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Twists of superconformal algebras

Chris Elliott, Owen Gwilliam, Matteo Lotito

TL;DR

This work develops a systematic framework for conformal twists of superconformal field theories across dimensions $3$–$6$ by studying square-zero odd elements $\mathcal{Q}$ in complexified superconformal algebras and their adjoint orbits. By analyzing the associated centralizers, exact subalgebras, and nilpotence varieties, the authors classify twists and describe their real and complex structures, linking twists to protected operator sectors and to mathematical structures such as vertex algebras and $E_n$-algebras via prefactorization algebras. They connect these algebraic twists to concrete constructions in physics literature (e.g., Donaldson–Witten and Schur twists) and provide a rigorous pathway from twisted observables to vertex algebras and Ek-algebras, including reality conditions and affine-patch techniques. Across dimensions, the paper also develops a unifying language to discuss twisted observables, subalgebras of closed/exact elements, and the geometric and group-theoretic data controlling the twisted theories. The results offer a cohesive, representation-theoretic and geometric blueprint for understanding protected sectors and higher-algebraic structures arising from superconformal twists, with broad implications for both mathematics and high-energy theory.

Abstract

We take first steps toward a theory of ``conformal twists'' for superconformal field theories in dimension 3 to 6, extending the well-known analysis of twists for supersymmetric theories. A conformal twist is a square-zero odd element in the superconformal Lie algebra, and we classify all twists and describe their orbits under the adjoint action of the superconformal group. We work mostly with the complexified superconformal algebras, unless explicitly stated otherwise; real forms of the superconformal algebra may have important physical implications, but we only discuss these subtleties in a few special cases. Conformal twists can give rise to interesting subalgebras and protected sectors of operators in a superconformal field theory, with the Donaldson--Witten topological field theory and the vertex operator algebras of 4-dimensional N=2 SCFTs being prominent examples. To obtain mathematical precision, we explain how to extract vertex algebras and E_n algebras from a twisted superconformal field theory using factorization algebras.

Twists of superconformal algebras

TL;DR

This work develops a systematic framework for conformal twists of superconformal field theories across dimensions – by studying square-zero odd elements in complexified superconformal algebras and their adjoint orbits. By analyzing the associated centralizers, exact subalgebras, and nilpotence varieties, the authors classify twists and describe their real and complex structures, linking twists to protected operator sectors and to mathematical structures such as vertex algebras and -algebras via prefactorization algebras. They connect these algebraic twists to concrete constructions in physics literature (e.g., Donaldson–Witten and Schur twists) and provide a rigorous pathway from twisted observables to vertex algebras and Ek-algebras, including reality conditions and affine-patch techniques. Across dimensions, the paper also develops a unifying language to discuss twisted observables, subalgebras of closed/exact elements, and the geometric and group-theoretic data controlling the twisted theories. The results offer a cohesive, representation-theoretic and geometric blueprint for understanding protected sectors and higher-algebraic structures arising from superconformal twists, with broad implications for both mathematics and high-energy theory.

Abstract

We take first steps toward a theory of ``conformal twists'' for superconformal field theories in dimension 3 to 6, extending the well-known analysis of twists for supersymmetric theories. A conformal twist is a square-zero odd element in the superconformal Lie algebra, and we classify all twists and describe their orbits under the adjoint action of the superconformal group. We work mostly with the complexified superconformal algebras, unless explicitly stated otherwise; real forms of the superconformal algebra may have important physical implications, but we only discuss these subtleties in a few special cases. Conformal twists can give rise to interesting subalgebras and protected sectors of operators in a superconformal field theory, with the Donaldson--Witten topological field theory and the vertex operator algebras of 4-dimensional N=2 SCFTs being prominent examples. To obtain mathematical precision, we explain how to extract vertex algebras and E_n algebras from a twisted superconformal field theory using factorization algebras.
Paper Structure (52 sections, 32 theorems, 161 equations, 1 table)

This paper contains 52 sections, 32 theorems, 161 equations, 1 table.

Table of Contents

  1. Introduction
  2. What Twisting Does to Superconformal Theories
  3. Superconformal Algebras and Twists
  4. The Conformal Case
  5. A more physical formulation
  6. The Superconformal Case
  7. The Question of Classifying Twists
  8. How Twists Appear in the Supersymmetric Case
  9. Associated Varieties of Super Lie Algebras
  10. How Twists Appear in the Conformal Case
  11. The Observables of a Twisted Superconformal Theory
  12. Subalgebras of Q-Closed and Q-Exact Elements
  13. Observables and BRST Operators
  14. The Maximal Case
  15. Working on an Affine Patch
  16. ...and 37 more sections

Key Result

Lemma 2.4

As a vector space, there is a direct sum decomposition where the summands correspond, in order, to the Lie algebras of the four subgroups listed above.

Theorems & Definitions (114)

  • Example 1.2: $\mathbb{E}_k$-algebras
  • Remark 1.3
  • Remark 1.4
  • Example 1.5: Vertex Algebras
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Remark 2.6
  • ...and 104 more