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 . 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.