Using on-shell amplitude methods, we explore 4-dimensional Electric-Magnetic duality and its double copy. We show explicitly that the on-shell scattering amplitudes know about `dual' photons (and dual gravitons), that the off-shell photon propagator double copies to the graviton propagator and that the magnetic part of the propagator is essential for the double copy to hold. We also show that there is an equivalent gravito-magnetic part of the graviton propagator which is essential in giving rise to solutions with either angular momentum or NUT charge. Furthermore, we comment on the so-called Weinberg paradox, which states that scattering amplitudes involving the mixing of electric and magnetic monopoles cannot be Lorentz invariant, and would seem to preclude the existence of the 't Hooft-Polyakov (topological) monopole. We trace this paradox to the magnetic part of the propagator, showing that it can be eliminated if one restricts to proper orthochronous Lorentz transformations. Finally, we compute the fully relativistic cross-section for arbitrary spin dyons using the recently formulated on-shell duality transformation and show that this is always fully Lorentz invariant.
We review the status of electric/magnetic duality for free gauge field theories in four space-time dimensions with emphasis on Maxwell theory and linearized Einstein gravity. Using the theory of vector and tensor spherical harmonics, we provide explicit construction of dual photons and gravitons by decomposing the fields into axial and polar configurations with opposite parity and interchanging the two sectors. When the theories are defined on AdS(4) space-time there are boundary manifestations of the duality, which for the case of gravity account for the energy-momentum/Cotton tensor duality …
The linearized Einstein equations in D spacetime dimensions can be written as twisted self-duality equations expressing that the linearized curvature tensor of the graviton described by a rank-two symmetric tensor, is dual to the linearized curvature tensor of the "dual graviton" described by a tensor of (D-3,1) Young symmetry type. In the case of 4 dimensions, both the graviton and its dual are rank-two symmetric tensors (Young symmetry type (1,1)), while in the case of 11 space-time dimensions relevant to M-theory, the dual graviton is described by a tensor of (8,1) Young symmetry type. We p…
Gravity is now understood to become simple on-shell. We sketch how it becomes simple also off-shell, when reformulated appropriately. Thus, we describe a simple Lagrangian for gravitons that makes use of a certain complex of differential operators. The Lagrangian is constructed analogously to that of Maxwell's theory, just using a different complex. The complex, and therefore also our description of gravitons, makes sense on any half-conformally flat four-dimensional manifold.
In this talk we review the work in [1,2,3] where we have studied the applicability of the color-kinematics duality to the scattering of two distinguishable scalar matter particles with one gluon emission in QCD, or one graviton emission in Einstein gravity. We have shown that the duality works well in the Regge limit under two different extensions of the gauge theory: the introduction of a new scalar contact interaction and the relaxation of the distinguishability of the scalars. Both modifications correspond to theories obtained by dimensional reduction from higher-dimensional pure gauge theo…
Entire Four-Graviton EFT from the Duality Between Color and Kinematics
Northwestern University（西北大学）；Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA)（天体物理跨学科探索与研究中心）；University of Chicago（芝加哥大学）
We review (and elaborate on) the `dual graviton problem' in the context of duality covariant formulations of M-theory (exceptional field theories). These theories require fields that are interpreted as components of the dual graviton in order to build complete multiplets under the exceptional groups E$_{d(d)}$, $d=2,\ldots, 9$. Circumventing no-go arguments, consistent theories for such fields have been constructed by incorporating additional covariant compensator fields. The latter fields are needed for gauge invariance but also for on-shell equivalence with $D=11$ and type IIB supergravity. …
An M2/M5 Duality from the Giant Graviton Expansion
National Taiwan University（国立台湾大学）；University of Cambridge（剑桥大学）；Osaka Metropolitan University（大阪公立大学）；Imperial College London（伦敦帝国学院）；Institute of Physics of the Czech Academy of Sciences（捷克科学院物理研究所）
We construct the off-shell recursion for gravity and the graviton current for the perturbative double field theory (DFT). We first formulate the perturbative DFT, which is equivalent but simpler to perturbative general relativity, to all-orders in fluctuations of generalised metric. The perturbative action and equations of motion (EoM) are derived to arbitrary order for pure gravity case. We then derive the graviton off-shell recursion, the gravity counterpart of the Berends-Giele recursion in Yang-Mills theory, through the so-called perturbiner method using the EoM of the perturbative DFT. We…
We argue that gravity theories in AdS4 are holographically dual to either of two three-dimensional CFT's: the usual Dirichlet CFT1 where the fixed graviton acts as a source for the stress-energy tensor, and a dual CFT2 with a fixed dual graviton which acts as a source for a dual stress-energy tensor. The dual stress-energy tensor is shown to be the Cotton tensor of the Dirichlet CFT. The two CFT's are related by a Legendre transformation generated by a gravitational Chern-Simons coupling. This duality is a gravitational version of electric-magnetic duality valid at any radius r, where the reno…
In the gauge-theoretic formulation of gravity the cubic vertex becomes simple enough for some graviton scattering amplitudes to be computed using Berends-Giele-type recursion relations. We present such a computation for the current with all same helicity on-shell gravitons. Once the recursion relation is set up and low graviton number cases are worked out, a natural guess for the solution in terms of a sum over trees presents itself readily. The solution can also be described either in terms of the half-soft function familiar from the 1998 paper by Bern, Dixon, Perelstein and Rozowsky or as a …
S and U-duality dictate that graviton scattering amplitudes in IIB superstring theory be automorphic functions on the appropriate fundamental domain which describe the inequivalent vacua of (compactified) theories. A constrained functional form of graviton scattering is proposed using Eisenstein series and their generalizations compatible with: a) two-loop supergravity, b) genus one superstring theory, c) the perturbative coupling dependence of the superstring, and d) with the unitarity structure of the massless modes. The form has a perturbative truncation in the genus expansion at a given or…
In a spacetime of dimension n, the dual graviton is characterised by a Young diagram with two columns, the first of length n-3 and the second of length one. In this paper we perform the off-shell dualisation relating the dual graviton to the double-dual graviton, displaying the precise off-shell field content and gauge invariances. We then show that one can further perform infinitely many off-shell dualities, reformulating linearised gravity in an infinite number of equivalent actions. The actions require supplementary mixed-symmetry fields which are contained within the generalised Kac-Moody …
We study the renormalization group of generic effective field theories that include gravity. We follow the on-shell amplitude approach, which provides a simple and efficient method to extract anomalous dimensions avoiding complications from gauge redundancies. As an invaluable tool we introduce a modified helicity $\tilde{h}$ under which gravitons carry one unit instead of two. With this modified helicity we easily explain old and uncover new non-renormalization theorems for theories including gravitons. We provide complete results for the one-loop gravitational renormalization of a generic mi…
In part 1 of this series compact integral representations had been obtained for the one-loop photon-graviton amplitude involving a charged spin 0 or spin 1/2 particle in the loop and an arbitrary constant electromagnetic field. In this sequel, we study the structure and magnitude of the various polarization components of this amplitude on-shell. Explicit expressions are obtained for a number of limiting cases.
We derive two new equations of quantum gravity and combine them with reinterpretations of previously proposed concepts of dark energy, inflation and black holes into a theory which may be a first step toward a comprehensive description of all three phenomena. The resulting theory also predicts new tests which can be experimentally checked within just a few years. The two new equations are : A) a creation equation to give stimulated emission for any surface filled with gravitons, pulling energy from a background, and B) the association of an outgoing soliton wave of gravitons, a "shell front" w…
We show that the trace anomaly for gravitons calculated using the usual effective action formalism depends on the choice of gauge when the background spacetime is not a solution of the classical equation of motion, that is, when off-shell. We then use the gauge independent Vilkovisky-DeWitt effective action to restore gauge independence to the off-shell case. Additionally we explicitly evaluate trace anomalies for some N-sphere background spacetimes.
We investigate tree-level five-point amplitudes in scalar-QCD expressed in terms of Sudakov variables and find the equivalent "gravitational" counterparts using the color-kinematics duality proposed by Bern, Carrasco, and Johannson. Taking the multi-Regge limit in the gravitational amplitudes, we show that those pieces in the coupling of two reggeized gravitons to one on-shell graviton directly stemming from the double copy of the vertex for two reggeized gluons to one on-shell gluon are universal and properly reproduced by the duality.
We provide a unified treatment of electric-magnetic duality, at the action level and with manifest Lorentz invariance, for massive, massless as well as partially-massless gravitons propagating in maximally symmetric spacetimes of any dimension n>3. For massive and massless fields, we complete previous analyses that use parent-action techniques by giving dual descriptions that enable direct counting of physical degrees of freedom in the flat and massless limit. The same treatment is extended to the partially-massless case, where the duality has been previously discussed in covariant form only a…
We continue to develop the pure connection formalism for gravity. We derive the Feynman rules for computing the connection correlation functions, as well as the prescription for obtaining the Minkowski space graviton scattering amplitudes from the latter. The present formalism turns out to be simpler than the metric based one in many aspects. Simplifications result from the fact that the conformal factor of the metric, a source of complications in the usual approach, does not propagate in the connection formulation even off-shell. This simplifies both the linearized theory and the interactions…
It has been speculated that the S-matrix elements of type IIB superstring theory satisfy the Ward identity associated with the S-duality. This indicates that a group of S-matrix elements at each loop level are invariant under the linear $SL(2,R)$ transformations. If one evaluates one element of such S-dual multiplet, then all other components can be found by the simple use of the linear $SL(2,R)$ transformations. In this paper, we calculate the disk-level S-matrix element of one graviton(dilaton), one B-field and one gauge boson on the word volume of D$_3$-brane. The S-dual multiplet correspon…
For the free massless spin-one and spin-two field theories one may write the action in a form which is manifestly invariant under electric-magnetic duality. This is achieved by introducing new potentials through solving the constraints of the Hamiltonian formulation. The price for making electric-magnetic duality invariance manifest through this direct procedure is losing manifest Lorentz invariance. Both theories admit supersymmetric extensions, which make the bosonic fields and their corresponding fermionic partners to be parts of the same geometrical object, a supermultiplet. We present in …
Modern on-shell S-matrix methods may dramatically improve our understanding of perturbative quantum gravity, but current foundations of on-shell techniques for General Relativity still rely on off-shell Feynman diagram analysis. Here, we complete the fully on-shell proof of Ref.~\cite{ST} that the recursion relations of Britto, Cachazo, Feng, and Witten (BCFW) apply to General Relativity tree amplitudes. We do so by showing that the surprising requirement of "bonus" $z^{-2}$ scaling under a BCFW shift directly follows from Bose-symmetry. Moreover, we show that amplitudes in generic theories su…
We compute the graviton two scalar off-shell interaction vertex at tree level in Type IIB superstring theory on the pp-wave background using the light-cone string field theory formalism. We then show that the tree level vertex vanishes when all particles are on-shell and conservation of p_{+} and p_{-} are imposed. We reinforce our claim by calculating the same vertex starting from the corresponding SUGRA action expanded around the pp-wave background in the light-cone gauge.
We extend the argument presented by Benincasa, Boucher-Veronneau, and Cachazo to show that graviton tree amplitudes are well behaved under large complex deformations of the momenta of a pair of like-helicity gravitons. This shows that BCFW recursion relations for gravity amplitudes can be constructed using such shifts, providing an alternative proof to the recent one by Arkani-Hamed and Kaplan. By using auxiliary recursion relations the cancellations which are hidden when using covariant Feynman diagrams become manifest.
We provide a new efficient diagrammatic tool, in the context of the scattering equations, for computation of covariant $D$-dimensional tree-level $n$-point amplitudes with pairs of spinning massive particles using compact exponential numerators. We discuss how this framework allows non-integer spin extensions of recurrence relations for amplitudes developed for integer spin. Our results facilitate the on-going program for generating observables in classical general relativity from on-shell tree amplitudes through the Kawai-Lewellen-Tye relations and generalized unitarity.
We demonstrate that the use of analytical on-shell methods involving calculation of the discontinuity across the t-channel cut associated with the exchange of a pair of massless particles (photons or gravitons) can be used to evaluate one-loop contributions to electromagnetic and gravitational scattering, with and without polarizability, reproducing via simple algebraic manipulations, results obtained previously, generally using Feynman diagram techniques. In the gravitational case the use of factorization permits a straightforward and algebraic calculation of higher order scattering without c…
We argue that if the Newtonian gravitational field of a body can mediate entanglement with another body, then it should also be possible for the body producing the Newtonian field to entangle directly with on-shell gravitons. Our arguments are made by revisiting a gedankenexperiment previously analyzed by Belenchia et al., which showed that a quantum superposition of a massive body requires both quantized gravitational radiation and local vacuum fluctuations of the spacetime metric in order to avoid contradictions with complementarity and causality. We provide a precise and rigorous descriptio…
We investigate quasilocal tachyon condensation by using gravity/gauge duality. In order to cure the IR divergence due to a tachyon, we introduce two regularization schemes: AdS space and a d=10 Schwarzschild black hole in a cavity. These provide stable canonical ensembles and thus are good candidates for the endpoint of tachyon condensation. Introducing the Cardy-Verlinde formula, we establish the on-shell gravity/gauge duality. We propose that the stringy geometry resulting from the off-shell tachyon dynamics matches onto the off-shell AdS black hole, where "off-shell" means non-equilibrium c…
It is well known that the classical equations of motion of Maxwell and Born-Infeld theories are invariant under a duality symmetry acting on the field strengths. We review the implementation of the SL(2,Z) duality in these theories as linear but non-local transformations of the potentials.
The low-energy limit of the massless two-loop five-point amplitudes for both type IIA and type IIB superstrings is computed with the pure spinor formalism and its overall coefficient determined from first principles. For the type IIB theory, the five-graviton amplitude is found to be proportional to its tree-level counterpart at the corresponding order in $α'$. Their ratio ties in with expectations based on S-duality since it matches the same modular function $E_{5/2}$ which relates the two-loop and tree-level four-graviton amplitudes. For R-symmetry violating states, the ratio between tree-le…
The non-linear duality relation between the gravity and dual gravity fields are found in E theory by carrying out $E_{11}$ variations of previously found duality relations. We also find the dual graviton equation of motion up to the addition of some very specific terms whose coefficients are not determined. Using the calculations in this paper this ambiguity was resolved in reference [15] where the full non-linear dual gravity equation was found. As a result the equations of motion in E theory have now been found at the full non-linear level up to, and including, level three, which contains th…
In this paper we study the coupling of scalar (Higgs) particles ($ϕ$) with gravitons ($h$) and their possible effects. The general form of the 3-point interaction $ϕ(p) h(1)h(2)$ can be derived using the scaling behavior of the spinor variables under the little group; the resulting vertices exhibit such simplicity, that some simplifications should be hidden in the expressions obtained from the extended scalar action. To investigate this, we study an extended Einstein-Hilbert action that besides the minimal coupling, it also includes terms of the form $ϕR^2$, $ϕR^{μν} R_{μν}$ and $ϕR^{μνρσ} R_{…
In this paper, we examine the Dirac monopole in the framework of Off-Shell Electromagnetism, the five dimensional U(1) gauge theory associated with Stueckelberg-Schrodinger relativistic quantum theory. After reviewing the Dirac model in four dimensions, we show that the structure of the five dimensional theory prevents a natural generalization of the Dirac monopole, since the theory is not symmetric under duality transformations. It is shown that the duality symmetry can be restored by generalizing the electromagnetic field strength to an element of a Clifford algebra. Nevertheless, the genera…
We give the manifestly T-dual formulation of the massless sector of the classical 3D Type II superstring in off-shell 3D N=2 superspace, including the action. It has a simple relation to the known superspace of 4D N=1 supergravity in 4D M-theory via 5D F-theory. The pre potential appears as part of the vielbein, without derivatives.
The classical equations of motion of Maxwell and Born-Infeld theories are known to be invariant under a duality symmetry acting on the field strengths. We implement the SL(2,Z) duality in these theories as linear but non-local transformations on the potentials. We show that the action and the partition function in the Hamiltonian formalism are modular invariant in any gauge. For the Born-Infeld theory we find that the longitudinal part of the fields have to be complexified.
We obtain T--duality invariant second order differential equations satisfied by the $D^8\mathcal{R}^4$ and $D^{10} \mathcal{R}^4$ interactions from the low energy expansion of the one loop four graviton amplitude in toroidally compactified type II string theory. The eigenvalues of these equations are completely determined by the structure of the one loop integrands. Unlike the BPS interactions, these non--BPS interactions satisfy Poisson equations having source terms that receive contributions from both the bulk and boundary of the worldsheet moduli space. We explicitly solve these equations i…
We discuss the calculation of one-loop amplitudes in N=8 supergravity using MHV diagrams. In contrast to MHV amplitudes of gluons in Yang-Mills, tree-level MHV amplitudes of gravitons are not holomorphic in the spinor variables. In order to extend these amplitudes off shell, and use them as vertices to build loops, we introduce certain shifts for the spinor variables associated to the loop momenta. Using this off-shell prescription, we rederive the four-point MHV amplitude of gravitons at one loop, in complete agreement with known results. We also discuss the extension to the case of one-loop …
We review equivalent formulations of nonlinear and higher derivatives theories of electromagnetism exhibiting electric-magnetic duality rotations symmetry. We study in particular on shell and off shell formulations of this symmetry, at the level of action functionals as well as of equations of motion. We prove the conjecture that the action functional leading to Born-Infeld nonlinear electromagnetism, that is duality rotation invariant off shell and that is known to be a root of an algebraic equation of fourth order, is a hypergeometric function.
It was established long ago that SO(2) electric-magnetic duality is an {\em off-shell} symmetry of the free Maxwell theory, i.e., that it leaves invariant the action and not just the equations of motion. We review here that analysis and extend it to the Maxwell field coupled to scalar fields defined on the $SL(2,\mathbb{R})/SO(2)$ coset space, showing that $SL(2,\mathbb{R})$ is in that case an {\em off-shell} symmetry. We also show how the result can be generalized to many Maxwell fields and $Sp(2n, \mathbb{R})$ duality symmetry - or a subgroup of it, recovering in particular the case of maxim…
A sigma model with four-dimensional target space parametrized by chiral and twisted chiral N=(2,2) superfields can be extended to N=(4,4) supersymmetry off-shell, but this is not true for a model of semichiral fields, where the N=(4,4) supersymmetry can only be realized on-shell. The two models can be related to each other by T-duality. In this paper we perform a duality transformation from a chiral and twisted chiral model with off-shell N=(4,4) supersymmetry to a semichiral model. We find that additional non-linear terms must be added to the original transformations to obtain a semichiral mo…
Dynamical origin of duality between gauge theory and gravity is studied using the dual transformation and the formation of graviton as a collective excitation of dual gauge bosons. In this manner, electric-magnetic duality in gauge theory is reduced to the duality between gauge theory and gravity.
In this paper, we study the tree amplitudes with gluons coupled to gravitons. We first study the relations among the mixed amplitudes. With BCFW on-shell recursion relation, we will show the color-order reversed relation, $U(1)$-decoupling relation and KK relation hold for tree amplitudes with gluons coupled to gravitons. We then study the disk relation which expresses mixed amplitudes by pure gluon amplitudes. More specifically we will prove the disk relation for mixed amplitudes with gluons coupled to one graviton. Using the disk relation and the properties of pure gluon amplitudes, the colo…
Many gauge theories possess a hidden duality between color and kinematics in their on-shell scattering amplitudes. An open problem is to formulate an off-shell realization of the duality, thus manifesting a kinematic algebra. We show that 3D Chern-Simons (CS) theory in Lorenz gauge obeys off-shell color-kinematics duality. This holds both for the gauge field and the BRST ghosts, and the duality is manifest in the Feynman rules. A kinematic algebra can be formulated through a second-order differential operator (Poisson bracket) acting on the off-shell fields, and it corresponds to 3D volume-pre…
We elaborate on the color-kinematics duality for off-shell diagrams in gauge theories coupled to matter, by investigating the scattering process $gg\to ss, q\bar q, gg$, and show that the Jacobi relations for the kinematic numerators of off-shell diagrams, built with Feynman rules in axial gauge, reduce to a color-kinematics violating term due to the contributions of sub-graphs only. Such anomaly vanishes when the four particles connected by the Jacobi relation are on their mass shell with vanishing squared momenta, being either external or cut particles, where the validity of the color-kinema…
We define a perturbatively calculable quantity--the on-shell correlator--which furnishes a unified description of particle dynamics in curved spacetime. Specializing to the case of flat and anti-de Sitter space, on-shell correlators coincide precisely with on-shell scattering amplitudes and boundary correlators, respectively. Remarkably, we find that symmetric manifolds admit a generalization of on-shell kinematics in which the corresponding momenta are literally the isometry generators of the spacetime acting on the external kinematic data. These isometric momenta are intrinsically non-commut…
Tree-level scattering amplitudes for a scalar particle coupled to an arbitrary number N of photons and a single graviton are computed. We employ the worldline formalism as the main tool to compute the irreducible part of the amplitude, where all the photons and the graviton are directly attached to the scalar line, then derive a tree replacement rule to construct the reducible parts of the amplitude which involve irreducible pure N-photon two-scalar amplitudes where one photon line emits the graviton. We test our construction by verifying the on-shell gauge and diffeomorphism Ward identities, …
