Periodic Reporting for period 1 - FlatHolo (Towards a holographic approach for gravity in asymptotically flat spacetimes)
Periodo di rendicontazione: 2023-09-01 al 2025-08-31
My proposed project broadly aimed to exploit the infinity of symmetries of gravity in four-dimensional asymptotically flat spacetimes (AFS) and the associated constraints to gain quantitative insight into non-perturbative aspects of gravitational scattering. AFS are interesting to study because they describe physical events in our universe on astrophysical timescales such as the gravitational radiation from a pair of merging black holes. Moreover, they were recently shown to host a set of precise equivalences relating spacetime symmetries to universal aspects of gravitational scattering. These ideas led to the Celestial Holography program, which aims to develop a holographic understanding of gravity in four-dimensional (4D) AFS. There is increasing evidence that dual theories in this case live on a 2D sphere at infinity (the celestial sphere) and that gravitational observables are encoded in correlation functions on the sphere (celestial amplitudes) obeying a wide range of constraints.
The project had two main goals. The first objective (WP1) was to develop the foundations for systematically incorporating massive particles and ultimately black holes into the AFS holographic framework. On the one had, I proposed to initiate a study of celestial amplitudes involving massive particles. Such amplitudes are expected to carry information about observables associated with black hole scattering, including gravitational waves. On the other hand, I proposed the study of celestial amplitudes in shockwave and black hole backgrounds. Especially in the context of the Anti-de-Sitter/conformal field theory (AdS/CFT) correspondence, shockwave spacetimes have in the past served as toy models for black hole physics and chaos. It is currently not known whether these aspects are universally (ie. also in AFS) captured by observables (correlation functions) in a CFT.
The second objective (WP2) was to derive certain aspects of celestial holography from a flat space limit of the much better understood holographic correspondence in asymptotically negatively curved spacetimes (AdS/CFT). The project proposed to first understand how celestial amplitudes emerge from correlation functions of a conformal field theory in one higher dimension. It then proposed to leverage some of the tools developed over many years in the study of AdS/CFT to gain new insights into flat space holography, including the implications of an infinity of symmetry constraints and the emergence of bulk subregions from the boundary celestial (C)CFT. Since the exterior of a black hole spacetime is a subregion, one of the aims of WP2 can be reformulated as searching for a missing entry in the flat space dictionary relating black hole observables to CCFT correlators.
The project had two main goals. The first objective (WP1) was to develop the foundations for systematically incorporating massive particles and ultimately black holes into the AFS holographic framework. On the one had, I proposed to initiate a study of celestial amplitudes involving massive particles. Such amplitudes are expected to carry information about observables associated with black hole scattering, including gravitational waves. On the other hand, I proposed the study of celestial amplitudes in shockwave and black hole backgrounds. Especially in the context of the Anti-de-Sitter/conformal field theory (AdS/CFT) correspondence, shockwave spacetimes have in the past served as toy models for black hole physics and chaos. It is currently not known whether these aspects are universally (ie. also in AFS) captured by observables (correlation functions) in a CFT.
The second objective (WP2) was to derive certain aspects of celestial holography from a flat space limit of the much better understood holographic correspondence in asymptotically negatively curved spacetimes (AdS/CFT). The project proposed to first understand how celestial amplitudes emerge from correlation functions of a conformal field theory in one higher dimension. It then proposed to leverage some of the tools developed over many years in the study of AdS/CFT to gain new insights into flat space holography, including the implications of an infinity of symmetry constraints and the emergence of bulk subregions from the boundary celestial (C)CFT. Since the exterior of a black hole spacetime is a subregion, one of the aims of WP2 can be reformulated as searching for a missing entry in the flat space dictionary relating black hole observables to CCFT correlators.
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(si apre in una nuova finestra). 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(si apre in una nuova finestra) 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(si apre in una nuova finestra) 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).
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(si apre in una nuova finestra). 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(si apre in una nuova finestra) 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(si apre in una nuova finestra) 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).
The main results of the research activities are:
-- A map between CFT$_d$ correlators with bulk-point kinematics and amplitudes in (d-1)-dimensional celestial CFT (CCFT$_{d-1}$). This proposed map is foundational for the future exploration of how flat space celestial observables emerge from holographic observables in AdS/CFT in a flat-space limit.
-- An explicit derivation of CCFT$_{d-1}$ two- and three-point amplitudes from two- and three-point correlation functions of primary operators in a CFT$_d$. The derivation illustrates how distributional, Poincare covariant celestial amplitudes emerge from unitary CFT correlators in one higher dimension. These results are foundational for future studies of the decomposition of four-point functions in a basis of conformal blocks. Since the building blocks of the latter are three-point functions, our results are expected to shed light on the appropriate basis to be used to decompose celestial four-point functions, which will further allow us to understand the spectrum of such theories, as well as how bulk unitarity is encoded in the CCFT.
-- A new entry in the holographic dictionary in asymptotically flat spacetimes, namely a relation between spacetime subregions, and their associated infrared contribution to entanglement, and CCFT. This new entry was proposed in the toy model of free Maxwell theory in (3+1)-dimensional Minkowski spacetime. The qualitative aspects of these results should carry over to the infrared sector of gravity, which shares many similarities with gauge theory. The generalization of this story in gravity is currently under investigation.
-- A dictionary between shockwave spacetimes and asymptotically flat spacetimes with infrared gravitational modes (to appear later in the summer). These results will have implications for further understanding the holographic description of scattering in asymptotically flat black hole backgrounds.
-- A map between CFT$_d$ correlators with bulk-point kinematics and amplitudes in (d-1)-dimensional celestial CFT (CCFT$_{d-1}$). This proposed map is foundational for the future exploration of how flat space celestial observables emerge from holographic observables in AdS/CFT in a flat-space limit.
-- An explicit derivation of CCFT$_{d-1}$ two- and three-point amplitudes from two- and three-point correlation functions of primary operators in a CFT$_d$. The derivation illustrates how distributional, Poincare covariant celestial amplitudes emerge from unitary CFT correlators in one higher dimension. These results are foundational for future studies of the decomposition of four-point functions in a basis of conformal blocks. Since the building blocks of the latter are three-point functions, our results are expected to shed light on the appropriate basis to be used to decompose celestial four-point functions, which will further allow us to understand the spectrum of such theories, as well as how bulk unitarity is encoded in the CCFT.
-- A new entry in the holographic dictionary in asymptotically flat spacetimes, namely a relation between spacetime subregions, and their associated infrared contribution to entanglement, and CCFT. This new entry was proposed in the toy model of free Maxwell theory in (3+1)-dimensional Minkowski spacetime. The qualitative aspects of these results should carry over to the infrared sector of gravity, which shares many similarities with gauge theory. The generalization of this story in gravity is currently under investigation.
-- A dictionary between shockwave spacetimes and asymptotically flat spacetimes with infrared gravitational modes (to appear later in the summer). These results will have implications for further understanding the holographic description of scattering in asymptotically flat black hole backgrounds.