Evidence has recently emerged for a hidden symmetry of scattering amplitudes in N=4 super Yang-Mills theory called dual conformal symmetry. At weak coupling the presence of this symmetry has been observed through five loops, while at strong coupling the symmetry has been shown to have a natural interpretation in terms of a T-dualized AdS_5. In this paper we study dual conformally invariant off-shell four-point Feynman diagrams. We classify all such diagrams through four loops and evaluate 10 new off-shell integrals in terms of Mellin-Barnes representations, also finding explicit expressions for their infrared singularities.
We derive a tensorial formula for a fourth-order conformally invariant differential operator on conformal 4-manifolds. This operator is applied to algebraic Weyl tensor densities of a certain conformal weight, and takes its values in algebraic Weyl tensor densities of another weight. For oriented manifolds, this operator reverses duality: For example in the Riemannian case, it takes self-dual to anti-self-dual tensors and vice versa. We also examine the place that this operator occupies in known results on the classification of conformally invariant operators, and we examine some related opera…
Starting from the classification of 6-dimensional Drinfeld doubles and their decomposition into Manin triples we construct 3-dimensional Poisson-Lie T-dual or more precisely T-plural sigma models. Of special interest are those that are conformally invariant. Examples of models that satisfy vanishing beta-function equations with zero dilaton are presented and their duals are calculated. It turns out that for "traceless" dual algebras they satisfy the beta-function equations as well but usually with rather nontrivial dilaton. We also present explicit examples of several kinds of obstacles and di…
We consider conformally coupled scalar with $ϕ^4$ coupling in AdS$_4$ and study its various boundary conditions on AdS boundary. We have obtained perturbative solutions of equation of motion of the conformally coupled scalar with power expansion order by order in $ϕ^4$ coupling $λ$ up to $λ^2$ order. In its dual CFT, we get 2,4 and 6 point functions by using this solution with Dirichlet and Neumann boundary conditions via AdS/CFT dictionary. We also consider marginal deformation on AdS boundary and get its on-shell and boundary effective actions.
We demonstrate separability of conformally coupled scalar field equation in general (off-shell) Kerr-NUT-AdS spacetimes in all dimensions. The separability is intrinsically characterized by the existence of a complete set of mutually commuting conformal wave operators that can be constructed from a hidden symmetry of the principal Killing-Yano tensor. By token of conformal symmetry, the separability also works for any Weyl rescaled (off-shell) metrics. This is especially interesting in four dimensions where it guarantees separability of a conformally coupled scalar field in the general Plebans…
It is showed by a conformal rescaling that the inflationary background can be dual to a slowly expanding background, which is almost Minkowski and described by a conformal field theory conformally coupled to gravity. It is proved that the primordial curvature perturbation and tensor perturbation generated during these two conformally equivalent backgrounds are completely equal, and the scale invariance of perturbations is determined by the conformal invariance of field theory in slowly expanding background. In dual slowly expanding scenario, the primordial universe is asymptotical to a static …
We look for 3-dimensional Poisson-Lie dualizable sigma models that satisfy the vanishing beta-function equations with constant dilaton field. Using the Poisson-Lie T-plurality we then construct 3-dimensional sigma models that correspond to various decompositions of Drinfeld double. Models with nontrivial dilaton field may appear. It turns out that for ``traceless'' dual algebras they satisfy the vanishing beta-function equations as well. In certain cases the dilaton cannot be defined in some of the dual models. We provide an explanation why this happens and give criteria predicting when it hap…
We present a generalization of the boundary state formalism for the bosonic string that allows us to calculate the overlap of the boundary state with arbitrary closed string states. We show that this generalization exactly reproduces world-sheet sigma model calculations, thus giving the correct overlap with both on- and off-shell string states, and that this new boundary state automatically satisfies the requirement for integrated vertex operators in the case of non-conformally invariant boundary interactions.
We suggest three new ${\cal N}=1$ conformal dual pairs. First, we argue that the ${\cal N}=2$ $E_6$ Minahan-Nemeschansky (MN) theory with a $USp(4)$ subgroup of the $E_6$ global symmetry conformally gauged with an ${\cal N}=1$ vector multiplet and certain additional chiral multiplet matter resides at some cusp of the conformal manifold of an $SU(2)^5$ quiver gauge theory. Second, we argue that the ${\cal N}=2$ $E_7$ MN theory with an $SU(2)$ subgroup of the $E_7$ global symmetry conformally gauged with an ${\cal N}=1$ vector multiplet and certain additional chiral multiplet matter resides at s…
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 propose a new off-shell formulation for N-extended conformal supergravity in three spacetime dimensions. Our construction is based on the gauging of the N-extended superconformal algebra in superspace. Covariant constraints are imposed such that the algebra of covariant derivatives is given in terms of a single curvature superfield which turns out to be the super Cotton tensor. An immediate corollary of this construction is that the curved superspace is conformally flat if and only if the super Cotton tensor vanishes. Upon degauging of certain local symmetries, our formulation is shown to r…
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…
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.
We give an alternative description of the physical content of general relativity that does not require a Lorentz invariant spacetime. Instead, we find that gravity admits a dual description in terms of a theory where local size is irrelevant. The dual theory is invariant under foliation preserving 3-diffeomorphisms and 3D conformal transformations that preserve the 3-volume (for the spatially compact case). Locally, this symmetry is identical to that of Horava-Lifshitz gravity in the high energy limit but our theory is equivalent to Einstein gravity. Specifically, we find that the solutions of…
The dual conformal box integral in Minkowski space is not fully determined by the conformal invariants $z$ and $\bar{z}$. Depending on the kinematic region its value is on a 'branch' of the Bloch-Wigner function which occurs in the Euclidean case. Dual special conformal transformations in Minkowski space can change the kinematic region in such a way that the value of the integral jumps to another branch of this function, encoding a subtle breaking of dual conformal invariance for the integral. We classify conformally equivalent configurations of four points in compactified Minkowski space. We …
We construct a purely frame-like parent action that allows to dualise, at the off-shell level, an arbitrary mixed-symmetry bosonic massless fields in Minkowski background of dimensions d. Starting from any massless mixed-symmetry gauge field in the standard Skvortsov frame-like formulation and following an off-shell dualisation procedure, we obtain dual theories which are on-shell related by so(d-2) Hodge duality. The Hodge dualisation can be done on any column of the Young diagram characterizing the generalised spin of the original frame-like field. Dualisation with respect to the first colum…
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.
We present a conformally invariant generalized form of the free particle action by connecting the wave and particle aspects through gravity. Conformal invariance breaking is introduced by choosing a particular configurat$ of dynamical variables. This leads to the geometrization of the quantum aspects of matter.
A canonical action describing the interaction of chiral gauge fields in D=6 Minkowski space-time is constructed. In a particular partial gauge fixing it reduces to the action found by Perry and Schwarz. The additional gauge symmetries are used to show the off-shell equivalence of the dimensional reduction to D=5 Minkowski space-time of the chiral gauge field canonical action and the Born-Infeld canonical action describing an interacting D=5 Abelian vector field. Its extension to improve the on-shell equivalence arguments of dual D-brane actions to off-shell ones is discussed.
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 define a non-commutative product for arbitrary gauge and B-field backgrounds in terms of correlation functions of open strings. While off-shell correlations are, of course, not conformally invariant, it turns out that, at least to first derivative order, our product has the trace property and is associative up to surface terms if the background fields are put on-shell. No on-shell conditions for the inserted functions are needed, but it is essential to include the full contribution of the Born-Infeld measure. We work with a derivative expansion and avoid any topological limit, which would e…
Università degli Studi Guglielmo Marconi（古列尔莫·马可尼大学）；Sezione INFN Roma Tor Vergata（罗马托维尔加塔INFN分部）；Instituto de Ciencias Exactas y Naturales, Universidad Arturo Prat（阿图罗·普拉特大学精确与自然科学研究院）；Facultad de Ciencias, Universidad Arturo Prat（阿图罗·普拉特大学理学院）；Departamento de Física y Astronimía, Facultad de Ciencias Exactas, Universidad Andres Bello（安德烈斯·贝略大学精确理学院物理与天文学系）
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 derive a generalization of the flat space Yang's and Newman's equations for self-dual Yang-Mills fields to (locally) conformally Kahler Riemannian 4-manifolds. The results also apply to Einstein metrics (whose full curvature is not necessarily self-dual). We analyse the possibility of hidden symmetries in the form of Backlund transformations, and we find a continuous group of hidden symmetries only for the case in which the geometry is conformally half-flat. No isometries are assumed.
Conformally Einstein anti-self-dual spaces and their generalisations
Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）；School of Mathematical and Physical Sciences, University of Sheffield（谢菲尔德大学数学与物理科学学院）
A simple conformal quantum mechanics model of a d-component variable is proposed, which exactly reproduces the retarded Green functions and conformal weights of conformally coupled scalar fields in de Sitter spacetime seen by a static patch observer. It is found that the action integral of this model is automatically expressed by a complex integral over the time variable t along a closed contour in a way which is typical to the Schwinger-Keldysh formalism of a thermofield theory. Hence this model is at finite temperature. The case of conformally coupled scalar fields in 3d Schwarzschild de Sit…
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…
A 0+1-dimensional candidate theory for the CFT$_1$ dual to AdS$_2$ is discussed. The quantum mechanical system does not have a ground state that is invariant under the three generators of the conformal group. Nevertheless, we show that there are operators in the theory that are not primary, but whose "non-primary character" conspires with the "non-invariance of the vacuum" to give precisely the correlation functions in a conformally invariant theory.
Lagrangians for several new off-shell 4D, N = 1 supersymmetric descriptions of massive superspin-1 and superspin-3/2 multiplets are described. Taken together with the models previously constructed, there are now four off-shell formulations for the massive gravitino multiplet (superspin-1) and six off-shell formulations for the massive graviton multiplet (superspin-3/2). Duality transformations are derived which relate some of these dynamical systems.
In d=4 de Sitter space, novel conformally invariant photon-like theories consistently couple to charged matter. We show that these higher spin, maximal depth, partially massless systems enjoy a Maxwellian, "electric-magnetic" duality.
We examine the NS-NS corrections to the type II supergravity given by Gross and Sloan, under the linear T-duality transformations. We show that the gravity and the B-field couplings are invariant under off-shell T-duality, whereas the dilaton couplings are invariant under on-shell T-duality. Requiring the compatibility of the dilaton couplings with off-shell T-duality as a guiding principle, we extend them to include the appropriate Ricci curvature and the divergence of the B-field strength.
Conformally invariant sigma models in $D=2n$ dimensions with target non-compact O(2n,1) groups are studied. It is shown that despite the non-compact nature of the O(2n,1) groups, the classical action and Hamiltonian are positive definite. Instanton field configurations are found to correspond geometrically to conformal ``stereographic'' mappings of $R^{2n}$ into the Euclidean signature $AdS_{2n}$ spaces. Zaikov's relationship between Self Dual $p$-branes and Chern-Simons $p'$-branes, provided $p=p'+1$ and the embedding $D=p+1$-dimensional manifold is Euclidean, is elaborated further. Branes ac…
We use compatibility of the second fundamental form corrections to DBI action at order $α'^2$ which includes trace of the second fundamental form, with T-duality and with the linear S-duality as guiding principles to find an off-shell D-brane action at order $α'^2$ in type II superstring theories in flat spcetime.
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 …
We study the compatibility of the AdS/CFT duality with the bulk and boundary causality, and derive a conformally invariant averaged null energy condition (CANEC) for quantum field theories in 3 and 5-dimensional curved boundaries. This is the generalization of the averaged null energy condition (ANEC) in Minkowski spacetime to curved boundaries, where null energy is averaged along the null line with appropriate weight for conformal invariance. For this purpose we take, as our guiding principle, the no-bulk-shortcut theorem of Gao and Wald, which essentially asserts that when going from one poi…
A new topological conformal field theory in four Euclidean dimensions is constructed from N=4 super Yang-Mills theory by twisting the whole of the conformal group with the whole of the R-symmetry group, resulting in a theory that is conformally invariant and has two conformally invariant BRST operators. A curved space generalisation is found on any Riemannian 4-fold. This formulation has local Weyl invariance and two Weyl-invariant BRST symmetries, with an action and energy-momentum tensor that are BRST-exact. This theory is expected to have a holographic dual in 5-dimensional de Sitter space.
The superspace formulation for four-dimensional N = 2 matter-coupled supergravity recently developed in arXiv:0805.4683 makes use of a new type of conformal compensator with infinitely many off-shell degrees of freedom: the so-called covariant weight-one polar hypermultiplet. In the present note we prove the duality of this formulation to the known minimal (40+40) off-shell realization for N = 2 Poincare supergravity involving the improved tensor compensator. Within the latter formulation, we present new off-shell matter couplings realized in terms of covariant weight-zero polar hypermultiplet…
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.
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…
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 computed the actions for the 1D N=5 sigma-models with respect to the two inequivalent (2,8,6) multiplets. 4 supersymmetry generators are manifest, while the constraint originated by imposing the 5-th supersymmetry automatically induces a full N=8 off-shell invariance. The resulting action coincides in the two cases and corresponds to a conformally flat 2D target satisfying a special geometry of rigid type. To obtain these results we developed a computational method (for Maple 11) which does not require the notion of superfields and is instead based on the nowadays available list of the ineq…
Form factors, as quantities involving both local operators and asymptotic particle states, contain information of both the spectrum of operators and the on-shell amplitudes. So far the studies of form factors have been mostly focused on the large Nc planar limit, with a few exceptions of Sudakov form factors. In this paper, we discuss the systematical construction of full color dependent form factors with generic local operators. We study the color decomposition for form factors and discuss the general strategy of using on-shell unitarity cut method. As concrete applications, we compute the fu…
The equilibration of matter and onset of hydrodynamics can be understood in the AdS/CFT context as a gravitational collapse process, in which "collision debris" create a horizon. In this paper we consider the simplest geometry possible, a flat shell (or membrane) falling in the holographic direction toward the horizon. The metric is a combination of two well known solutions: thermal AdS above the shell and pure AdS below, while motion of the shell is given by the Israel junction condition. Furthermore, when the shell motion can be considered slow, we were able to solve for two-point functions …
We study the thermodynamics and the phase structures of Reissner-Nordstrom and Born-Infeld black holes in AdS space by constructing ``off-shell'' free energies using thermodynamic quantities derived directly from the action. We then use these results to propose ``off-shell'' effective potentials for the respective boundary gauge theories. The saddle points of the potentials describe all the equilibrium phases of the gauge theories.
We show that the primordial gravitational wave with scale-invariant spectrum might emerge from a nearly Minkowski space, in which the gravity is asymptotic-past free. We illustrate it with a model, in which the derivative of background scalar field nonminimally couples to gravity. We also show that since here the tensor perturbation is dominated by its growing mode, mathematically our slowly expanding background is conformally dual to the matter contraction, but there is no the anisotropy problem.
We review the application of a duality-symmetric approach to gravity and supergravity with emphasizing benefits and disadvantages of the formulation. Contents of these notes includes: 1) Introduction with putting the accent on the role of dual gravity within M-theory; 2) Dualization of gravity with a cosmological constant in D = 3; 3) On-shell description of dual gravity in D > 3; 4) Construction of the duality-symmetric action for General Relativity with/without matter fields; 5) On-shell description of dual gravity in linearized approximation; 6) Brief summary of the paper.
