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).


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 $\hat{\mathfrak{g}}^{\vee}$.


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.


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


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.

  • $\begingroup$ Many thanks for this very helpful answer. I'll wait for other (if there are) answers before validating. $\endgroup$ – DamienC Nov 4 '11 at 22:40
  • $\begingroup$ Just a small comment/question: I don't know how to get actual correlation functions of WZW (in the spirit of the original AGT conjecture) in this way except the correlation function on the torus with one puncture. So I would be glad to see the precise statement relating the partition function on $S^4$ with correlation functions of WZW. Is such a statement known? $\endgroup$ – Alexander Braverman Nov 6 '11 at 16:56
  • $\begingroup$ The relation of the partition function on $S^4$ to correlation functions of WZW is just a speculation. The precise statement has not been made yet even in physics literature. $\endgroup$ – Satoshi Nawata Nov 6 '11 at 17:58

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