AGT conjecture and WZW model

Question

In 2009 Alday, Gaiotto and Tachikawa conjectured an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of $N=2$ super-conformal field theory.

I would like to know if anything similar shows up for WZW conformal blocks.

---

A much more elementary (but related) question is the following:

Is there any elliptic recursion for 4-point conformal blocks in WZW theory, like in Liouville theory (see e.g. these papers).

From a mathematician view-point, this relation between 4-point spheric and 1-point toric conformal blocks is not so surprising. Indeed, the moduli space $\mathcal M_{1,1}$ of elliptic curves and the unordered moduli space $\mathcal M_{0,[4]}$ of 4-punctured rational curves are the same.

I would be anyway very interested in learning about recent advances in the WZW side of the story (if it exists).

Answer 1

As far as the first question is concerned, I can make comments on it.

In the piorineering work by Braverman, it was shown that the intersection cohomology groups $\oplus IH_{T\times (\mathbb{C}^*)^2}({\cal U}_{G,B})$ of the Uhlenbeck compactification ${\cal U}_{G,B}$ of the moduli space $Bun_{G,B}$ of parabolic $G$-bundle have an action of affine Lie algebra $\hat{\mathfrak{g}}^{\vee}$.

http://arxiv.org/abs/math/0401409

This result was translated in physics language by Alday and Tachikawa that the instanton partition function of $SU(2)$ gauge theory with a full surface operator is equal to the conformal blocks of the affine $\hat{\mathfrak{sl}}_2$ algebra.

http://arxiv.org/abs/1005.4469

The extension to $\hat{\mathfrak{sl}}_N$ has been discussed by Wyllard et al.

http://arxiv.org/abs/1008.1412

Therefore, it is expected that the partition function of $\cal{N}=2$ gauge theory on $S^4$ with a full surface operator is equivalent to the correlation function of the $SL(N)$ WZW theory.