Hitchin systems, N=2 gauge theories and W-gravity

TL;DR

This work advances an M-theory-based derivation of the AGT duality between 2D CFTs and 4D ${\cal N}=2$ gauge theories by showing that the moduli of $N$ M5-branes wrapped on a Riemann surface are quantized by the $A_{N-1}$ Toda theory, i.e., W-gravity. It identifies ${\cal N}=2$ chiral ring operators with integrated $W_j$ currents in the Toda/CFT, supported by OPE constraints that reproduce Nekrasov's partition function through $W$-geometry. The construction relies on a Hitchin system for the M5-brane setup, with the Miura transform linking holomorphic differentials $W_j$ to the Cartan-valued field $\Phi$, and it recovers Liouville theory as the $N=2$ case. The results suggest a universal framework for higher-rank (ADE) theories and point toward further links with surface operators, central charge computations from M5 anomalies, and relations to integrable hierarchies and Chern-Simons approaches to W-gravity.

Abstract

We propose some arguments supporting an M-theory derivation of the duality recently discovered by Alday, Gaiotto and Tachikawa between two-dimensional conformal field theories and N=2 superconformal gauge theories in four dimensions. We find that A_{N-1} Toda field theory is the simplest two-dimensional conformal field theory quantizing the moduli of N M5-branes wrapped on a Riemann surface. This leads us to identify chiral operators of the N=2 gauge theories with W-algebra currents. As a check of this correspondence we study some relevant OPE's obtaining that Nekrasov's partition function satisfies W-geometry constraints.