Twisted Chiral Algebras of Class $${\mathcal {S}}$$ and Mixed Feigin–Frenkel Gluing

Abstract

AbstractThe correspondence between four-dimensional $${\mathcal {N}}=2$$ N = 2 superconformal field theories and vertex operator algebras, when applied to theories of class $${\mathcal {S}}$$ S , leads to a rich family of VOAs that have been given the monicker chiral algebras of class$${\mathcal {S}}$$ S . A remarkably uniform construction of these vertex operator algebras has been put forward by Tomoyuki Arakawa in Arakawa (Chiral algebras of class $${\mathcal {S}}$$ S 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 $${\mathfrak {g}}$$ g , and applies equally well regardless of whether $${\mathfrak {g}}$$ g 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 $${{{\mathcal {S}}}}$$ S 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 $${{{\mathcal {S}}}}$$ S with outer automorphism twist lines. We show that our definition passes some consistency checks and point out some important open problems.