EVERY CFT$_3$ HAS AN $ \mathcal{L}_Λw_{1+\infty}$ SYMMETRY

Abstract: Recently a one-parameter family of deformed $ \mathcal{L} w_{1+\infty}$ soft symmetry algebras, denoted $ \mathcal{L}Λw{1+\infty}$, acting on tree-level gravitational theories in AdS$4$ has been discovered. Here we show that all CFT$_3$s, including those dual to quantum gravity on AdS$_4$, admit an $\mathcal{L}Λw_{1+\infty}$ action generated by the ANEC operator, its conformal descendants and their commutators. This extends the previous tree-level results on these soft symmetries to the strongly-coupled quantum regime.

• The paper shows that every CFT₃ exhibits an infinite LΛw₁+∞ symmetry realized through the ANEC and its conformal descendants.
• It employs light ray operators and Fourier mode expansions to construct a closed wedge algebra consistent with SO(3,2) isometries.
• These findings bridge quantum gravity in AdS₄ with holography, offering deep structural insights into the universal symmetry of 3D CFTs.

$\mathcal{L}_\Lambda w_{1+\infty}$ Symmetry in Every CFT$_3$

Introduction and Motivation

The research establishes that every three-dimensional conformal field theory (CFT$_3$)—including strongly coupled and quantum CFTs—realizes an explicit action of the deformed soft symmetry algebra $\mathcal{L}_\Lambda w_{1+\infty}$, previously discovered in the context of tree-level AdS$_4$ Einstein gravity and its celestial holographic correspondence. In the bulk AdS$_4$, soft graviton symmetries are described by a $\Lambda$-deformed version of the $w_{1+\infty}$ algebra, which incorporates a non-trivial Jacobi-satisfying extension compatible with SO(3,2) isometries but not with conventional conformal invariance due to the presence of a nonzero cosmological constant.

The work is motivated by a succession of developments in flat holography, asymptotic symmetries, and celestial CFT, where soft gauge or graviton modes generate infinite-dimensional symmetry algebras at null infinity. Recently, these ideas have been mapped conformally to AdS$_4$, but a major open problem remained: whether the corresponding quantum (not just tree-level) boundary duals—including for arbitrary CFT$_3$—realize these infinitely generated symmetry algebras.

Light Ray Operators and Algebraic Structure

A key technical element is the analysis and organization of light ray operators, particularly the family generated by the averaged null energy condition (ANEC) operator and its conformal descendants in CFT$_3$0.

The ANEC operator in flat space is defined as:

$_3$1

with generalizations $_3$2 and $_3$3 that integrate various moments and stress tensor components. Under conformal mapping from flat $_3$4 to the $_3$5 Einstein cylinder (EC$_3$6), these operators correspond to integrals along null trajectories connecting antipodal points on the boundary, providing a geometric parameterization by an angle on the $_3$7.

Figure 1: The set of all light rays (null $_3$8) on the $_3$9 Einstein cylinder, forming a Cauchy surface relevant to boundary operator construction.

Mode expansions (Fourier transforms in the $_3$0 coordinate) lead to a basis $_3$1. Their commutators, computed for general CFT$_3$2, yield a closed subalgebra:

$_3$3

$_3$4

These are extended by systematically taking conformal descendants and iterated commutators, leading to a wedge algebra forming a closed sector isomorphic to (a subalgebra of) $_3$5.

Crucially, the algebra is labeled by three integer indices $_3$6 reflecting $_3$7 weights and with support on a specific wedge in this index lattice, where the lowest weights correspond to ANEC modes. The complete construction is visualized as a two-dimensional lattice.

Figure 2: Lattice of states in the wedge algebra: green dots depict the ANEC modes at the extremal edge, while the shaded region is generated by conformal transformations and commutators.

Conformal Action and Wedge Algebra

The action of the full $_3$8 isometry group on light ray operators is worked out in detail, constructing a module whereby the entire wedge algebra is generated by conformal descendants of the ANEC. The set of operators with quantum numbers in the wedge

$_3$9

close under the extended commutator, which in the mode basis reads:

$\mathcal{L}_\Lambda w_{1+\infty}$0

The commutator structure is unique for each operator in the wedge and is constructed recursively from the ANEC using the conformal raising/lowering operators. The full constructive algorithm is demonstrated with explicit formulas and visualized in the lattice figure.

Figure 3: Construction overview—green dots represent ANEC modes, vertical strips are built from commutators and conformal transformations, illustrating the allowed and forbidden transitions in the lattice.

This derivation confirms all CFT$\mathcal{L}_\Lambda w_{1+\infty}$1 models possess this infinite symmetry algebra at the quantum level, extending previous results that held only for tree-level perturbative gravity in AdS$\mathcal{L}_\Lambda w_{1+\infty}$2.

Implications and Future Directions

This result implies a universal hidden symmetry structure in all unitary 3D conformal field theories, regardless of their coupling, spectrum, or integrability properties. Since the ANEC and its descendants are physical, state-independent operators in CFT$\mathcal{L}_\Lambda w_{1+\infty}$3 (tied to causality and positivity of energy), the presence of this wedge algebra is a deep structural property of quantum field theory in three dimensions.

On the holographic side, the result strongly supports the conjecture that the deformed $\mathcal{L}_\Lambda w_{1+\infty}$4 symmetry in (quantum) gravity on AdS$\mathcal{L}_\Lambda w_{1+\infty}$5 is not an artifact of perturbation theory, but rather a robust, non-perturbative feature encoded at the operator level in the boundary CFT$\mathcal{L}_\Lambda w_{1+\infty}$6.

The explicit construction also provides a powerful technical tool for studying constraints on correlation functions, OPE data, and modular Hamiltonians and for investigating the interplay between boundary and bulk symmetries, including effects of boundary conditions not captured in the self-dual sector.

A future problem is the extension to higher-dimensional CFTs, generalizations to other bulk boundary dualities involving positive cosmological constant (de Sitter), and the explicit construction of bulk analogs respecting standard AdS$\mathcal{L}_\Lambda w_{1+\infty}$7 boundary conditions.

Conclusion

This work demonstrates that every 3D conformal field theory possesses a non-trivial, infinite-dimensional symmetry described by the wedge subalgebra of the deformed $\mathcal{L}_\Lambda w_{1+\infty}$8 algebra, generated by the ANEC light ray operator and its conformal descendants. The construction unifies and rigorously extends celestial holographic symmetry ideas into the deep quantum regime and establishes a correspondence between non-abelian asymptotic symmetries in gravity and operator algebras in arbitrary CFT$\mathcal{L}_\Lambda w_{1+\infty}$9 (2603.26459).

The identified symmetry should have lasting consequences for the study of holography, quantum gravity, and the structure of conformal field theories going forward.