Para-Liouville/Toda central charges from M5-branes

TL;DR

The paper addresses how para-W symmetry and para-Toda structures arise in 2d theories engineered by N M5-branes on $\mathbb{R}^4/\mathbb{Z}_m$ with Nekrasov deformations. It derives the 2d central charge from the 6d anomaly polynomial, decomposing the 2d theory into a free boson, $\widehat{\mathrm{SU}}(m)_N$, and an $m$-th para-Toda sector, with $c = N m + \frac{N^3 - N}{m} \left(b + \frac{1}{b}\right)^2$ and a refined splitting $c = 1 + c(\widehat{\mathrm{SU}}(m)_N) + \left[ \frac{m(N^2 - 1)}{m+N} + \frac{N^3 - N}{m} \left(b + \frac{1}{b}\right)^2 \right]$. This reproduces known results for $m=1$ (affine Toda/W_N) and $m=N=2$ (super-Virasoro) and generalizes to a framework where RCFTs and para-Toda algebras encode the 2d dynamics, with speculative extensions to other gauge groups $G$ and ALE orbifolds. The work provides a coherent string-theoretic route to para-W/Toda symmetry realization in 2d, linking 6d anomaly physics, instanton moduli spaces, and RCFT/CFT structures, and suggesting broader implications for level-rank dualities and orbifold generalizations.

Abstract

We propose that N M5-branes, put on R^4/Z_m with deformation parameters epsilon_{1,2}, realize two-dimensional theory with SU(m)_N symmetry and m-th para-W_N symmetry. This includes the standard W_N symmetry for m=1 and super-Viraroro symmetry for m=N=2. We provide a small check of this proposal by calculating the central charge of the 2d theory from the anomaly polynomial of the 6d theory.