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-dimen
sional 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.
The 4D 4-point scattering amplitude of massless scalars via a massive exchange is expressed in a basis of conformal primary particle wavefunctions. This celestial amplitude is expanded in a basis of 2D conformal partial waves on the unitary principal
series, and then rewritten as a sum over 2D conformal blocks via contour deformation. The conformal blocks include intermediate exchanges of spinning light-ray states, as well as scalar states with positive integer conformal weights. The conformal block prefactors are found as expected to be quadratic in the celestial OPE coefficients.
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 powerf
ul 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 asymptotics are shown to be erased by a flat space analog of the AdS resolution of the bulk point singularity. The residues of the poles on the negative axis are identified with operator coefficients in the IR effective action. Far along the real positive axis, the scattering is argued to grow exponentially according to the black hole area law. Exclusive amplitudes are shown to simply factorize into conformally hard and conformally soft factors. The soft factor contains all IR divergences and is given by a celestial current algebra correlator of Goldstone bosons from spontaneously broken asymptotic symmetries. The hard factor describes the scattering of hard particles together with the boost-eigenstate clouds of soft photons or gravitons required by asymptotic symmetries. These provide an IR safe $mathcal{S}$-matrix for the scattering of hard particles.
The operator product expansion (OPE) on the celestial sphere of conformal primary gluons and gravitons is studied. Asymptotic symmetries imply recursion relations between products of operators whose conformal weights differ by half-integers. It is sh
own, for tree-level Einstein-Yang-Mills theory, that these recursion relations are so constraining that they completely fix the leading celestial OPE coefficients in terms of the Euler beta function. The poles in the beta functions are associated with conformally soft currents.
Asymptotic particle states in four-dimensional celestial scattering amplitudes are labelled by their $SL(2,mathbb{C})$ Lorentz/conformal weights $(h,bar{h})$ rather than the usual energy-momentum four-vector. These boost eigenstates involve a superpo
sition of all energies. As such, celestial gluon (or photon) scattering cannot obey the usual (energetically) soft theorems. In this paper we show that tree-level celestial gluon scattering, in theories with sufficiently soft UV behavior, instead obeys conformally soft theorems involving $h to 0$ or $bar{h} to 0$. Unlike the energetically soft theorem, the conformally soft theorem cannot be derived from low-energy effective field theory.
The color memory effect is the non-abelian gauge theory analog of the gravitational memory effect, in which the passage of color radiation induces a net relative SU(3) color rotation of a pair of nearby quarks. It is proposed that this effect can be
measured in the Regge limit of deeply inelastic scattering at electron-ion colliders.
It is shown that there is a universal gravitational memory effect measurable by inertial detectors in even spacetime dimensions $dgeq 4$. The effect falls off at large radius $r$ as $r^{3-d}$. Moreover this memory effect sits at one corner of an infr
ared triangle with the other two corners occupied by Weinbergs soft graviton theorem and infinite-dimensional asymptotic symmetries.
A transient color flux across null infinity in classical Yang-Mills theory is considered. It is shown that a pair of test `quarks initially in a color singlet generically acquire net color as a result of the flux. A nonlinear formula is derived for t
he relative color rotation of the quarks. For weak color flux the formula linearizes to the Fourier transform of the soft gluon theorem. This color memory effect is the Yang-Mills analog of the gravitational memory effect.
Recently it has been shown that the vacuum state in QED is infinitely degenerate. Moreover a transition among the degenerate vacua is induced in any nontrivial scattering process and determined from the associated soft factor. Conventional computatio
ns of scattering amplitudes in QED do not account for this vacuum degeneracy and therefore always give zero. This vanishing of all conventional QED amplitudes is usually attributed to infrared divergences. Here we show that if these vacuum transitions are properly accounted for, the resulting amplitudes are nonzero and infrared finite. Our construction of finite amplitudes is mathematically equivalent to, and amounts to a physical reinterpretation of, the 1970 construction of Faddeev and Kulish.
Recently a boundary energy-momentum tensor $T_{zz}$ has been constructed from the soft graviton operator for any 4D quantum theory of gravity in asymptotically flat space. Up to an anomaly which is one-loop exact, $T_{zz}$ generates a Virasoro action
on the 2D celestial sphere at null infinity. Here we show by explicit construction that the effects of the IR divergent part of the anomaly can be eliminated by a one-loop renormalization that shifts $T_{zz}$.
