We generalize the analysis of the asymptotic higher spin symmetries developed in the first three parts of this series by considering the minimal coupling of Einstein Gravity and Yang-Mills theory. We show that there exist symmetry parameters that satisfy a collection of dual equations of motion, which allow the construction of an infinite collection of charges that are conserved in the absence of radiation. These Noether charges act on the Einstein Yang-Mills phase space canonically and non-linearly. Their action defines a symmetry algebroid which reduces to a symmetry algebra at non-radiative cuts and generalizes the celestial $\textit{sw}$ algebra. The corresponding symmetry bracket is shown to satisfy the Jacobi identity and an interesting cross-product structure, which is analyzed in details.

Canada

Nicolas Cresto

Laurent Freidel

Asymptotic higher spin symmetries. Part IV. Einstein-Yang-Mills theory

In this work, we demonstrate that quantizing gravity on a null hypersurface leads to the emergence of a CFT associated with each null ray. This result stems from the ultralocal nature of null physics and is derived through a canonical analysis of the Raychaudhuri equation, interpreted as a constraint generating null time reparametrizations. The CFT exhibits a non-zero central charge, providing a mechanism for the quantum emergence of time in gravitational systems and an associated choice of vacuum state. Our analysis reveals that the central charge quantifies the degrees of freedom along each null ray. Throughout our investigation, the area element of a cut plays a crucial role, necessitating its treatment as a quantum operator due to its dynamic nature in phase space or because of quantum backreaction. Furthermore, we show that the total central charge diverges in a perturbative analysis due to the infinite number of null generators. This divergence is resolved if there is a discrete spectrum for the area form operator. We introduce the concept of ‘embadons’ to denote these localized geometric units of area, the fundamental building blocks of geometry at a mesoscopic quantum gravity scale.

Luca Ciambelli

United States

Robert Leigh

Quantum null geometry and gravity

The membrane paradigm displays underlying connections between a timelike stretched horizon and a null boundary (such as a black hole horizon) and bridges the gravitational dynamics of the horizon with fluid dynamics. In this work, we revisit the membrane viewpoint of a finite-distance null boundary and present a unified geometrical treatment of the stretched horizon and the null boundary based on the rigging technique of hypersurfaces. This allows us to provide a unified geometrical description of null and timelike hypersurfaces, which resolves the singularity of the null limit appearing in the conventional stretched horizon description. We also extend the Carrollian fluid picture and the geometrical Carrollian description of the null horizon, which have been recently argued to be the correct fluid picture of the null boundary, to the stretched horizon. To this end, we draw a dictionary between gravitational degrees of freedom on the stretched horizon and the Carrollian fluid quantities and show that Einstein’s equations projected onto the horizon are the Carrollian hydrodynamic conservation laws. Lastly, we report that the gravitational pre-symplectic potential of the stretched horizon can be expressed in terms of conjugate variables of Carrollian fluids and also derive the Carrollian conservation laws and the corresponding Noether charges from symmetries.

Japan

Puttarak Jai-akson

Carrollian hydrodynamics and symplectic structure on stretched horizons

Celestial holography provides a reformulation of scattering amplitudes in four dimensional asymptotically flat spacetimes in terms of conformal correlators of operators on the two dimensional celestial sphere in a basis of boost eigenstates. A basis of massless particle states has been previously identified in terms of conformal primary wavefunctions labeled by a boost weight ∆ = 1+iλ with λ ∈ ℝ. Here we show that a discrete orthogonal and complete basis exists for ∆ ∈ ℤ. This new basis consists of a tower of discrete memory and Goldstone observables, which are conjugate to each other and allow to reconstruct gravitational signals belonging to the Schwartz space. We show how generalized dressed states involving the whole tower of Goldstone operators can be constructed and evaluate the higher spin Goldstone 2-point functions. Finally, we recast the tower of higher spin charges providing a representation of the w 1+∞  loop algebra (in the same helicity sector) in terms of the new discrete basis.

Italy

Daniele Pranzetti

Netherlands

Ana-Maria Raclariu

A discrete basis for celestial holography

We initiate a study of gravity focusing on generic null hypersurfaces, non-perturbatively in the Newton coupling. We present an off-shell account of the extended phase space of the theory, which includes the expected spin-2 data as well as spin-0, spin-1 and arbitrary matter degrees of freedom. We construct the charges and the corresponding kinematic Poisson brackets, employing a Beltrami parameterization of the spin-2 modes. We explicitly show that the constraint algebra closes, the details of which depend on the non-perturbative mixing between spin-0 and spin-2 modes. Finally we show that the spin zero sector encodes a notion of a clock, called dressing time, which is dynamical and conjugate to the constraint.

Null Raychaudhuri: canonical structure and the dressing time

Similar to gravity, an infinite tower of symmetries generated by higher-spin charges has been identified in Yang-Mills theory by studying collinear limits or celestial operator products of gluons. This work aims to recover this loop symmetry in terms of charge aspects constructed on the gluonic Fock space. We propose an explicit construction for these higher spin charge aspects as operators which are polynomial in the gluonic annihilation and creation operators. The core of the paper consists of a proof that the charges we propose form a closed loop algebra to quadratic order. This closure involves using the commutator of the cubic order expansion of the charges with the linear (soft) charge. Quite remarkably, this shows that this infinite-dimensional symmetry constrains the non-linear structure of Yang-Mills theory. We provide a similar all spin proof in gravity for the so-called global quadratic (hard) charges which form the loop wedge subalgebra of w 1+∞.

On infinite symmetry algebras in Yang-Mills theory

The failure to calculate the vacuum energy is a central problem in theoretical physics. Presumably the problem arises from the insistent use of effective field theory reasoning in a context that is well beyond its intended scope. If one follows this path, then one is led inevitably to statistical or anthropic reasoning for observations. It appears that a more palatable resolution of the vacuum energy problem requires some form of UV/IR feedback. In this paper we take the point of view that such feedback can be thought of as arising by defining a notion of quantum space-time. We reformulate the regularized computation of vacuum energy in such a way that it can be interpreted in terms of a sum over elementary phase space volumes, that we identify with a ground state degeneracy. This observation yields a precise notion of UV/IR feedback, while leaving a scale unfixed. Here we argue that holography can be thought to provide a key piece of information: we show that equating this microscopic ground state degeneracy with macroscopic gravitational entropy yields a prediction for the vacuum energy that can easily be consistent with observations. Essentially, the smallness of the vacuum energy is tied to the large size of the Universe. We discuss how within this scenario notions of effective field theory can go so wrong.

Poland

Jerzy Kowalski-Glikman

Robert G. Leigh

Djordje Minic

Vacuum energy density and gravitational entropy

Gravitational subsystems with boundaries carry the action of an infinite-dimensional symmetry algebra, with potentially profound implications for the quantum theory of gravity. We initiate an investigation into the quantization of this corner symmetry algebra for the phase space of gravity localized to a region bounded by a 2-dimensional sphere. Starting with the observation that the algebra   <math> <mtext>sdiff</mtext> </math>  $$ \mathfrak{sdiff} $$ (S 2) of area-preserving diffeomorphisms of the 2-sphere admits a deformation to the finite-dimensional algebra   <math> <mi>su</mi> </math>  $$ \mathfrak{su} $$ (N), we derive novel finite-N deformations for two important subalgebras of the gravitational corner symmetry algebra. Specifically, we find that the area-preserving hydrodynamical algebra   <math> <mtext>sdiff</mtext> <mfenced> <msup> <mi>S</mi> <mn>2</mn> </msup> </mfenced> <msub> <mo>⊕</mo> <mi>L</mi> </msub> <msup> <mi>ℝ</mi> <msup> <mi>S</mi> <mn>2</mn> </msup> </msup> </math>  $$ \mathfrak{sdiff}\left({S}^2\right){\oplus}_{\mathcal{L}}{\mathbb{R}}^{S^2} $$  arises as the large-N limit of   <math> <mi>sl</mi> <mfenced> <mi>N</mi> <mi>ℂ</mi> </mfenced> <mo>⊕</mo> <mi>ℝ</mi> </math>  $$ \mathfrak{sl}\left(N,\mathbb{C}\right)\oplus \mathbb{R} $$  and that the full area-preserving corner symmetry algebra   <math> <mtext>sdiff</mtext> <mfenced> <msup> <mi>S</mi> <mn>2</mn> </msup> </mfenced> <msub> <mo>⊕</mo> <mi>L</mi> </msub> <mi>sl</mi> <msup> <mfenced> <mn>2</mn> <mi>ℝ</mi> </mfenced> <msup> <mi>S</mi> <mn>2</mn> </msup> </msup> </math>  $$ \mathfrak{sdiff}\left({S}^2\right){\oplus}_{\mathcal{L}}\mathfrak{sl}{\left(2,\mathbb{R}\right)}^{S^2} $$  is the large-N limit of the pseudo-unitary group   <math> <mi>su</mi> </math>  $$ \mathfrak{su} $$ (N, N). We find matching conditions for the Casimir elements of the deformed and continuum algebras and show how these determine the value of the deformation parameter N as well as the representation of the deformed algebra associated with a quantization of the local gravitational phase space. Additionally, we present a number of novel results related to the various algebras appearing, including a detailed analysis of the asymptotic expansion of the   <math> <mi>su</mi> </math>  $$ \mathfrak{su} $$ (N) structure constants, as well as an explicit computation of the full   <math> <mtext>diff</mtext> </math>  $$ \mathfrak{diff} $$ (S 2) structure constants in the spherical harmonic basis. A consequence of our work is the definition of an area operator which is compatible with the deformation of the area-preserving corner symmetry at finite N.

William Donnelly

Seyed Moosavian

Antony Speranza

Matrix quantization of gravitational edge modes

In this paper we extract from a large-<math><mi>r</mi></math> expansion of the vacuum Einstein’s equations a dynamical system governing the time evolution of an infinity of higher-spin charges. Upon integration, we evaluate the canonical action of these charges on the gravity phase space. The truncation of this action to quadratic order and the associated charge conservation laws yield an infinite tower of soft theorems. We show that the canonical action of the higher spin charges on gravitons in a conformal primary basis, as well as conformally soft gravitons reproduces the higher spin celestial symmetries derived from the operator product expansion. Finally, we give direct evidence that these charges form a canonical representation of a <math><msub><mi>w</mi><mrow><mn>1</mn><mo>+</mo><mi>∞</mi></mrow></msub></math> loop algebra on the gravitational phase space.

Higher spin dynamics in gravity and <math><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>∞</mi></mrow></msub></mrow></math> celestial symmetries

We identify in Einstein gravity an asymptotic spin-2 charge aspect whose conservation equation gives rise, after quantization, to the sub-subleading soft theorem. Our treatment reveals that this spin-2 charge generates a non-local spacetime symmetry represented at null infinity by pseudo-vector fields. Moreover, we demonstrate that the non-linear nature of Einstein’s equations is reflected in the Ward identity through collinear corrections to the sub-subleading soft theorem. Our analysis also provides a unified treatment of the universal soft theorems as conservation equations for the spin-0,-1,-2 canonical generators, while highlighting the important role played by the dual mass.