions (seemingly plural, though they might be linked) of a $q$-deformed Virasoro algebra. But is there any link of these to a $q$-deformed Heisenberg algebra? (Classically there is an expression for $L_n$ in terms of a normal ordered quadratic in the generators of the Heisenberg algebra.) I would be grateful for a reference to somewhere that discusses this in terms understandable to a non-expert in conformal field theory. However anything to shed light on the current status of the q-Virasoro and q-Heisenberg algebras, or relates them to the undeformed theory, would be of interest.

Apologies for my ignorance on this subject, as I pointed out I am certainly not an expert in CFT. I am currently trying to sort this out as an example of the action of vector fields in noncommutative geometry...


1 Answer 1


The main sources are Awata et al or Frenkel-Reshetikhin. In http://arxiv.org/pdf/q-alg/9507034v5.pdf section 4, you can see the q,t case. You can also look at http://arxiv.org/pdf/q-alg/9505025v1.pdf where the introduction gives more references to how this relates to the undeformed case. This gives the classical Virasoro as functions on opers on the punctured disc.

  • $\begingroup$ Thanks - the second reference is useful. Is there a more symmetric formalism for the deformed virasoro in this reference? It would be easier to see how it relates to other possible definitions. $\endgroup$ Oct 9, 2015 at 11:40
  • $\begingroup$ In particular, there seems to be no reference to $s_{n+m}$ in the commutator of $\{s_n,s_m\}$. $\endgroup$ Oct 9, 2015 at 12:28

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