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Chiral higher-spin symmetry of the celestial twistor sphere

Tung Tran

TL;DR

The paper constructs and analyzes chiral higher‑spin symmetry algebras $\mathfrak{ca}$ on the celestial twistor sphere via Koszul duality from holomorphic twistorial higher‑spin theories. Classically, $\mathfrak{ca}$ is associative and identifies with color‑kinematic algebras of 4d chiral/self‑dual theories; quantum corrections can break associativity unless the spectrum is in the anomaly‑protected sets or axionic currents are added. The authors compute OPEs and a range of correlation functions, showing that tree and loop amplitudes are rational and highly constrained, with nontrivial higher‑spin form factors appearing mainly in Yang–Mills‑like setups; in many cases, loop amplitudes vanish due to quantum integrability. They also formulate defect CFTs sourcing the currents, including chiral boson/fermion models and matrix‑valued cases, and discuss extensions with Weyl fermions and axions. Overall, the work delineates when the celestial chiral algebras remain associative at one loop and what sorts of higher‑spin correlators are allowed, shedding light on possible unconstructed higher‑spin theories and their observable signatures on the celestial sphere.

Abstract

We study the chiral higher-spin symmetry algebras $\mathfrak{ca}$ of various twistorial higher-spin theories. These symmetries play the roles of asymptotic symmetries on the celestial twistor sphere, which constrain the observables of twistorial theories. To first order in quantum correction, we show that the chiral algebras associated with anomaly-free holomorphic twistorial higher-spin theories are associative themselves. On the other hand, the chiral algebras associated with anomalous holomorphic twistorial higher-spin theories only become associative upon including suitable axionic currents. When computing $4d$ form factors in terms of correlation functions between higher-spin currents on the celestial twistor sphere, we observe that there are some non-vanishing higher-spin form factors. This observation, however, is only well justified for the case of theories with Yang-Mills-like interactions. We also give some brief comments on the case of higher-derivative interactions.

Chiral higher-spin symmetry of the celestial twistor sphere

TL;DR

The paper constructs and analyzes chiral higher‑spin symmetry algebras on the celestial twistor sphere via Koszul duality from holomorphic twistorial higher‑spin theories. Classically, is associative and identifies with color‑kinematic algebras of 4d chiral/self‑dual theories; quantum corrections can break associativity unless the spectrum is in the anomaly‑protected sets or axionic currents are added. The authors compute OPEs and a range of correlation functions, showing that tree and loop amplitudes are rational and highly constrained, with nontrivial higher‑spin form factors appearing mainly in Yang–Mills‑like setups; in many cases, loop amplitudes vanish due to quantum integrability. They also formulate defect CFTs sourcing the currents, including chiral boson/fermion models and matrix‑valued cases, and discuss extensions with Weyl fermions and axions. Overall, the work delineates when the celestial chiral algebras remain associative at one loop and what sorts of higher‑spin correlators are allowed, shedding light on possible unconstructed higher‑spin theories and their observable signatures on the celestial sphere.

Abstract

We study the chiral higher-spin symmetry algebras of various twistorial higher-spin theories. These symmetries play the roles of asymptotic symmetries on the celestial twistor sphere, which constrain the observables of twistorial theories. To first order in quantum correction, we show that the chiral algebras associated with anomaly-free holomorphic twistorial higher-spin theories are associative themselves. On the other hand, the chiral algebras associated with anomalous holomorphic twistorial higher-spin theories only become associative upon including suitable axionic currents. When computing form factors in terms of correlation functions between higher-spin currents on the celestial twistor sphere, we observe that there are some non-vanishing higher-spin form factors. This observation, however, is only well justified for the case of theories with Yang-Mills-like interactions. We also give some brief comments on the case of higher-derivative interactions.

Paper Structure

This paper contains 62 sections, 6 theorems, 242 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review
  3. Twistor and higher spins
  4. Holomorphic twistorial higher-spin theories
  5. Holomorphic Chern-Simons theories
  6. Holomorphic BF theories
  7. Holomorphic Green-Schwarz anomaly cancellation mechanism
  8. Anomaly-free theories.
  9. Green-Schwarz anomaly cancellation on twistor space.
  10. Chiral higher-spin algebras of the celestial twistor sphere
  11. Kozsul duality and chiral higher-spin algebra
  12. Koszul duality.
  13. Chiral algebra.
  14. Higher-spin current OPEs
  15. Chiral higher-spin algebras at classical level
  16. ...and 47 more sections

Key Result

Corollary 3.1

The chiral higher-spin algebra $\mathfrak{ca}$ can be classically identified with a color-kinematic algebra where the generators $\mathcal{J}$ obey the Jacobi's identities for

Figures (4)

  • Figure 1: The coupling between a bulk field $\mathtt{A}$ near a defect $\mathbb{P}^1_{\mathtt{p}}$ and a current $\mathtt{J}[\Delta;\mathtt{H}]$.
  • Figure 2: The Koszul couplings between a bulk axion $\varrho$ and two currents $\mathtt{U},\mathtt{V}$.
  • Figure 3: Starting with the non-commutative algebra $(\mathtt{Coh},\star)$, we have constructed $\mathfrak{ca}$ via Koszul duality as in Costello:2022wso. After the OPE between holomorphic currents are determined, we can now bootstrap the correlation functions $\langle \mathtt{J}\ldots\mathtt{J}\rangle$ up to some number of loops. These correlators can be interpreted as form factors in some $4d$ spacetime theories.
  • Figure 4: Some of the many two-loop diagrams for chiral higher-spin theories.

Theorems & Definitions (11)

  • Corollary 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 1 more