# SciPost Commentary Page

### Original publication:

Title: | Quantum q-Langlands Correspondence |

Author(s): | Mina Aganagic, Edward Frenkel, Andrei Okounkov |

As Contributors: | (none claimed) |

arxiv Link: | http://arxiv.org/abs/1701.03146v1 |

### Abstract:

We formulate a two-parameter generalization of the geometric Langlands correspondence, which we prove for all simply-laced Lie algebras. It identifies the q-conformal blocks of the quantum affine algebra and the deformed W-algebra associated to two Langlands dual Lie algebras. Our proof relies on recent results in quantum K-theory of the Nakajima quiver varieties. The physical origin of the correspondence is the 6d little string theory. The quantum Langlands correspondence emerges in the limit in which the 6d string theory becomes the 6d conformal field theory with (2,0) supersymmetry.

## Sylvain Ribault on 2017-01-17 [id 87]

How interesting is this article from the point of view of conformal field theory? The authors do take CFT limits of their $q$-deformed CFT results, let's consider what they obtain.

There are two types of known relations between conformal blocks of the affine algebra $\widehat{s\ell}_2$ and of the Virasoro algebra: expressing Virasoro blocks in terms of affine blocks, or the reverse. Neither type is bijective: either we get generic Virasoro blocks in terms of particular affine blocks, or generic affine blocks in terms of particular Virasoro blocks that involve degenerate fields. Relations of the second kind are more useful, because affine blocks are a priori more complicated than Virasoro blocks. For example, the $H_3^+$ model has an $\widehat{s\ell}_2$ affine symmetry algebra, and was solved using the expression for $\widehat{s\ell}_2$ blocks in terms of Virasoro blocks, which led to an expression for $H_3^+$ correlation functions in terms of Liouville theory correlation functions.

The article by Aganagic, Frenkel and Okounkov generalizes $\widehat{s\ell}_2$-Virasoro relations to higher affine algebras and $W$-algebras. In Remark 6.7 page 79, at the very end of Section 6, they explain that their relation is of the first kind, i.e. involves only particular affine blocks. So a priori it is not possible to get generic affine blocks from their relation.

On the other hand, the limitation that their $W$-algebra blocks are not quite arbitrary and must have integral representations, is only technical and might conceivably be overcome, leading to a generalized relation that would involve truly generic $W$-algebra blocks. Overcoming this limitation in the case $q$-deformed blocks would however be more difficult, because there is no known definition of generic $q$-deformed blocks (even in the $q$-Virasoro case), only definitions of particular blocks that have integral representations and/or obey $q$-difference equations.

## Sylvain Ribault on 2017-01-23 [id 89]

(in reply to Sylvain Ribault on 2017-01-17 [id 87])Actually, generic $q$-deformed blocks are better understood than I thought: see for example the article by Negut, https://arxiv.org/abs/1608.08613 . (Thanks to Joerg Teschner for recommending that article.) In that article there is no limitation that blocks have integral representations, only that some fields are semi-degenerate, as usual in the AGT-$W$ relation. Even this limitation can probably be overcome, see the article by Mitev and Pomoni https://arxiv.org/abs/1409.6313 . I am still not sure that global conformal symmetry of $q$-deformed CFT is well-understood, though.