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Lectures on Celestial Holography
TL;DR: In this paper, a lecture on celestial holography was given at the Pre-Strings school in 2019, where the authors reviewed how semiclassically, the sub-leading soft graviton theorem implies an enhancement of the Lorentz symmetry of scattering in four-dimensional asymptotically flat gravity to Virasoro.
Abstract: These notes consist of 3 lectures on celestial holography given at the Pre-Strings school 2021. We start by reviewing how semiclassically, the subleading soft graviton theorem implies an enhancement of the Lorentz symmetry of scattering in four-dimensional asymptotically flat gravity to Virasoro. This leads to the construction of celestial amplitudes as $\mathcal{S}$-matrices computed in a basis of boost eigenstates. Both massless and massive asymptotic states are recast as insertions on the celestial sphere transforming as global conformal primaries under the Lorentz SL$(2, \mathbb{C})$. We conclude with an overview of celestial symmetries and the constraints they impose on celestial scattering.
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TL;DR: In this article, a free scalar theory in two dimensions exhibiting BMS$_3$ symmetry was studied, which can also be understood as the ultra-relativistic limit of a CFT$_2.
Abstract: The BMS (Bondi-van der Burg-Metzner-Sachs) symmetry arises as the asymptotic symmetry of flat spacetime at null infinity. In particular, the BMS algebra for three dimensional flat spacetime (BMS$_3$) is generated by the super-rotation generators which form a Virasoro sub-algebra with central charge $c_L$, together with mutually-commuting super-translation generators. The super-rotation and super-translation generators have non-trivial commutation relations with another central charge $c_M$. In this paper, we study a free scalar theory in two dimensions exhibiting BMS$_3$ symmetry, which can also be understood as the ultra-relativistic limit of a free scalar CFT$_2$. Upon canonical quantization on the highest weight vacuum, the central charges are found to be $c_L=2$ and $c_M=0$. Because of the vanishing central charge $c_M=0$, the theory features novel properties: there exist primary states which form a multiplet, and the Hilbert space can be organized by an enlarged version of BMS modules dubbed the staggered modules. We further calculate correlation functions and the torus partition function, the later of which is also shown explicitly to be modular invariant.
15 citations
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TL;DR: In this paper, the authors provide a brief overview of the subject of asymptotic symmetries of gauge and gravity theories in space-time spacetimes as background material for celestial holography.
Abstract: The aim of these Lectures is to provide a brief overview of the subject of asymptotic symmetries of gauge and gravity theories in asymptotically flat spacetimes as background material for celestial holography.
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TL;DR: In this paper, the manifestation of asymptotic symmetries via soft theorems, their organization into currents in a celestial CFT, aspects of the holographic dictionary, a literature guide, and accompanying exercises are discussed.
Abstract: Lecture notes prepared for the 2021 SAGEX PhD School in Amplitudes hosted by the University of Copenhagen August 10th through 13th. Topics covered include: the manifestation of asymptotic symmetries via soft theorems, their organization into currents in a celestial CFT, aspects of the holographic dictionary, a literature guide, and accompanying exercises.
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TL;DR: In this article, it was shown that the tree level MHV graviton scattering amplitudes are invariant under the infinite dimensional Virasoro algebra as is expected for a $2$-D CFT.
Abstract: In this paper we continue our study of the tree level MHV graviton scattering amplitudes from the point of view of celestial holography. In arXiv:2008.04330 we showed that the celestial OPE of two gravitons in the MHV sector can be written as a linear combination of $\overline{SL(2,\mathbb C)}$ current algebra and supertranslation descendants. In this note we show that the OPE is in fact manifestly invariant under the infinite dimensional Virasoro algebra as is expected for a $2$-D CFT. This is consistent with the conjecture that the holographic dual in $4$-D asymptotically flat space time is a $2$-D CFT. Since we get only one copy of the Virasoro algebra we can conclude that the holographic dual theory which computes the MHV amplitudes is a chiral CFT with a host of other infinite dimensional global symmetries including $\overline{SL(2,\mathbb C)}$ current algebra, supertranslations and subsubleading soft graviton symmetry. We also discuss some puzzles related to the appearance of the Virasoro symmetry.
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TL;DR: In this paper, Euclidean integral blocks are constructed for the OPE of scalar primaries, which can alternatively be fixed using Poincare symmetry acting on both massless and massive states.
Abstract: Starting from the defining two-point and three-point functions of Celestial CFTs, Euclidean integral blocks are constructed for the OPE of scalar primaries. In their integral form they can alternatively be fixed using Poincare symmetry acting on both massless and massive states. Subsequently, an analytic continuation is done to define the Lorentzian version of the correlation functions and the OPE blocks as valued on the $(1,1)$ cylinder, the universal cover of the recently studied celestial torus. The continuation is essentially the same that is used in the derivation of the KLT relations for string amplitudes. It is shown explicitly that the continued OPE blocks encode the contributions from massive primary states as well as their shadow and light-transformed partners of continuous spin. The corresponding pairings are also studied, thus the construction provides the fundamental relation between correlation functions and OPE coefficients in the scalar CCFT.
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TL;DR: In this article, it was shown that the Kaluza-Klein modes of Type IIB supergravity on $AdS_5\times {\bf S}^5$ match with the chiral operators of the super Yang-Mills theory in four dimensions.
Abstract: Recently, it has been proposed by Maldacena that large $N$ limits of certain conformal field theories in $d$ dimensions can be described in terms of supergravity (and string theory) on the product of $d+1$-dimensional $AdS$ space with a compact manifold. Here we elaborate on this idea and propose a precise correspondence between conformal field theory observables and those of supergravity: correlation functions in conformal field theory are given by the dependence of the supergravity action on the asymptotic behavior at infinity. In particular, dimensions of operators in conformal field theory are given by masses of particles in supergravity. As quantitative confirmation of this correspondence, we note that the Kaluza-Klein modes of Type IIB supergravity on $AdS_5\times {\bf S}^5$ match with the chiral operators of ${\cal N}=4$ super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the ${\cal N}=4$ theory has a large $N$ phase transition related to the thermodynamics of $AdS$ black holes.
14,084 citations
TL;DR: In this paper, the holographic correspondence between field theories and string/M theory is discussed, focusing on the relation between compactifications of string theory on anti-de Sitter spaces and conformal field theories.
5,610 citations
TL;DR: It is argued that the entanglement entropy in d + 1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS(d+2), analogous to the Bekenstein-Hawking formula for black hole entropy.
Abstract: A holographic derivation of the entanglement entropy in quantum (conformal) field theories is proposed from anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We argue that the entanglement entropy in d + 1 dimensional conformal field theories can be obtained from the area of d dimensional minimal surfaces in AdS(d+2), analogous to the Bekenstein-Hawking formula for black hole entropy. We show that our proposal agrees perfectly with the entanglement entropy in 2D CFT when applied to AdS(3). We also compare the entropy computed in AdS(5)XS(5) with that of the free N=4 super Yang-Mills theory.
4,395 citations
TL;DR: In this article, the effects of particle growth with momentum on information spreading near black hole horizons were investigated. But the authors only considered the earliest times of the propagation of information near the horizon.
Abstract: According to ’t Hooft the combination of quantum mechanics and gravity requires the three‐dimensional world to be an image of data that can be stored on a two‐dimensional projection much like a holographic image. The two‐dimensional description only requires one discrete degree of freedom per Planck area and yet it is rich enough to describe all three‐dimensional phenomena. After outlining ’t Hooft’s proposal we give a preliminary informal description of how it may be implemented. One finds a basic requirement that particles must grow in size as their momenta are increased far above the Planck scale. The consequences for high‐energy particle collisions are described. The phenomenon of particle growth with momentum was previously discussed in the context of string theory and was related to information spreading near black hole horizons. The considerations of this paper indicate that the effect is much more rapid at all but the earliest times. In fact the rate of spreading is found to saturate the bound fro...
4,047 citations
TL;DR: In this paper, it was shown that the super-conformal field theory on the brane decouples from the bulk of the field theory in a low energy limit.
Abstract: We show that the large $N$ limit of certain conformal field theories in various dimensions include in their Hilbert space a sector describing supergravity on the product of Anti-deSitter spacetimes, spheres and other compact manifolds. This is shown by taking some branes in the full M/string theory and then taking a low energy limit where the field theory on the brane decouples from the bulk. We observe that, in this limit, we can still trust the near horizon geometry for large $N$. The enhanced supersymmetries of the near horizon geometry correspond to the extra supersymmetry generators present in the superconformal group (as opposed to just the super-Poincare group). The 't Hooft limit of 4-d ${\cal N} =4$ super-Yang-Mills at the conformal point is shown to contain strings: they are IIB strings. We conjecture that compactifications of M/string theory on various Anti-deSitter spacetimes are dual to various conformal field theories. This leads to a new proposal for a definition of M-theory which could be extended to include five non-compact dimensions.
3,136 citations