Conformal basis for flat space amplitudes
TL;DR: In this article, the authors studied scalar conformal primary wavefunctions on the principal continuous series of the Klein-Gordon, Maxwell, and linearized Einstein equations under the Lorentz group.
Abstract: We study solutions of the Klein-Gordon, Maxwell, and linearized Einstein equations in ${\mathbb{R}}^{1,d+1}$ that transform as $d$-dimensional conformal primaries under the Lorentz group $SO(1,d+1)$. Such solutions, called conformal primary wavefunctions, are labeled by a conformal dimension $\mathrm{\ensuremath{\Delta}}$ and a point in ${\mathbb{R}}^{d}$, rather than an on-shell ($d+2$)-dimensional momentum. We show that the continuum of scalar conformal primary wavefunctions on the principal continuous series $\mathrm{\ensuremath{\Delta}}\ensuremath{\in}\frac{d}{2}+i\mathbb{R}$ of $SO(1,d+1)$ spans a complete set of normalizable solutions to the wave equation. In the massless case, with or without spin, the transition from momentum space to conformal primary wavefunctions is implemented by a Mellin transform. As a consequence of this construction, scattering amplitudes in this basis transform covariantly under $SO(1,d+1)$ as $d$-dimensional conformal correlators.
Citations
More filters
Posted Content•
TL;DR: A transcript of a course given by Strominger at Harvard in spring semester 2016 as discussed by the authors contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes.
Abstract: This is a redacted transcript of a course given by the author at Harvard in spring semester 2016. It contains a pedagogical overview of recent developments connecting the subjects of soft theorems, the memory effect and asymptotic symmetries in four-dimensional QED, nonabelian gauge theory and gravity with applications to black holes. The lectures may be viewed online at this https URL. Please send typos or corrections to strominger@physics.this http URL.
599 citations
TL;DR: In this paper, the authors compute low-point, tree-level gluon scattering amplitudes in the space of spin-one primary wave functions in the Lorentz group.
Abstract: Recently, spin-one wave functions in four dimensions that are conformal primaries of the Lorentz group $SL(2,\mathbb{C})$ were constructed. We compute low-point, tree-level gluon scattering amplitudes in the space of these conformal primary wave functions. The answers have the same conformal covariance as correlators of spin-one primaries in a $2d$ CFT. The Britto--Cachazo--Feng--Witten (BCFW) recursion relation between three- and four-point gluon amplitudes is recast into this conformal basis.
230 citations
Cites background or methods from "Conformal basis for flat space ampl..."
...It was shown in [10] that the conformal primary wavefunctions are a complete and delta-function-normalizable...
[...]
...We will restrict ourselves to four-dimensional Minkowski space R1,3 with spacetime coordinates Xµ (µ = 0, 1, 2, 3)....
[...]
...Recently it has emerged [10] from the study of two-point functions, that the unitary principal continuous series (which has appeared in a variety of CFT studies [11–14]) of the Lorentz group plays a central role....
[...]
...A comprehensive survey of conformal primary wavefunctions with or without spin in arbitrary spacetime dimensions was performed in [10]....
[...]
...The explicit expression for the conformal primary wavefunctions was given in terms of the spin-one bulk-to-boundary propagator in the three-dimensional hyperbolic space H3 in [3, 10]....
[...]
TL;DR: In this article, the Mellin-Barnes representation of correlators in Fourier space was used for boundary correlation functions in both the anti-de Sitter and de Sitter spaces.
Abstract: We develop a Mellin space approach to boundary correlation functions in anti-de Sitter (AdS) and de Sitter (dS) spaces. Using the Mellin-Barnes representation of correlators in Fourier space, we show that the analytic continuation between AdSd+1 and dSd+1 is encoded in a collection of simple relative phases. This allows us to determine the late-time tree-level three-point correlators of spinning fields in dSd+1 from known results for Witten diagrams in AdSd+1 by multiplication with a simple trigonometric factor. At four point level, we show that Conformal symmetry fixes exchange four-point functions both in AdSd+1 and dSd+1 in terms of the dual Conformal Partial Wave (which in Fourier space is a product of boundary three-point correlators) up to a factor which is determined by the boundary conditions. In this work we focus on late-time four-point correlators with external scalars and an exchanged field of integer spin-l. The Mellin-Barnes representation makes manifest the analytic structure of boundary correlation functions, providing an analytic expression for the exchange four-point function which is valid for general d and generic scaling dimensions, in particular massive, light and (partially-)massless fields. It moreover naturally identifies boundary correlation functions for generic fields with multi-variable Meijer-G functions. When d = 3 we reproduce existing explicit results available in the literature for external conformally coupled and massless scalars. From these results, assuming the weak breaking of the de Sitter isometries, we extract the corresponding correction to the inflationary three-point function of general external scalars induced by a general spin- l field at leading order in slow roll. These results provide a step towards a more systematic understanding of de Sitter observables at tree level and beyond using Mellin space methods.
179 citations
TL;DR: In this paper, a large class of conformally-covariant differential operators and a crossing equation that they obey are introduced, which dramatically simplify calculations involving operators with spin in conformal field theories.
Abstract: We introduce a large class of conformally-covariant differential operators and a crossing equation that they obey. Together, these tools dramatically simplify calculations involving operators with spin in conformal field theories. As an application, we derive a formula for a general conformal block (with arbitrary internal and external representations) in terms of derivatives of blocks for external scalars. In particular, our formula gives new expressions for “seed conformal blocks” in 3d and 4d CFTs. We also find simple derivations of identities between external-scalar blocks with different dimensions and internal spins. We comment on additional applications, including deriving recursion relations for general conformal blocks, reducing inversion formulae for spinning operators to inversion formulae for scalars, and deriving identities between general 6j symbols (Racah-Wigner coefficients/“crossing kernels”) of the conformal group.
158 citations
TL;DR: In this paper, the soft limit of tree-level Minkowskian scattering amplitudes in a conformal group on the celestial two-sphere at null infinity is considered.
Abstract: Four-dimensional Minkowskian scattering amplitudes can also be presented in a basis of a different group ($S\phantom{\rule{0}{0ex}}L(2,C)$), which acts as the conformal group on the celestial two-sphere at null infinity. This type of representation is useful for understanding the flat-space holography. The authors propose a procedure for taking the (conformally) soft limit of tree-level scattering amplitudes in this basis.
157 citations
References
More filters
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
Posted Content•
TL;DR: In this article, a correspondence between conformal field theory observables and those of supergravity was proposed, where correlation functions in conformal fields are given by the dependence of the supergravity action on the asymptotic behavior at infinity.
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 $\N=4$ super Yang-Mills theory in four dimensions. With some further assumptions, one can deduce a Hamiltonian version of the correspondence and show that the $\N=4$ theory has a large $N$ phase transition related to the thermodynamics of $AdS$ black holes.
8,751 citations
"Conformal basis for flat space ampl..." refers background or result in this paper
...Similar to the scalar massless conformal primary wavefunction, its spin-one analog has been obtained [4] from the spin-one bulk-to-boundary propagator in Hd+1 [36,47]....
[...]
...One last ingredient we need is the scalar bulk-to-boundary propagator G∆(p̂; ~ w) in Hd+1 [36],...
[...]
TL;DR: In this paper, Fourier transform the scattering amplitudes from momentum space to twistor space, and argue that the transformed amplitudes are supported on certain holomorphic curves, which is a consequence of an equivalence between the perturbative expansion of = 4 super Yang-Mills theory and the D-instanton expansion of a certain string theory.
Abstract: Perturbative scattering amplitudes in Yang-Mills theory have many unexpected properties, such as holomorphy of the maximally helicity violating amplitudes. To interpret these results, we Fourier transform the scattering amplitudes from momentum space to twistor space, and argue that the transformed amplitudes are supported on certain holomorphic curves. This in turn is apparently a consequence of an equivalence between the perturbative expansion of = 4 super Yang-Mills theory and the D-instanton expansion of a certain string theory, namely the topological B model whose target space is the Calabi-Yau supermanifold
1,626 citations
TL;DR: In this paper, a general definition of local symmetries on the manifold of field configurations is given that encompasses, as special cases, the usual gauge transformations of Yang-Mills theory and general relativity.
Abstract: The general relationship between local symmetries occurring in a Lagrangian formulation of a field theory and the corresponding constraints present in a phase space formulation are studied. First, a prescription—applicable to an arbitrary Lagrangian field theory—for the construction of phase space from the manifold of field configurations on space‐time is given. Next, a general definition of the notion of local symmetries on the manifold of field configurations is given that encompasses, as special cases, the usual gauge transformations of Yang–Mills theory and general relativity. Local symmetries on phase space are then defined via projection from field configuration space. It is proved that associated to each local symmetry which suitably projects to phase space is a corresponding equivalence class of constraint functions on phase space. Moreover, the constraints thereby obtained are always first class, and the Poisson bracket algebra of the constraint functions is isomorphic to the Lie bracket algebra of the local symmetries on the constraint submanifold of phase space. The differences that occur in the structure of constraints in Yang–Mills theory and general relativity are fully accounted for by the manner in which the local symmetries project to phase space: In Yang–Mills theory all the ‘‘field‐independent’’ local symmetries project to all of phase space, whereas in general relativity the nonspatial diffeomorphisms do not project to all of phase space and the ones that suitably project to the constraint submanifold are ‘‘field dependent.’’ As by‐products of the present work, definitions are given of the symplectic potential current density and the symplectic current density in the context of an arbitrary Lagrangian field theory, and the Noether current density associated with an arbitrary local symmetry. A number of properties of these currents are established and some relationships between them are obtained.
833 citations
TL;DR: In this paper, it was shown that there is a bijective correspondence between equivalence classes of asymptotic reducibility parameters and (n−2)-forms in the context of Lagrangian gauge theories.
744 citations