Spectrum of local boundary operators from boundary form factor bootstrap

Using the recently introduced boundary form factor bootstrap equations, we map the complete space of their solutions for the boundary version of the scaling Lee–Yang model and sinh-Gordon theory. We show that the complete space of solutions, graded by the ultraviolet behaviour of the form factors can be brought into correspondence with the spectrum of local boundary operators expected from boundary conformal field theory, which is a major evidence for the correctness of the boundary form factor bootstrap framework.     

Conformal field theory representation and background-independent action for superstring field theory

The two-dimensional conformal field theory representation of Witten's open superstring field theory is discussed. We argue that the previously suggested cubic action for the superstring is actually dependent on the spacetime background, and suggest a modification which formally removes this dependence. The transformation from the new version of the superstring cubic action to Witten's action for open string states is discussed. A class of linearized solutions to the equations of motion of the open superstring cubic action are exhibited. These solutions are in one-to-one correspondence with the physical excitations (both massless and massive) of the open and closed type I superstrings.     

Complex symmetries of electrodynamics

We prove that a set of nonsingular free solutions of Maxwell's equations forms a representation of the group obtained by analytic continuation of the Poincaré group to complex values of the group parameters, and that a set of singular solutions forms a representation of the group obtained by analytic continuation of the conformal group to complex values of the group parameters. These results are obtained by constructing a theory governing 2 × 2 complex matrix fields defined for complex values of position and time; the equations of this theory are invarient with respect to complex Poincaré transformations and complex conformal transformations, but the set of nonsingular solutions is in one-to-one correspondence with a set of nonsingular solutions of Maxwell's equations, and a similar correspondence exists for the singular solutions. Certain collections of solutions of Maxwell's equations for the field of a current form representations of these complex groups if both magnetic and electric currents are permitted, in which case complex transformations provide a natural connection between electric and magnetic charge. A class of complex transformations also yield natural relations between sources moving slower than light and sources moving faster than light.     

The Conformal Metric Associated with the U(1) Gauge of the Stueckelberg–Schrödinger Equation

We review the relativistic classical and quantum mechanics of Stueckelberg, and introduce the compensation fields necessary for the gauge covariance of the Stueckelbert–Schrödinger equation. To achieve this, one must introduce a fifth, Lorentz scalar, compensation field, in addition to the four vector fields with compensate the action of the space-time derivatives. A generalized Lorentz force can be derived from the classical Hamilton equations associated with this evolution function. We show that the fifth (scalar) field can be eliminated through the introduction of a conformal metric on the spacetime manifold. The geodesic equation associated with this metric coincides with the Lorentz force, and is therefore dynamically equivalent. Since the generalized Maxwell equations for the five dimensional fields provide an equation relating the fifth field with the spacetime density of events, one can derive the spacetime event density associated with the Friedmann–Robertson–Walker solution of the Einstein equations. The resulting density, in the conformal coordinate space, is isotropic and homogeneous, decreasing as the square of the Robertson–Walker scale factor. Using the Einstein equations, one see that both for the static and matter dominated models, the conformal time slice in which the events which generate the world lines are contained becomes progressively thinner as the inverse square of the scale factor, establishing a simple correspondence between the configurations predicted by the underlying Friedmann–Robertson–Walker dynamical model and the configurations in the conformal coordinates.     

On the Geometry and Mass of Static,Asymptotically AdS Spacetimes,and the Uniqueness of the AdS Soliton

We prove two theorems, announced in [6], for static spacetimes that solve Einstein's equation with negative cosmological constant. The first is a general structure theorem for spacetimes obeying a certain convexity condition near infinity, analogous to the structure theorems of Cheeger and Gromoll for manifolds of non-negative Ricci curvature. For spacetimes with Ricci-flat conformal boundary, the convexity condition is associated with negative mass. The second theorem is a uniqueness theorem for the negative mass AdS soliton spacetime. This result lends support to the new positive mass conjecture due to Horowitz and Myers which states that the unique lowest mass solution which asymptotes to the AdS soliton is the soliton itself. This conjecture was motivated by a nonsupersymmetric version of the AdS/CFT correspondence. Our results add to the growing body of rigorous mathematical results inspired by the AdS/CFT correspondence conjecture. Our techniques exploit a special geometric feature which the universal cover of the soliton spacetime shares with familiar ``ground state' spacetimes such as Minkowski spacetime, namely, the presence of a null line, or complete achronal null geodesic, and the totally geodesic null hypersurface that it determines. En route, we provide an analysis of the boundary data at conformal infinity for the Lorentzian signature static Einstein equations, in the spirit of the Fefferman-Graham analysis for the Riemannian signature case. This leads us to generalize to arbitrary dimension a mass definition for static asymptotically AdS spacetimes given by Chruciel and Simon. We prove equivalence of this mass definition with those of Ashtekar-Magnon and Hawking-Horowitz.     

QCD and a holographic model of hadrons

We propose a five-dimensional framework for modeling low-energy properties of QCD. In the simplest three parameter model we compute masses, decay rates and couplings of the lightest mesons. The model fits experimental data to within 10%. The framework is a holographic version of the QCD sum rules, motivated by the anti-de Sitter/conformal field theory correspondence. The model naturally incorporates properties of QCD dictated by chiral symmetry, which we demonstrate by deriving the Gell-Mann-Oakes-Renner relationship for the pion mass.     

On AGT relations with surface operator insertion and a stationary limit of beta-ensembles

We present a summary of what is currently known about of the AGT relations for conformal blocks with the additional insertion of the simplest degenerate operator, and a special choice of the corresponding intermediate dimension, in which the conformal blocks satisfy hypergeometric-type differential equations in the position of the degenerate operator. Special attention is devoted to the representation of the conformal block through using the beta-ensemble resolvents and to its asymptotics in the limit of large dimensions (both external and intermediate) taken asymmetrically in terms of the deformation epsilon-parameters. The next-to-leading term in the asymptotics defines the generating differential in the Bohr–Sommerfeld representation of the one-parameter deformed Seiberg–Witten prepotentials, (whose full two-parameter deformation leads to Nekrasov functions). This generating differential is also shown to be the one-parameter version of the single-point resolvent for the corresponding beta-ensemble, and its periods in the perturbative limit of the gauge theory are expressed through the ratios of the Harish–Chandra function. The Schrödinger/Baxter equations, considered earlier in this context, directly follow from the differential equations for the degenerate conformal block. This approach provides a powerful method for the evaluation of the single-deformed prepotentials, and even for the Seiberg–Witten prepotentials themselves. We primarily concentrate on the representative case of the insertion into the four-point block on a sphere and the one-point block on a torus.     

On the origin of supergravity boundary terms in the AdS /CFT correspondence

The standard formulation of the AdS/CFT correspondence is incomplete since it requires adding to a supergravity action some a priori unknown boundary terms. We suggest a modification of the correspondence principle based on the Hamiltonian formulation of the supergravity action, which does not require any boundary terms. Then all the boundary terms of the standard formulation naturally appear by passing from the Hamiltonian version to the Lagrangian one. As examples the graviton part of the supergravity action on the product of AdS_d+1 with a compact Einstein manifold ϵ and fermions on AdS_d+1, are considered. We also discuss conformal transformations of gravity fields on the boundary of AdS and show that they are induced by the isometrics of AdS.     

Induced gravity and cosmology

We propose a fully conformal invariant theory describing gravity as a spontaneously broken theory. Newton's constant is automatically generated. We find through the study of classical solutions of the equations of motion that the breakdown of conformal symmetry can take place at the tree approximation without introducing arbitrary forms for the scalar potential. Using cosmological metrics, which we find natural from the physical point of view, some conclusions can be drawn regarding the nature of those metrics. The case of constant scalar curvature is particularly interesting, and gives rise to a gravitational version of the Goldstone theorem.     

Lie algebras of conservation laws of variational ordinary differential equations

We establish a new version of the first Noether Theorem, according to which the (equivalence classes of) first integrals of given Euler–Lagrange equations in one independent variable are in exact one-to-one correspondence with the (equivalence classes of) vector fields satisfying two simple geometric conditions, namely they simultaneously preserve the holonomy distribution of the jets space and the action from which the Euler–Lagrange equations are derived.     

Global Solutions of Einstein—Dirac Equation on the Conformal Space

The difference between the Riemann and Lorentz spinor manifolds of four dimensions is that the Dirac operator of the former is elliptic and that of the latter is hyperbolic.Moreover the spinor group of the former is a compact group and that of the latter is a noncompact group,which is isomorphic to SL(2,C).Hence the results and their interpretation coming from the two theories would be different.In this short note we study only the Lorentz spinor manifold and,especially,the solutions of Einstein-Dirac equations on the conformal space,which is closely related to the AdS/CFT correspondence.     

Notes on the hidden conformal symmetry in the near horizon geometry of the Kerr black hole

Toward the Kerr/CFT correspondence for the generic non-extremal Kerr black hole, the analysis of scattering amplitudes by near extremal Kerr provides a clue. This pursuit reveals a hidden conformal symmetry in the low frequency wave equation for a scalar field in a certain spacetime region referred to as the near region. For extremal case, the near region is expected to be the near horizon region in which the correspondence via the asymptotic symmetry is studied. We investigate the hidden conformal symmetry in the near horizon limit and consider the relation between the hidden conformal symmetry and the asymptotic symmetry in the near horizon limit. By using an appropriate definition of the quasi-local charge, we obtain the deviation of the entropy from the extremality.     

Gravitational interpretation of the Hitchin equations

By referring to theorems of Donaldson and Hitchin, we exhibit a rigorous AdS/CFT-type correspondence between classical 2+1-dimensional vacuum general relativity theory on Σ×R$\Sigma \times \mathbb{R}$ and SO(3) Hitchin theory (regarded as a classical conformal field theory) on the spacelike past boundary Σ$\Sigma$, a compact, oriented Riemann surface of genus greater than 1. Within this framework we can interpret the 2+1-dimensional vacuum Einstein equation as a decoupled “dual” version of the two-dimensional SO(3) Hitchin equations.     

Conformal invariance and conserved quantity of third-order Lagrange equations for non-conserved mechanical systems

This paper studies conformal invariance and conserved quantity of third-order Lagrange equations for non-conserved mechanical systems. Third-order Lagrange equations, the definition and a determining equation of conformal invariance of the system are presented. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The necessary and sufficient condition that conformal invariance of the system would have Lie symmetry under single-parameter infinitesimal transformations is obtained. The corresponding conserved quantity of conformal invariance is derived with the aid of a structure equation. Lastly, an example is given to illustrate the application of the results.     

Gravity duals for nonrelativistic conformal field theories

We attempt to generalize the anti-de Sitter/conformal field theory correspondence to nonrelativistic conformal field theories which are invariant under Galilean transformations. Such systems govern ultracold atoms at unitarity, nucleon scattering in some channels, and, more generally, a family of universality classes of quantum critical behavior. We construct a family of metrics which realize these symmetries as isometries. They are solutions of gravity with a negative cosmological constant coupled to pressureless dust. We discuss realizations of the dust, which include a bulk superconductor. We develop the holographic dictionary and find two-point correlators of the correct form. A strange aspect of the correspondence is that the bulk geometry has two extra noncompact dimensions.     

Massless Fields on Dirac Six-Cone and De Sitter Ambient Space

We have proceeded to obtain manifestly conformally invariant (CI) equations for thinkable graviton fields in de Sitter (dS) space-time. The tensor fields are originally considered in 4+2 dimensional conformal space or Dirac’s six-cone and then project to dS space which is embedded in 4+1 dimensional ambient space. It will be shown that, by projecting these tensor fields there exists a correspondence between the massless fields on the cone and dS space. Also, we have shown that for rank-2 tensor field the divergenceless condition, which is necessary when we attempt to correspond the tensor field with the unitary irreducible representations (UIRs) of dS group, is not really a condition at all, it is a consequence of ambient space property. Due to the combined occurrences of corresponding fields and divergenceless property, the appropriate CI field equations have obtained in a fairly simple way and without imposing any extra condition.     

Theory of the conformally invariant mass

By interpreting the conformal transformations as space-time-dependent change of units and introducing the concept of the conformally invariant mass and charge, we develop new conformally invariant Maxwell equations with source terms and equations of motion for massive particles. Although the usual equations of motion with mass terms break the conformal symmetry, it is shown that the Minkowski space is not the most general framework to describe physical processes and there exists a wider consistent dynamics in which conformal invariance is exact. New results also include the general transformation laws of the electromagnetic fields, of currents and force densities. The theory leads naturally to an affine connection and to the 21-parameter inhomogeneous conformal group, ISO(4, 2).     

Hidden conformal symmetry and pair production near the cosmological horizon in Kerr-Newman-Taub-NUT-de Sitter spacetime

We show that the study of the hidden conformal symmetry that is associated with the Kerr/CFT correspondence can also apply to the cosmological horizon in the Kerr-Newman-Taub-NUT-de Sitter spacetime. This symmetry allows employing some two dimensional conformal field theory methods to understand the properties of the cosmological horizon. The entropy can be understood by using the Cardy formula, and the equation for the scattering process in the near region is in agreement with that obtained from a two point function in the two-dimensional conformal field theory. We also show that pair production can occur near the cosmological horizon in Kerr-Newman-Taub-NUT-de Sitter for near extremal conditions.