Celestial Lw1+∞ Symmetries in 4D Quantum Gravity

• Celestial Lw1+∞ symmetries are infinite-dimensional chiral algebras that organize soft graviton, gluon, and supermultiplet currents in celestial holography.
• They are realized through explicit CFT and Hamiltonian formalisms on the celestial (super)sphere, with clear ties to twistor and phase space geometries.
• Supersymmetric extensions and Λ-deformations highlight their applicability to both flat and curved backgrounds in 4D quantum gravity and supergravity.

Celestial $Lw_{1+\infty}$ symmetries constitute the infinite-dimensional chiral algebra emerging naturally in the celestial holographic descriptions of four-dimensional quantum gravity and supergravity. These symmetries extend the standard asymptotic symmetry groups, encompassing and organizing all conformally soft graviton, gluon, and supermultiplet charges into a unified current algebra on the celestial (super)sphere. The structure is sharply defined via explicit realizations in both CFT and Hamiltonian formalisms, and possesses geometric interpretations in twistor and phase space frameworks. The algebra admits a direct generalization to supersymmetric extensions and to curved (A)dS backgrounds.

1. Structure of the Bosonic $Lw_{1+\infty}$ Algebra

The classical celestial $Lw_{1+\infty}$ algebra arises from a tower of chiral currents $w^q_m(z)$ of holomorphic spin $h = q-1$ with

$$q \in \{1,\tfrac32,2,\tfrac52,\dots\},\quad m\in\{1-q,\dots,q-1\}.$$

The mode expansion is

$$w^q_m(z) = \sum_{n\in\mathbb{Z}+h} z^{-n-h} w_{m,n}^q.$$

The defining commutator of the wedge subalgebra is

$$\bigl[w_{m_1,n_1}^{q_1},\,w_{m_2,n_2}^{q_2}\bigr] = \bigl((q_2 - 1)m_1 - (q_1 - 1)m_2\bigr) w^{q_1+q_2-2}_{m_1+m_2,\,n_1+n_2}. \tag{1}$$

Central extensions, such as

$$c\,\delta_{q_1+q_2-2,1}\,\delta_{m_1+m_2,0}\,\delta_{n_1+n_2,0},$$

can in principle be added, but are absent at tree level in gravitational celestial holography.

This algebra organizes all the holomorphic conformally soft graviton currents and, via light (Fourier) transforms and mode expansion, generates the tower of soft graviton Ward identities on celestial CFT correlators. Virasoro and supertranslation currents are identified as low-spin subalgebras, while higher-spin elements encode subleading and sub-subleading soft theorems (Tropper, 17 Dec 2024).

2. Supersymmetric Extensions and the Celestial Supersphere

For bulk theories with $\mathcal N$ supersymmetries, the algebra enhances to the loop algebra of the wedge of the super-$w_{1+\infty}$ algebra, denoted $L\bigl(w_{1+\infty}^\wedge\bigr)$. The additional generators are fermionic towers: $$\left(w^{I_1\cdots I_A}\right)^q_m(z),\quad A=0,1,\dots,\mathcal N,\quad I_a=1,\dots,\mathcal N,$$ combined into superfields

$\mathds{W}^q_m(z|\eta) = \sum_{A=0}^\mathcal N \frac1{A!}\eta_{I_A}\cdots\eta_{I_1} (w^{I_1\cdots I_A})^q_m(z).$

The commutation relations extend (1) by fermionic grading: $$\bigl[(w^{I_1\cdots I_A})^{q_1}_{m_1,n_1}, \,(w^{J_1\cdots J_B})^{q_2}_{m_2,n_2}\bigr] = \bigl((q_2 - 1)m_1 - (q_1 - 1)m_2\bigr)\, (w^{I_1\cdots I_A J_1\cdots J_B})^{q_1+q_2-2}_{m_1+m_2,\,n_1+n_2}. \tag{2}$$ Or in superspace,

$\bigl[\,\mathds{W}^{q_1}_{m_1,n_1}(\eta_1),\;\mathds{W}^{q_2}_{m_2,n_2}(\eta_2)\,\bigr] = \bigl((q_2-1)m_1-(q_1-1)m_2\bigr) \mathds{W}^{q_1+q_2-2}_{m_1+m_2,n_1+n_2}(\eta_1+\eta_2). \tag{3}$

These superalgebras arise from manifestly four-dimensional super-Poincaré-invariant constructions, as opposed to intrinsic 2d superconformal algebras, and are naturally supported on the celestial supersphere—a supermanifold extension of the celestial sphere (Tropper, 17 Dec 2024).

The $\frak{bms}_4$ algebra of asymptotic symmetries extends to the supersymmetric $\frak{sbms}_{4|\mathcal N}$, manifesting both gravitational and supermultiplet soft theorems on the celestial boundary.

3. Hamiltonian Vector Field Realization and Twistor Geometry

There exists a canonical realization of the (super-)$Lw_{1+\infty}^\wedge$ algebra as the Lie algebra of Hamiltonian vector fields on the (super)space $\mathbb{C}^{2|\mathcal N}$. For the purely bosonic sector ($\mathcal{N}=0$), take $\mu^\alpha$ ($\alpha=0,1$), with Poisson bracket

$$\Pi^{\alpha\beta} = \varepsilon^{\alpha\beta} = \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix}.$$

Define

$w^q_m = \tfrac12 (\mu^0)^{q-1+m} (\mu^1)^{q-1-m},$

with Hamiltonian vector fields $X_{w^q_m}(f) = \{f, w^q_m\}$, closing under

$$[X_{w^{q_1}_{m_1}}, X_{w^{q_2}_{m_2}}] = X_{\{w^{q_1}_{m_1}, w^{q_2}_{m_2}\}} = X_{[(q_2-1)m_1-(q_1-1)m_2]\,w^{q_1+q_2-2}_{m_1+m_2}}.$$

On the supermanifold $\mathbb{C}^{2|\mathcal N}$, the Poisson bivector generalizes as

$$\Pi = \tfrac\partial{\partial\mu^\alpha} \wedge \tfrac\partial{\partial\mu_\alpha} + 0.$$

If a cosmological constant $\Lambda\neq0$ is introduced, a fermionic term proportional to $\sqrt\Lambda$ appears: $$\Pi_{\rm fermionic} \sim \sqrt\Lambda\,\varepsilon^{IJ}\tfrac\partial{\partial\theta^I} \wedge \tfrac\partial{\partial\theta^J}.$$ In this setting, basis elements

$$(w^{I_1\cdots I_A})^q_m = \tfrac1{2^{1-A}} (\mu^0)^{q-1+m} (\mu^1)^{q-1-m} \theta^{I_1}\cdots\theta^{I_A},$$

generate Hamiltonian vector fields whose Lie brackets reproduce the super-$w_{1+\infty}^\wedge$ algebraic relations. This geometric realization is consistent with the twistor space expectation and establishes a mathematically rigorous underpinning for the appearance of these infinite-dimensional symmetries in gravitational and supergravity models with celestial holographic duals (Tropper, 17 Dec 2024).

4. Algebraic Deformations: Cosmological Constant and Curved Backgrounds

For bulk backgrounds with non-zero cosmological constant $\Lambda$, the $Lw_{1+\infty}$ algebra admits a $\Lambda$-deformation. The deformed commutators are

$$[w_{m_1,n_1}^{q_1},\,w_{m_2,n_2}^{q_2}]_\Lambda = A\,w^{q_1+q_2-2}_{m_1+m_2,\,n_1+n_2} \;-\; \Lambda B\,w^{q_1+q_2-1}_{m_1+m_2,n_1+n_2},$$

where

$$A=(q_2-1)m_1-(q_1-1)m_2,\quad B=(q_2-2)n_1-(q_1-2)n_2. \tag{4}$$

In twistor coordinates $(\mu^\alpha, \lambda_\alpha)$,

$$\Pi^{(\Lambda)} = \tfrac\partial{\partial\mu^\alpha}\wedge \tfrac\partial{\partial\mu_\alpha} + \Lambda\,\tfrac\partial{\partial\lambda_\alpha}\wedge\tfrac\partial{\partial\lambda^\alpha},$$

with basis elements

$$w^q_{m,n} = (\mu^0)^{q-1+m}(\mu^1)^{q-1-m} \lambda_0^{q-n-2}\lambda_1^{q+n-2},$$

one finds

$$\{w^{q_1}_{m_1,n_1}, w^{q_2}_{m_2,n_2}\}_\Lambda = A\,w^{q_1+q_2-2}_{m_1+m_2,\,n_1+n_2} - \Lambda B\,w^{q_1+q_2-1}_{m_1+m_2,\,n_1+n_2}.$$

The supersymmetric extension includes a $\sqrt\Lambda$-dependent fermionic contribution: $[\mathds{W}^{q_1}_{m_1,n_1}(\eta_1), \mathds{W}^{q_2}_{m_2,n_2}(\eta_2)]_\Lambda = A\,\mathds{W}^{q_1+q_2-2}_{m_1+m_2,n_1+n_2}(\eta_1+\eta_2) - \Lambda B\,\mathds{W}^{q_1+q_2-1}_{m_1+m_2,n_1+n_2}(\eta_1+\eta_2) + 2\sqrt\Lambda\,\delta^{IJ}\,\eta_{2,I}\,\eta_{1,J}\,\mathds{W}^{q_1+q_2-1}_{m_1+m_2,n_1+n_2}(\eta_1+\eta_2).$ These deformations encode the $\Lambda$-corrected OPEs found in (A)dS celestial holography (Tropper, 17 Dec 2024).

5. Relation to Soft Theorems, OPEs, and Bulk Holography

The $Lw_{1+\infty}$ and its supersymmetric extension govern the correlation functions, OPEs, and conformally soft algebraic structures of the celestial CFT duals to bulk gravitational and supergravity theories. The tree-level holomorphic OPEs of celestial (super)primaries are

$$\mathbb{O}_{a,\Delta}(z|\eta)\,\mathbb{O}_{b,\Delta'}(0|\eta') \sim -\frac{1}{z} \sum_c g_{ab}^{~c}\,C_p(h,h')\,\mathbb{O}_{c,\Delta+\Delta'+p}(0|\eta+\eta'),$$

with the mode algebra closing under the commutators (1)–(3).

Supersymmetry generators $Q_{I,\alpha}$ act on the modes by

$[Q_{I,\alpha},\mathds{W}^q_{m,n}(\eta)] = z^\alpha \partial_{\eta_I} \mathds{W}^{q+\frac12}_{m-\frac12,n}(\eta),\quad [Q^I_\alpha,\mathds{W}^q_{m,n}(\eta)] = -z^\alpha \eta_I \mathds{W}^q_{m,n}(\eta),$

and satisfy the Jacobi identities, ensuring consistency of the superalgebraic structure (Tropper, 17 Dec 2024).

These relations ensure that celestial OPEs, factorization of soft theorems, and chiral symmetry algebra cohere into a closed algebraic system, both for the bosonic sector and in the manifestly $\mathcal N$-extended supersymmetric context.

6. Universality, Theoretical Significance, and Outlook

The emergence of celestial $Lw_{1+\infty}$ symmetries, both in their bosonic and supersymmetric forms, is universal and independent of the underlying microscopic details of the gravity or supergravity theory, provided the theory admits a consistent celestial CFT dual with four-dimensional (super-)Poincaré symmetry. The algebraic structure of the celestial soft currents, their OPEs, and the associated Hamiltonian vector fields on (super)twistor space provide the unifying framework for understanding the infrared and asymptotic symmetry structure of quantum gravity, both in flat and curved spacetime, and both at tree level and in certain deformations.

The celestial supersphere construction and the loop algebraic techniques outlined are expected to play a key role in future investigations of higher-point correlators, loop corrections, and the emergence of quantum-corrected central extensions in celestial holography. These geometric and algebraic structures are foundational for ongoing explorations of the bulk/boundary correspondence, quantum gravity memory effects, and the interplay between infrared dynamics and infinite-dimensional symmetries (Tropper, 17 Dec 2024).