The research proposal described two work packages (WP1 and WP2) outlined above. The packages were supposed to be conducted in parallel, with a first deliverable due towards the end of the first year of the project, and three other deliverables to follow during the second year, as outlined in the Gantt chart (Figure 1 of the research proposal). Unfortunately, the project was terminated after 6 months because I started a permanent position in the theoretical physics group of the mathematics department at King's College London in March 2024. Nevertheless, during the initial 6 months, the project has been extremely fruitful. The activities and its main scientific outcomes are described below.
Part I of W2 proposed a derivation of celestial amplitudes from correlators in a holographic, higher-dimensional conformal field theory (CFT). This was achieved and the results are reported in
Deliverable 2.1(öffnet in neuem Fenster). The idea is as follows. Consider a conformal correlator of primary operators in a d-dimensional CFT (CFT$_d$). Such correlators admit an integral representation in terms of AdS$_{d+1}$ Witten diagrams, which are AdS counterparts of Feynman diagrams in flat space. The key observation is that for special kinematics of the boundary operators, the associated AdS-Witten diagrams reduce to flat space amplitudes in a conformal primary basis. This allows one to define a map between CFT$_d$ correlators with ``bulk-point'' kinematics and amplitudes in (d-1)-dimensional celestial CFT (CCFT$_{d-1}$). The map involves a certain rescaling of the CFT$_d$ operators and a shift in their dimensions.
In
https://arxiv.org/pdf/2405.07972(öffnet in neuem Fenster) we have illustrated this construction by showing that two- and three-point functions of primary operators in Lorentzian CFT$_d$ become celestial amplitudes in CCFT$_{d-1}$ in this kinematic limit. The calculation required a careful treatment of the distributional components of correlators in the bulk-point limit, as well as consideration of other singularities emerging in the Lorentzian regime. A remarkable match with the CCFT$_{d-1}$ results was obtained. One can also understand these outcomes by considering a particular contraction of the conformal symmetries of CFT$_d$. The latter admit an infinite dimensional enhancement whose generators act on cuts of the $d$-dimensional Lorentzian cylinder. A geometric picture of the emergence of the celestial $(d-1)$-dimensional sphere from CFT$_d$ was thereby obtained, as promised. We are currently exploring the implications of these results for the celestial operator product expansion as outlined in Part II of WP2. This part is on track to be completed by Q6 of the proposed funding period (which was however terminated in March 2024).
The second main outcome of the project was to develop some of the foundations for WP1. In the process, we proposed a new entry in the holographic dictionary in asymptotically flat spacetimes, namely a relation between spacetime subregions, in particular a certain infrared contribution to their entanglement, and CCFT. Such a relation was promised as part of deliverable 2.2 of WP2. The results are reported in
https://arxiv.org/pdf/2403.13913(öffnet in neuem Fenster) which has in part been conducted and completed during the funding period. In this work, we define ``soft'' charges associated with bulk subregions in free Maxwell theory in (3+1)-dimensional Minkowski space. We show that correlated fluctuations in these charges lead to a non-trivial contribution to the vacuum entanglement entropy in these theories. Furthermore, these fluctuations were proposed to be computed by an infrared, on-shell action of ``edge-modes'' living on a codimension-2 celestial sphere. The edge modes were explicitly constructed as classical solutions to the free Maxwell equations that carry non-trivial large-gauge charges. We also showed that the same charges may be turned on by physical charged particles propagating \textit{outside} the spacetime region of interest in an embedding of the spacetime inside the conformal cylinder. In this embedding picture, the non-trivial entanglement arises due to a modification of the Gauss law constraint to account for these sources.
During the funding period we started to develop a similar picture for gravity. Shockwave spacetimes have in the past served as toy models for black hole physics and chaos. We are in the process of building a dictionary between shockwave spacetimes and asymptotically flat spacetimes with infrared gravitational modes, with a first series of results to be reported in a publication later this summer. This work will build the foundations for developing a complete picture of shockwave scattering in the celestial CFT (Deliverable 1.2 of WP1) and studying how chaos manifests in celestial observables (Deliverable 2.2 of WP1).
