Liouville/Toda central charges from M5-branes

TL;DR

Problem: derive Liouville/Toda central charges from the 6d anomaly data of M5-branes via a compactification on a Riemann surface with equivariant deformation. Approach: reduce the 6d anomaly eight-form $I_8[G]$ on a four-manifold $X_4$ with U(1)^2 action and a topological twist, yielding a 2d anomaly $I_4$ whose coefficients give $c_R$ and $c_L$; for $X_4=\mathbb{R}^4$ with equivariant parameters $\epsilon_{1,2}$ and $b^2=\epsilon_1/\epsilon_2$, obtain $c_L = c_{\text{Toda}}[G] = r_G + (b+1/b)^2\,d_G h_G$ and $c_R$ vanishing after the twist. Findings: the construction reproduces the ADE Toda central charges and aligns with Nekrasov's partition function as the chiral half of Liouville/Toda correlators; the method extends to ADE ($A,D,E$) Sicilian theories. Significance: provides a geometric, 6d anomaly-based derivation of 2d CFT central charges and suggests a unified framework connecting M5-brane anomalies, equivariant localization, and AGT-type correspondences.

Abstract

We show that the central charge of the Liouville and ADE Toda theories can be reproduced by equivariantly integrating the anomaly eight-form of the corresponding six-dimensional N=(0,2) theories, which describe the low-energy dynamics of M5-branes.