W Symmetry in six dimensions
Christopher Beem, Leonardo Rastelli, Balt C. van Rees
TL;DR
The paper identifies a protected two-dimensional chiral algebra within six-dimensional (2,0) SCFTs, demonstrating it is the ${ m W}_{ rak g}$ algebra with central charge $c_{2d}=4 d_{ rak g} h^ vee_{ rak g}+r_{ rak g}$. This chiral sector arises from cohomology of a nilpotent supercharge on a fixed plane, yielding meromorphic correlators and a chiral OPE algebra that encodes protected operator data, including half-BPS generators corresponding to Casimir invariants of $ rak g$. For the abelian theory, the chiral algebra is a ${ m u}(1)$ current with $c_{2d}=1$, and a universal relation ties $c_{2d}$ to 6d Weyl anomaly coefficients. In the non-abelian case, checks using the unrefined superconformal index and large-$n$ holography support the ${ m W}_{ rak g}$ conjecture, and the framework extends to codimension-two defects where the defect chiral algebra matches a quantum Drinfeld-Sokolov reduction at critical level, connecting to AGT and the geometric Langlands program.
Abstract
Six-dimensional conformal field theories with $(2,0)$ supersymmetry are shown to possess a protected sector of operators and observables that are isomorphic to a two-dimensional chiral algebra. We argue that the chiral algebra associated to a $(2,0)$ theory labelled by the simply-laced Lie algebra $\mathfrak{g}$ is precisely the W algebra of type $\mathfrak{g}$, for a specific value of the central charge. Simple examples of observables that are made accessible by this correspondence are the three-point functions of half-BPS operators. For the $A_n$ series, we compare our results at large $n$ to those obtained using the holographic dual description and find perfect agreement. We further find protected chiral algebras that appear on the worldvolumes of codimension two defects in $(2,0)$ SCFTs. This construction has likely implications for understanding the microscopic origin of the AGT correspondence.