Special Conformal Transformations Research Articles

Bubble universe from flat spaces

We show by matching two flat spaces one in Minkowski coordinates ( empty space) and the other in Minkowski coordinates after a special conformal transformation (also empty space) through a bubble with positive and constant surface tension, that the motion of the bubble is hyperbolic. If the surface tension is very big the initial size of the bubble is as small as we wish, so that we can indeed obtain an infinite universe out of empty spaces. The induced space in the bubble is de Sitter type.

From Homogeneous and Isotropic Universes to Braneworlds with Dynamical Tension Strings

Cosmological solutions are studied in the context of the modified measure formulation of string theory, then the string tension is a dynamical variable and the string the tension is an additional dynamical degree of freedom and its value is dynamically generated. These tensions are then not universal, rather each string generates its own tension which can have a different value for each of the string world sheets and in an ensemble of strings the values of the tensions can have a certain dispersion. We consider a new background field that can couple to these strings, the “tension scalar” which is capable of changing locally along the world sheet and then the value of the tension of the string changes accordingly. When many types of strings probing the same region of space are considered this tension scalar is constrained by the requirement of quantum conformal invariance. For the case of two types of strings probing the same region of space with different dynamically generated tensions, there are two different metrics, associated to the different strings. Each of these metrics have to satisfy vacuum Einstein’s equations and the consistency of these two Einstein’s equations determine the tension scalar. The universal metric, common to both strings generically does not satisfy Einstein’s equation. The two string dependent metrics considered here are flat space in Minkowski space and Minkowski space after a special conformal transformation. The limit where the two string tensions are the same is studied, it leads to a well defined solution. If the string tension difference between the two types of strings is very small but finite, the approximately homogeneous and isotropic cosmological solution lasts for a long time, inversely proportional to the string tension difference and then the homogeneity and isotropy of the cosmology disappears and the solution turns into an expanding Braneworld where the strings are confined between two expanding bubbles separated by a very small distance at large times.

Even the photon propagator must break de Sitter symmetry

The propagator for the massless vector field in de Sitter space cannot maintain de Sitter invariance in the general covariant gauge, except in the exactly transverse limit. This is due to a previously overlooked Ward-Takahashi identity that the propagator must satisfy. We construct the propagator that satisfies all the necessary conditions, and that preserves cosmological symmetries and dilation, but breaks spatial special conformal transformations. Our propagator vanishes in the infrared, contrary to previously reported de Sitter invariant results, which is relevant for loop computations.

Interpolating boundary conditions on AdS2

We consider two instances of boundary conditions for massless scalars on AdS2 that interpolate between the Dirichlet and Neumann cases while preserving scale invariance. Assessing invariance under the full SL(2; ℝ) conformal group is not immediate given their non-local nature. To further clarify this issue, we compute holographically 2- and 4-point correlation functions using the aforementioned boundary conditions and study their transformation properties. Concretely, motivated by the dual description of some multi-parametric families of Wilson loops in ABJM theory, we look at the excitations of an open string around an AdS2 ⊂ AdS4 × ℂℙ3 worldsheet, thus obtaining correlators of operators inserted along a 1-dimensional defect in mathcal{N} = 6 super Chern-Simons-matter theory at strong coupling. Of the two types of boundary conditions analyzed, only one leads to the expected functional structure for conformal primaries; the other exhibits covariance under translations and rescalings but not under special conformal transformations.

Fluid-electromagnetic helicities and knotted solutions of the fluid-electromagnetic equations

In this paper we consider an Euler fluid coupled to external electromagnetism. We prove that the Hopfion fluid-electromagnetic knot, carrying fluid and electromagnetic (EM) helicities, solves the fluid dynamical equations as well as the Abanov Wiegmann (AW) equations for helicities, which are inspired by the axial-current anomaly of a Dirac fermion. We also find a nontrivial knot solution with truly interacting fluid and electromagnetic fields. The key ingredients of these phenomena are the EM and fluid helicities. An EM dual system, with a magnetically charged fluid, is proposed and the analogs of the AW equations are written down. We consider a fluid coupled to a nonlinear generalizations for electromagnetism. The Hopfions are shown to be solutions of the generalized equations. We write down the formalism of fluids in 2+1 dimensions, and we dimensionally reduce the 3+1 dimensional solutions. We determine the EM knotted solutions, from which we derive the fluid knots, by applying special conformal transformations with imaginary parameters on un-knotted null constant EM fields.

Large gauge transformations, local coordinates and cosmological observables

In recent years new types of coordinate transformations have appeared in cosmology on top of the standard gauge transformations, such as the dilatations and special conformal transformations, or the ones leading to (conformal) Fermi coordinates. Some of these can remove effects that are invariant under the standard gauge transformations and also affect asymptotic boundary conditions, thus introducing a non-trivial ambiguity in our cosmological modeling. In this short note we point out that this ambiguity is irrelevant for the quantities we use to compare our model with observations—the cosmological observable relations—as they are invariant under all of these transformations. Importantly, this invariance holds only if one takes into account all the relativistic contributions to an observable, which is not the case in the literature in general. We finally also show that the practically-relevant property of conformal Fermi coordinates (a Friedmann-Lemaître-Robertson-Walker (FLRW) metric up to second order in distance) can be achieved through a globally-defined standard gauge transformation.

Life of the homogeneous and isotropic universe in dynamical string tension theories

Cosmological solutions are studied in the context of the modified measure formulation of string theory , then the string tension is a dynamical variable and the string the tension is an additional dynamical degree of freedom and its value is dynamically generated. These tensions are then not universal, rather each string generates its own tension which can have a different value for each of the string world sheets and in an ensemble of strings. The values of the tensions can have a certain dispersion in the ensemble. We consider a new background field that can couple to these strings, the “tension scalar” which is capable of changing locally along the world sheet and then the value of the tension of the string changes accordingly. When many types of strings probing the same region of space are considered this tension scalar is constrained by the requirement of quantum conformal invariance. For the case of two types of strings probing the same region of space with different dynamically generated tensions, there are two different metrics, associated to the different strings. Each of these metrics have to satisfy vacuum Einstein’s equations and the consistency of these two Einstein’s equations determine the tension scalar. The universal metric, common to both strings generically does not satisfy Einstein’s equation . The two string dependent metrics considered here are flat space in Minkowski space and Minkowski space after a special conformal transformation. The limit where the two string tensions are the same is studied, it leads to a well defined solution. If the string tension difference between the two types of strings is very small but finite, the approximately homogeneous and isotropic cosmological solution lasts for a long time, inversely proportional to the string tension difference and then the homogeneity and and isotropy of the cosmological disappears and the solution turns into an expanding braneworld where the strings are confined between two expanding bubbles separated by a very small distance at large times. The same principle is applied to the static end of the universe wall solution that lasts a time inversely proportional to the dispersion of string tensions. This suggest a scenario where quantum fluctuations of the cosmological or static solutions induce the evolution towards braneworld scenarios and decoherence between the different string tension states.

Brane world creation from flat or almost flat space in dynamical tension string theories

There is great interest in the construction of brane worlds, where matter and gravity are forced to be effective only in a lower dimensional surface, the brane . How these could appear as a consequence of string theory is a crucial question and this has been widely discussed. Here we will examine a distinct scenario that appears in dynamical tension theories and where string tension is positive between two surfaces separated by a short distance and at the two surfaces themselves the string tensions become infinite, therefore producing an effective confinement of the strings and therefore of all matter and gravity to the space between these to surfaces, which is in fact a new type of stringy brane world scenario. The basic formulation for obtaining this scenario consist of assuming two types of strings characterized by a different constant of integration related to the spontaneous string tension generation. These string tension multiplied by the embedding metric define conformally related metrics that both satisfy Einsteins equation. The braneworlds appear very naturally when these two metrics are both flat spaces related by a special conformal transformation. The two types of string tensions are determined and they blow up at two close expanding surfaces. A puzzling aspect appears then: the construction is based on flat spaces, but then there are also strings with very large tension near the boundaries of the braneworld,so can the back reaction from the infinite tension strings destroy the flat space background? Fortunatelly that can be resolved using the mechanism Universe creation from almost flat (or empty) spaces, which incorporates a gas of very large string tensions in a membrane, studied before in 1+1 membranes in a 2+1 embedding space and now is generalized for a 1+(D-2) membrane moving in a 1+(D-1) space.

Spinors and conformal correlators

In conformal field theory, momentum eigenstates can be parameterized by a pair of real spinors, in terms of which special conformal transformations take a simpler form. This well-known fact allows to express 2-point functions of primary operators in the helicity basis, exposing the consequences of unitarity. What is less known is that the same pair of spinors can be used, together with a pair of scalar quantities, to parameterize 3-point functions. We develop this formalism in 3 dimensions and show that it provides a simple realization of the operator product expansion (OPE) for scalar primary operators acting on the vacuum.

Light like segment compactification and braneworlds with dynamical string tension

There is great interest in the construction of brane worlds, where matter and gravity are forced to be effective only in a lower dimensional surface, the brane . How these could appear as a consequence of string theory is a crucial question and this has been widely discussed. Here we will examine a distinct scenario that appears in dynamical string tension theories and where string tension is positive between two surfaces separated by a short distance and at the two surfaces themselves the string tensions become infinite, therefore producing an effective confinement of the strings and therefore of all matter and gravity to the space between these to surfaces, which is in fact a new type of stringy brane world scenario. The specific model studied is in the context of the modified measure formulation the string where tension appear as an additional dynamical degree of freedom and these tensions are not universal, but rather each string generates its own tension, which can have a different value for each string. We consider a new background field that can couple to these strings, the tension scalar is capable then of changing locally along the world sheet and then the value of the tension of the extended object changes accordingly. When many types of strings probing the same region of space are considered this tension scalar is constrained by the requirement of quantum conformal invariance. For the case of two types of strings probing the same region of space with different dynamically generated tensions, there are two different metrics, associated to the different strings, that have to satisfy vacuum Einsteins equations and the consistency of these two Einsteins equations determine the tension scalar. The universal metric, common to both strings generically does not satisfy Einsteins equation . The two metrics considered here are flat space in Minkowshi space and flat space after a special conformal transformation and the tension field behaves in such a way that strings are confined inside a light like Segment or alternatively as expanding Braneworlds where the strings are confined between two expanding bubbles separated by a very small distance at large times.

Restricted Weyl symmetry and spontaneous symmetry breakdown of conformal symmetry

In this paper, we elucidate the relation between the restricted Weyl symmetry and spontaneous symmetry breakdown of conformal symmetry. Using a scalar–tensor gravity, we show that the restricted Weyl symmetry leads to spontaneous symmetry breakdown of a global scale symmetry when the vacuum expectation value of a scalar field takes a nonzero value. It is then shown that this spontaneous symmetry breakdown induces spontaneous symmetry breakdown of special conformal symmetry in a flat Minkowski spacetime, but the resultant Nambu–Goldstone boson is not an independent physical mode but expressed in terms of the derivative of the dilaton which is the Nambu–Goldstone boson of the global scale symmetry. In other words, the theories which are invariant under the general coordinate transformation and the restricted Weyl transformation exhibit a Nambu–Goldstone phase where both special conformal transformation and dilatation are spontaneously broken while preserving the Poincaré symmetry.

On the conformal symmetry of exceptional scalar theories

The DBI and special galileon theories exhibit a conformal symmetry at unphysical values of the spacetime dimension. We find the Lagrangian form of this symmetry. The special conformal transformations are non-linearly realized on the fields, even though conformal symmetry is unbroken. Commuting the conformal transformations with the extended shift symmetries, we find new symmetries, which when taken together with the conformal and shift symmetries close into a larger algebra. For DBI this larger algebra is the conformal algebra of the higher dimensional bulk in the brane embedding view of DBI. For the special galileon it is a real form of the special linear algebra. We also find the Weyl transformations corresponding to the conformal symmetries, as well as the necessary improvement terms to make the theories Weyl invariant, to second order in the coupling in the DBI case and to lowest order in the coupling in the special galileon case.

The dual conformal box integral in Minkowski space

The dual conformal box integral in Minkowski space is not fully determined by the conformal invariants z and 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 show that starting with any configuration, one can reach up to four branches of the integral using dual special conformal transformations. We also show that most configurations with real z and z¯ can be conformally mapped to a configuration in the same kinematic region with two points at infinity, where the box integral can be calculated directly in Minkowski space using only the residue theorem.

Coset Space Construction for the Conformal Group: Spontaneously Broken Phase and Inverse Higgs Phenomenon

We establish a mathematically rigorous way to construct effective theories resulting from the spontaneous breaking of conformal invariance. We show that the Namby-Goldstone field corresponding to spontaneously broken generators of special conformal transformations is always a nondynamical degree of freedom. We prove that the developed approach and the standard approach including application of the inverse Higgs mechanism are equivalent.

Electric charge in hyperbolic motion: the special conformal transformation solution

As a simple application of special conformal transformations, we derive the electromagnetic field produced by an electric charge in hyperbolic motion. Unlike other purely algebraic derivations, here we develop a more intuitive geometrical description, based on Minkowski diagrams.

Two-loop evolution equation for the B -meson distribution amplitude

We derive the two-loop evolution equation of the B-meson light-cone distribution amplitude which is the last missing element for the next-to-next-to-leading logarithmic resummation of QCD corrections to B decays in QCD factorization. We argue that the evolution kernel to all orders in perturbation theory can be written as a logarithm of the generator of special conformal transformations times the cusp anomalous dimension, up to a scheme-dependent overall constant. Up to this constant term, the evolution kernel to a given order in perturbation theory can be obtained from the calculation of special conformal anomaly at one order less.

General relativity as a biconformal gauge theory

We consider the conformal group of a space of dimn=p+q, with SO(p,q) metric. The quotient of this group by its homogeneous Weyl subgroup gives a principal fiber bundle with 2n-dim base manifold and Weyl fibers. The Cartan generalization to a curved 2n-dim geometry admits an action functional linear in the curvatures. Because symmetry is maintained between the translations and the special conformal transformations in the construction, these spaces are called biconformal; this same symmetry gives biconformal spaces overlapping structures with double field theories, including manifest T-duality. We establish that biconformal geometry is a form of double field theory, showing how general relativity with integrable local scale invariance arises from its field equations. While we discuss the relationship between biconformal geometries and the double field theories of T-dual string theories, our principal interest is the study of the gravity theory. We show that vanishing torsion and vanishing co-torsion solutions to the field equations overconstrain the system, implying a trivial biconformal space. With co-torsion unconstrained, we show that (1) the torsion-free solutions are foliated by copies of an n-dim Lie group, (2) torsion-free solutions generically describe locally scale-covariant general relativity with symmetric, divergence-free sources on either the co-tangent bundle of n-dim (p,q)-spacetime or the torus of double field theory, and (3) torsion-free solutions admit a subclass of spacetimes with n-dim non-abelian Lie symmetry. These latter cases include the possibility of a gravity-electroweak unification. It is notable that the field equations reduce all curvature components to dependence only on the solder form of an n-dim Lagrangian submanifold, despite the increased number of curvature components and doubled number of initial independent variables.

Inflationary soft theorems revisited: a generalized consistency relation

We reconsider the derivation of soft theorems associated with nonlinearly-realized symmetries in cosmology. Utilizing the path integral, we derive a generalized consistency relation that relates a squeezed (N+1)-point correlation function to an N-point function, where the relevant soft mode is at early rather than late time. This generalized (early-late-time) version has wider applicability than the standard consistency relation where all modes are evaluated at late times. We elucidate the conditions under which the latter follows from the former. A key ingredient is the physical mode condition: that the nonlinear part of the symmetry transformation must match the time dependence of the dominant, long wavelength physical mode. This is closely related to, but distinct from, the adiabatic mode condition. Our derivation sheds light on a number of otherwise puzzling features of the standard consistency relation: (1) the underlying nonlinearly-realized symmetries (such as dilation and special conformal transformation SCT) originate as residual gauge redundancies, yet the consistency relation has physical content—for instance, it can be violated; (2) the standard consistency relation is known to fail in ultra-slow-roll inflation, but since dilation and SCT remain good symmetries, there should be a replacement for the standard relation; (3) in large scale structure applications, it is known that the standard consistency relation breaks down if the long wavelength power spectrum is too blue. The early-late-time consistency relation helps address these puzzles. We introduce a toy model where explicit checks of this generalized consistency relation are simple to carry out. Our methodology can be adapted to cases where violations of the standard consistency relation involve additional light degrees of freedom beyond the inflaton.

4D spin-2 fields from 5D Chern-Simons theory

We consider a 5-dimensional Chern-Simons gauge theory for the isometry group of Anti-de-Sitter spacetime, AdS4+1 ≃ SO(4, 2), and invoke different dimensional reduction schemes in order to relate it to 4-dimensional spin-2 theories. The AdS gauge algebra is isomorphic to a parametrized 4-dimensional conformal algebra, and the gauge fields corresponding to the generators of non-Abelian translations and special conformal transformations reduce to two vierbein fields in D = 4. Besides these two vierbeine, our reduction schemes leave only the Lorentz spin connection as an additional dynamical field in the 4-dimensional theories. We identify the corresponding actions as particular generalizations of Einstein-Cartan theory, conformal gravity and ghost-free bimetric gravity in first-order form.

On the Application of the Method of Induced Representations to the Conformal Group

The method of induced representations is applied to the conformal group. By its means, it is shown that the coset space to be used within the coset space technique for the construction of conformally-invariant theories must include the “Nambu–Goldstone” fields for special conformal transformations. They turn out to be non-dynamical fields whose dependence on the coordinates is fixed by the symmetries.