Superconformal algebras and holomorphic field theories

Ingmar Saberi, Brian R Williams

Research output: Working paper


We compute the holomorphic twists of four-dimensional superconformal algebras, and argue thatthe resulting algebras act naturally by holomorphic vectorfields on holomorphically twisted superconformaltheories. For various standard examples of holomorphic twists, we demonstrate that this symmetry enhancesto the action of an infinite-dimensional local Lie algebra, the Dolbeault resolution of all holomorphic vectorfields on the punctured superspace (C2|N−1)×. Analogously, as discovered recently, global symmetries bya Lie algebragenhance to the Dolbeault resolution of holomorphic functions valued ing; at the classicallevel, both of these higher symmetry algebras act naturallyon the holomorphic twist of any Lagrangiantheory, whether superconformal or not. We show that these algebras are related to the two-dimensionalchiral algebras extracted from four-dimensional superconformal theories by Beem and collaborators; furtherdeforming the differential by their superconformal term induces the Koszul resolution of a plane inC2, andthe cohomology of the higher symmetry algebras are the usualchiral algebras of holomorphic vector fieldsandg-valued functions onC×—i.e., Virasoro and Kac–Moody. We show that the central charges of theirchiral algebras arise from recently studied central extensions of the higher symmetry algebras. However, thehigher algebras admit many further deformations not originating in the global superconformal algebra; weargue that these deformations can, for example, localize toany smooth complex curve inC2, resolving theholomorphic vector fields there, and expect that they will lead to even more exotic behavior in the case ofsingular or nonreduced curves. We consider explicit examples ofN= 2 gauge theories, and demonstratethat an anomaly to realizing the higher symmetry algebra at the quantum level vanishes precisely whenthe theory is, in fact, superconformal; for such theories, we also give an explicit description of the chiralalgebras that result after further deformation. Direct study of the representation theory of these highersymmetry algebras should lead to a decomposition of the superconformal index in terms of characters,and has the potential to generalize many familiar features of two-dimensional conformal theories to a moregeneral higher-dimensional setting.
Original languageEnglish
Number of pages62
Publication statusPublished - 9 Oct 2019

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