Twisted Chiral Algebras of Class and Mixed Feigin-Frenkel Gluing

The correspondence between four-dimensional $$ superconformal field theories and vertex operator algebras, when applied to theories of class $$ , leads to a rich family of VOAs that have been given the monicker chiral algebras of class $$ . A remarkably uniform construction of these vertex operator algebras has been put forward by Tomoyuki Arakawa in Arakawa (Chiral algebras of class $$ and Moore-Tachikawa symplectic varieties, 2018. arXiv:1811.01577 [math.RT]). The construction of Arakawa (2018) takes as input a choice of simple Lie algebra $$ , and applies equally well regardless of whether $$ is simply laced or not. In the non-simply laced case, however, the resulting VOAs do not correspond in any clear way to known four-dimensional theories. On the other hand, the standard realisation of class $$ theories involving non-simply laced symmetry algebras requires the inclusion of outer automorphism twist lines, and this requires a further development of the approach of Arakawa (2018). In this paper, we give an account of those further developments and propose definitions of most chiral algebras of class $$ with outer automorphism twist lines. We show that our definition passes some consistency checks and point out some important open problems.