Unravelling the Holomorphic Twist II: Anomalies and Extended Supersymmetry

Abstract: Twists of four-dimensional supersymmetric quantum field theories (SQFTs) isolate protected sectors with rich algebraic structures. We develop a unified framework for analyzing symmetries and anomalies in four-dimensional holomorphically twisted SQFTs, combiningcomplex-geometric and algebraic perspectives. This approach clarifies the connections between existing formulations in the literature and resolves several open questions left unanswered in the first installment of this series. We place particular emphasis on theories with extended supersymmetry, where the holomorphic twist gives rise to enhanced algebraic and geometric structures. We explain how these features emerge and govern the organization of the twisted theory. Furthermore, we demonstrate how a superconformal deformation of the twisted theory naturally leads to the associated vertex operator algebra, clarifying how the higher algebraic structures of the holomorphically twisted theory give rise to vertex algebras structures.