Unravelling the Holomorphic Twist: Central Charges

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>N</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>λ</mml:mi></mml:math> -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 <i>a</i> and <i>c</i> 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 <i>c</i> is encoded in the central extension of the Virasoro algebra.