Sabrina Pasterski
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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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 c…
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.
Celestial holography provides a promising avenue to studying bulk scattering in flat spacetime from the perspective of boundary celestial conformal field theory (CCFT). A key ingredient in connecting the two sides is the celestial amplitude, which is given by the Mellin transform of momentum space scattering amplitude in energy. As such, celestial amplitudes can be identified with the correlation functions in celestial conformal field theory. In this paper, we introduce the further notion of celestial Mellin amplitude, which is given by the Mellin transform of celestial amplitude in coordinate…
Massless scattering amplitudes in four-dimensional Minkowski spacetime can be Mellin transformed to correlation functions on the celestial sphere at null infinity called celestial amplitudes. We study various properties of massless four-point scalar and gluon celestial amplitudes such as conformal partial wave decomposition, crossing relations and optical theorem. As a byproduct, we derive the analog of the single and double soft limits for all gluon celestial amplitudes.
Massive celestial amplitudes and celestial amplitudes beyond four points
该研究计算了涉及重标量的三点天体振幅，并通过共形块展开发现五点和六点天体振幅中存在双粒子和三粒子算子，推测通过特定展开可观察到多粒子算子交换。
Celestial and momentum space amplitudes for massless particles are related to each other by a change of basis provided by the Mellin transform. Therefore properties of celestial amplitudes have counterparts in momentum space amplitudes and vice versa. In this paper, we study the celestial avatar of dual superconformal symmetry of $\mathcal{N}=4$ Yang-Mills theory. We also analyze various differential equations known to be satisfied by celestial $n$-point tree-level MHV amplitudes and identify their momentum space origins.
Regular celestial amplitudes
本文提出常规 celestial 幅度以解决传统 celestial 幅度在非 MHV 矛盾中的矛盾，证明其在树级与 celestial OPE 一致，修正了 CCFT 联合子的定义。
We study scattering amplitudes in the shadow conformal primary basis, which satisfies the same defining properties as the original conformal primary basis and has many advantages over it. The shadow celestial amplitudes exhibit locality manifestly on the celestial sphere, and behave like correlation functions in conformal field theory under the operator product expansion (OPE) limit. We study the OPE limits for three-point shadow celestial amplitude, and general $2\to n-2$ shadow celestial amplitudes from a large class of Feynman diagrams. In particular, we compute the conformal block expansio…
Celestial amplitudes provide holographic imprints of four-dimensional scattering processes in terms of conformal correlation functions on a two-dimensional sphere describing Minkowski space at null infinity. We construct the generators of Poincare and conformal groups in the celestial representation and discuss how these symmetries are manifest in the amplitudes.
Scattering amplitudes of massless particles in Minkowski space can be expressed in a conformal basis by Mellin transforming the momentum space amplitudes to correlation functions on the celestial sphere at null infinity. In this paper, we study celestial amplitudes of loop level gluons and gravitons. We focus on the rational amplitudes that carry all-plus and single-minus external helicities. Because these amplitudes are finite, they provide a concrete example of celestial amplitudes of Yang- Mills and gravity theory beyond tree level. We give explicit examples of four and five point functions…
In this paper, we study the implications of bulk locality on the celestial amplitude. In the context of the four-point amplitude, the fact that the bulk S-matrix factorizes locally in poles of Mandelstam variables is reflected in the imaginary part of the celestial amplitude. In particular, on the real axis in the complex plane of the boost weight, the imaginary part of the celestial amplitude can be given as a positive expansion on the Poincaré partial waves, which are nothing but the projection of flat-space spinning polynomials onto the celestial sphere. Furthermore, we derive the celestial…
Amplitude from crossing-symmetric celestial OPE
提出一种仅依赖CFT技术、利用交叉对称天球OPE从单个共形块系数重建四点无质量散射振幅的方法，并成功应用于纯杨-米尔斯、纯引力和爱因斯坦-杨-米尔斯理论的MHV振幅。
In this paper we evaluate the modified celestial amplitude for gravitons and gluons, as defined in arXiv:1801.10171[hep-th]. We find that the modified (tree) amplitude is finite for gravitons in Einstein gravity. The modified amplitude behaves like correlation function of operators inserted at various points of null-infinity in the Minkowski space-time. Therefore, unlike the standard celestial amplitudes, these are three dimensional objects. We also show that this amplitude admits conformal soft factorization recently studied in the literature.
Celestial amplitude for 2d theory
本文研究二维理论的 celestial 振幅，证明其为 S-矩阵的傅里叶变换，并检验交叉性和幺正性，发现仅靠这些条件不足以 bootstrap，需额外项；同时探讨引力 dressing 对极点的影响。
Off-shell celestial amplitudes with both time-like and space-like external legs are defined. The Feynman rules for scalar amplitudes, viewed as a set of recursion relations for off-shell momentum space amplitudes, are transformed to the celestial sphere using the split representation. For four-point celestial amplitudes, the Feynman expansion is shown to be equivalent to a conformal partial wave decomposition, providing an interpretation of conformal partial wave expansion coefficients as integrals over off-shell three-point structures. A conformal partial wave decomposition for a simple four-…
Remarks on celestial amplitudes and Liouville theory
该研究探讨天球全息与刘维尔场论的关联，发现刘维尔关联函数库仑气体实现中塞尔伯格型积分的对偶关系可导出维度平移的天球振幅间的系列关系，为相关领域研究提供了新的理论联系。
Pasterski, Shao and Strominger have recently proposed that massless scattering amplitudes can be mapped to correlators on the celestial sphere at infinity via a Mellin transform. We apply this prescription to arbitrary $n$-point tree-level gluon amplitudes. The Mellin transforms of MHV amplitudes are given by generalized hypergeometric functions on the Grassmannian $Gr(4,n)$, while generic non-MHV amplitudes are given by more complicated Gelfand $A$-hypergeometric functions.
The analytic structures of scattering amplitudes in gauge theory and gravity are examined on the celestial sphere. The celestial amplitudes in the two theories - computed by employing a regulated Mellin transform - can be compared at low multiplicity. It is established by direct computation that up to five external particles, the double copy relations of Kawai, Lewellen and Tye continue to hold identically, modulo certain multiplicative factors which are explicitly determined. Supersymmetric representations of the amplitudes are utilized throughout, manifesting the double copy structure betwee…
Celestial amplitude is a new reformulation of momentum space scattering amplitude and offers a promising way for flat holography. In this paper, we study the celestial amplitude in $\mathcal N=4$ Super-Yang-Mills (SYM) theory aiming at understanding the role of superconformal symmetry in celestial holography. We first construct the superconformal generators acting on the celestial superfield which assembles all the on-shell fields in the multiplet together in terms of celestial variables and Grassmann parameters. These generators satisfy the superconformal algebra of $\mathcal N=4$ SYM theory.…
Celestial Mellin Amplitudes
本文基于Mack的梅林表示，提出天体梅林振幅作为研究闵可夫斯基空间天体球上全息关联子的工具，并给出标量场论中树图和圈图示例，同时探讨其非微扰结构。
Recently, scattering amplitudes in four-dimensional Minkowski spacetime have been interpreted as conformal correlation functions on the two-dimensional celestial sphere, the so-called celestial amplitudes. In this note we consider tree-level scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory and present a Grassmannian formulation of their celestial counterparts. This result paves the way towards a geometric picture for celestial superamplitudes, in the spirit of positive geometries.
Celestial Leaf Amplitudes
本文研究天体叶振幅，证明Kleinian三点MHV振幅的叶子振幅除光锥奇点外均光滑，且完整平移不变天体振幅为叶振幅在胶子总权重为3处极点的留数，在相差光锥接触项下为零，并还原先前Mellin变换结果。
Singularity Structure of the Four Point Celestial Leaf Amplitudes
本文研究无质量标量和MHV胶子四点天球叶振幅，利用体尺度不变性约束证明其在$z=\ar z$处具有简单极点奇异性，并通过类时与类空楔形叶振幅相加恢复天球振幅的分布性质，同时验证了MHV胶子叶振幅满足此前由软胶子定理和OPE展开得到的微分方程。
We consider Yang-Mills theory with the coupling constant and theta angle determined by the vacuum expectation values of a dynamical (complex) dilaton field. We discuss the tree-level N-gluon MHV scattering amplitudes in the presence of a nontrivial background dilaton field and construct the corresponding celestial amplitudes by taking Mellin transforms with respect to the lightcone energies. In this way, we obtain two-dimensional CFT correlators of primary fields on the celestial sphere. We show that the celestial Yang-Mills amplitudes evaluated in the presence of a spherical dilaton shockwave…
We start by constructing a conformally covariant improvement of the celestial light transform which keeps track of the mixing between incoming and outgoing states under finite Lorentz transformations in $\mathbb{R}^{2,2}$. We then compute generic 2, 3 and 4-point celestial amplitudes for massless external states in the ambidextrous basis prepared by composing this $\mathrm{SL}(2,\mathbb{R})$ intertwiner with the usual celestial map between momentum and boost eigenstates. The results are non-distributional in the celestial coordinates $(z,\bar{z})$ and conformally covariant in all scattering ch…
Both celestial and momentum space amplitudes in four dimensions are beset by divergences resulting from spacetime translation and sometimes scale invariance. In this paper we consider a (linearized) marginal deformation of the celestial CFT for Yang-Mills theory which preserves 2D conformal invariance but breaks both spacetime translation and scale invariance and involves a chirally coupled massive scalar. The resulting celestial amplitudes are completely finite (apart from the usual soft and collinear divergences) and take the canonical CFT form. Moreover, we show they can be simply rewritten…
Integral Transformations for Conformally Invariant Celestial Amplitudes
本文提出一种将 celestial 坐标映射到新复变量的积分变换，用于 celestial 胶子幅，揭示其与闭弦散射幅的结构关系，并展示该变换对共形不变性必要的约束条件。
The formalism of relativistic partial wave expansion is developed for four-point celestial amplitudes of massless external particles. In particular, relativistic partial waves are found as eigenfunctions to the product representation of celestial Poincaré Casimir operators with appropriate eigenvalues. The requirement of hermiticity of Casimir operators is used to fix the corresponding integral inner product, and orthogonality of the obtained relativistic partial waves is verified explicitly. The completeness relation, as well as the relativistic partial wave expansion follow. Example celestia…
Using the ambitwistor string, we compute tree-level celestial amplitudes for biadjoint scalars, Yang-Mills and gravity to all multiplicities. They are presented in compact CHY-like formulas with operator-valued scattering equations and numerators acting on a generalized hypergeometric function. With these we extend the celestial double copy to tree-level amplitudes with arbitrary number of external states. We also show how color-kinematics duality is implemented in celestial amplitudes and its interpretation in terms of a generalized twisted cohomology theory.
Celestial Berends-Giele current
本文研究天球上的 Berends–Giele 递推，用其计算典型天体振幅，分析 BG 流的 OPE 行为，并把缝合程序推广到天体情形。
Goldstone bosons on celestial sphere and conformal soft theorems
本文通过梅林变换将软极限行为映射到天球关联函数的奇异性，研究了戈德斯通玻色子的天球振幅与共形软定理，并给出了UV完备模型中的示例，扩展了软自举方法。
An $AdS_{3}$ Dual for Supersymmetric MHV Celestial Amplitudes
本文在超对称AdS₃空间的全纯延拓上推广WZNW模型，首次为𝒩=8超引力和𝒩=4 SYM中的树级MHV天球振幅构造出全息对偶描述。
Multicollinear Singularities in Celestial CFT
本文通过研究天体振幅的全纯多共线极限并推导三类理论的三共线分裂函数，以 $ϕ^3$ 理论中的分支割线揭示双重留数条件受阻及 OPE 结合性问题。
Conformal blocks from celestial graviton amplitudes
本文提出一种解析的天体引力子关联函数，通过计算含影子算符的四点关联函数并进行共形块分解，反演求得单值天体引力子振幅，发现其具有双拷贝结构。
Missing Corner in the Sky: Massless Three-Point Celestial Amplitudes
本文首次计算了无质量标量、光子、胶子和引力子的三点天球振幅，证明其非零且可表示为二维共形场论关联函数，揭示了软粒子与守恒流及超平移流之间的深刻联系。
The functional structure of celestial amplitudes as constrained by Poincaré symmetry is investigated in $2,3,$ and $4$-point cases for massless external particles of various spin, as well as massive external scalars. Functional constraints and recurrence relations are found (akin to the findings in arXiv:1901.01622) that must be obeyed by the respective permissible correlator structures and function coefficients. In specific three-point cases involving massive scalars the resulting recurrence relations can be solved, e.g. reproducing purely from symmetry a three-point function coefficient know…
We discuss the properties of recently constructed "single-valued" celestial four-gluon amplitudes. We show that the amplitude factorizes into the "current" part and the "scalar" part. The current factor is given by the group-dependent part of the Wess-Zumino-Witten correlator of four holomorphic currents with a non-vanishing level of Kač-Moody algebra. The scalar factor can be expressed in terms of a complex integral of the Koba-Nielsen form, similar to the integrals describing four-point correlators in Coulomb gas models and, more generally, in the infinite central charge limit of Liouville t…
Scalar-Graviton Amplitudes and Celestial Holography
本文计算含一个有质量标量和二至四个引力子的散射振幅，发现令标量共形维数为零可得到无发散、非分布且符合标准CFT形式的天体关联函数，并修正软定理表述。
In a recent paper, here referred to as part I, we considered the celestial four-gluon amplitude with one gluon represented by the shadow transform of the corresponding primary field operator. This correlator is ill-defined because it contains branch points related to the presence of conformal blocks with complex spin. In this work, we adopt a procedure similar to minimal models and construct a single-valued completion of the shadow correlator, in the limit when the shadow is "soft." By following the approach of Dotsenko and Fateev, we obtain an integral representation of such a single-valued c…
The all-loop resummation of SU$(N)$ gauge theory amplitudes is known to factorize into an IR-divergent (soft and collinear) factor and a finite (hard) piece. The divergent factor is universal, whereas the hard function is a process-dependent quantity. We prove that this factorization persists for the corresponding celestial amplitudes. Moreover, the soft/collinear factor becomes a scalar correlator of the product of renormalized Wilson lines defined in terms of celestial data. Their effect on the hard amplitude is a shift in the scaling dimensions by an infinite amount, proportional to the cus…
A Celestial Description of Planar Super-Yang-Mills Theory
通过将天体的RSVW形式论扩展到迷你扭量超空间，利用Drummond-Henn公式构造了树级N^k-MHV天体叶振幅，并展示了其迷你扭量-傅里叶变换由迷你扭量线族模空间上的积分给出，同时通过半经典关联函数和迷你扭量sigma模型提供了两种动力学表述。
Partial Differential Equations for MHV Celestial Amplitudes in Liouville Theory
本文推导了Liouville耦合常数b参数化的n点MHV celestial amplitudes的偏微分方程，用于计算b的扰动展开，并探讨了其在纯Yang-Mills和爱因斯坦引力中的树级MHV函数的 semiclassical 极限。
We show that the Mellin transform of an $n$-point tree level MHV gluon scattering amplitude, also known as the celestial amplitude in pure Yang-Mills theory, satisfies a system of $(n-2)$ linear first order partial differential equations corresponding to $(n-2)$ positive helicity gluons. Although these equations closely resemble Knizhnik-Zamolodchikov equations for $SU(N)$ current algebra there is also an additional "correction" term coming from the subleading soft gluon current algebra. These equations can be used to compute the leading term in the gluon-gluon OPE on the celestial sphere. Sim…
Recent attempts at the construction of holography for asymptotically flat spacetimes have taken two different routes. Celestial holography, involving a two dimensional (2d) CFT dual to 4d Minkowski spacetime, has generated novel results in asymptotic symmetry and scattering amplitudes. A different formulation, using Carrollian CFTs, has been principally used to provide some evidence for flat holography in lower dimensions. Understanding of flatspace scattering has been lacking in the Carroll framework. In this work, using ideas from Celestial holography, we show that 3d Carrollian CFTs living …
Celestial strings: field theory, conformally soft limits, and mapping the worldsheet onto the celestial sphere
本文计算了 type I 超弦理论中树图级 5-胶子振幅的天体关联子，研究了如何在天体基中恢复场论极限，并探讨了共形软定理、有效场论展开及世界面到天体球的映射等性质。
On Positive Celestial Geometry: ABHY in the Sky
本文通过正几何描述 celestial 幅度，揭示了 celestial 树级 ϕ³ 幅度与 associahedron 的关系，并统一了不同 bulk 理论中的 celestial 幅度描述。
Celestial Mellin Amplitudes II: Spin-1 and Spin-2 Exchange, and Beyond
该研究发展天体梅林振幅以描述带自旋粒子交换，推导自旋1、2交换的闭式振幅与OPE数据，通过梅林空间移位算子将结果推广到任意整数自旋，为带自旋场的关联函数研究提供方法。
Celestial Regge theory
本文提出天体Regge理论，通过分析散射幅的解析性质，定义了天体Regge极限，并确定了与本体幅极点和不连续性相关的CFT数据。
An Integer Basis for Celestial Amplitudes
本文提出3+1维闵可夫斯基空间中无质量Klein-Gordon方程的离散整数共形维度基，并证明其完备性，使Wightman函数可表示为基的二次求和。
Celestial self-dual Yang-Mills theory: a new formula and the OPE limit
本文基于天体Berends-Giele流提出新公式计算天体色有序自对偶杨-米尔斯振幅，使领头阶OPE极限显式，并探讨了高阶OPE项。
The Cachazo-Strominger subleading soft graviton theorem for a positive helicity soft graviton is equivalent to the Ward identities for $\overline{SL(2,\mathbb C)}$ currents. This naturally gives rise to a $\overline{SL(2,\mathbb C)}$ current algebra living on the celestial sphere. The generators of the $\overline{SL(2,\mathbb C)}$ current algebra and the supertranslations, coming from a positive helicity leading soft graviton, form a closed algebra. We find that the OPE of two graviton primaries in the Celestial CFT, extracted from MHV amplitudes, is completely determined in terms of this alge…
We show that associativity of the tree-level OPE in a celestial CFT imposes constraints on the coupling constants of the corresponding bulk theory. These constraints are the same as those derived in arXiv:2111.11356 from the Jacobi identity of the algebra of soft modes. The constrained theories are interesting as apparently well-defined celestial CFTs with a deformed $w_{1+\infty}$ symmetry algebra. We explicitly work out the ramifications of these constraints on scattering amplitudes involving gluons, gravitons and scalars in these theories. We find that all four-point amplitudes constructibl…
Celestial Twistor Amplitudes
本文建立 YM 与引力中的天体扭量振幅表述，推导三点、四点振幅及天体扭量 BCFW 递推关系，并揭示相关全息对应。
Celestial amplitudes represent 4D scattering of particles in boost, rather than the usual energy-momentum, eigenstates and hence are sensitive to both UV and IR physics. We show that known UV and IR properties of quantum gravity translate into powerful constraints on the analytic structure of celestial amplitudes. For example the soft UV behavior of quantum gravity is shown to imply that the exact four-particle scattering amplitude is meromorphic in the complex boost weight plane with poles confined to even integers on the negative real axis. Would-be poles on the positive real axis from UV as…
A Celestial Dual for MHV Amplitudes
本文提出由 Liouville 理论、Kac-Moody 代数和自由费米子构成的二维 CFT 可全息生成四维 MHV 振幅，并通过大 N 与大 c 极限将叶振幅组合为天球振幅。
Celestial closed strings at one-loop
本文在共形基下研究闭超弦理论中一圈阶四引力子散射，用天体变量写出表达式，确认α'因子化，并证明场论极限与梅林变换交换，表明开弦胶子振幅的普遍性质也适用于闭弦引力子天体振幅。
We propose a new program for computing a certain integrand of scattering amplitudes of four-dimensional gauge theories which we call the \textit{form factor integrand}, starting from 6d holomorphic theories on twistor space. We show that the form factor integrands can be expressed as sums of products of 1.) correlators of a 2d chiral algebra, related to the algebra of asymptotic symmetries uncovered recently in the celestial holography program, and 2.) OPE coefficients of a 4d non-unitary CFT. We prove that conformal blocks of the chiral algebras are in one-to-one correspondence with local ope…
Split representation in celestial holography
本文通过切割闵可夫斯基空间费曼图内部线，在天体全息中发展了分裂表示，并用于计算无质量标量和光子四点函数的共形分波及共形块展开，发现t-通道中标量振幅存在交错模的新型交换。
We study the celestial description of the $O(N)$ sigma model in the large $N$ limit as introduced by Coleman, Jackiw and Politzer. Focusing on three dimensions, we analyze the implications of a UV complete, all-loop order 4-point amplitude of pions in terms of correlation functions defined on the celestial circle. We find these retain many key features from the previously studied tree-level case, such as their relation to Generalized Free Field theories and crossing-symmetry, but also incorporate new properties such as IR/UV softness and S-matrix metastable states. In particular, to understand…
Challenges to Understanding Celestial Holography from the Bottom Up
本文通过Sinh-Gordon模型检验了从底层构建天体隐秘性的方法，发现直接转换和展开的顺序会影响结果，揭示了量子场论中天体转换的复杂性。
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