Unravelling the Holomorphic Twist: Central Charges

Abstract

The holomorphic twist provides a powerful framework to study minimally protected sectors in supersymmetric quantum field theories. We investigate the algebraic structure underlying the holomorphic twist of $\mathcal{N} = 1$ superconformal field theories in four dimensions. In particular, in holomorphically twisted theories the flavour and conformal symmetry algebras are enhanced to infinite-dimensional higher Kac Moody and higher Virasoro symmetry algebras respectively. We explicitly compute the binary and ternary $\lambda$-brackets and clarify their relation with the underlying infinite-dimensional symmetry algebra. Doing so we show that the central extensions of said symmetry algebras precisely encode the conformal anomalies $a$ and $c$ as well as the flavour central charges of the physical four-dimensional theory. This parallels the familiar story in two dimensions where the conformal anomaly $c$ is encoded in the central extension of the Virasoro algebra.