Conformal Gauge Research Articles

Surface operators in {mathcal N}=2 SQCD and Seiberg Duality

We study half-BPS surface operators in mathcal {N}=2 supersymmetric asymptotically conformal gauge theories in four dimensions with SU(N) gauge group and 2N fundamental flavours using localization methods and coupled 2d/4d quiver gauge theories. We show that contours specified by a particular Jeffrey–Kirwan residue prescription in the localization analysis map to particular realizations of the surface operator as flavour defects. Seiberg duality of the 2d/4d quivers is mapped to contour deformations of the localization integral which in this case involves a residue at infinity. This is reflected as a modified Seiberg duality rule that shifts the Lagrangian of the purported dual theory by non-perturbative terms. The new rules, that depend on the 4d gauge coupling, lead to a match between the low energy effective twisted chiral superpotentials for any pair of dual 2d/4d quivers.

Discrete Painlevé system and the double scaling limit of the matrix model for irregular conformal block and gauge theory

We study the partition function of the matrix model of finite size that realizes the irregular conformal block for the case of the N=2 supersymmetric SU(2) gauge theory with Nf=2. This model has been obtained in arXiv:1008.1861 [hep-th] as the massive scaling limit of the β-deformed matrix model representing the conformal block. We point out that the model for the case of β=1 can be recast into a unitary matrix model with log potential and show that it is exhibited as a discrete Painlevé system by the method of orthogonal polynomials. We derive the Painlevé II equation, taking the double scaling limit in the vicinity of the critical point which is the Argyres–Douglas type point of the corresponding spectral curve. By the 0d-4d dictionary, we obtain the time variable and the parameter of the double scaled theory respectively from the sum and the difference of the two mass parameters scaled to their critical values.

Conformal to harmonic gauge for bosonic strings

We consider the Polyakov theory of bosonic strings in conformal gauge which are used to study the conformal anomaly. However, it exhibits ghost number anomaly. We show how this anomaly can be avoided by connecting this theory to that of in background covariant harmonic gauge which is known to be free from conformal and ghost number current anomaly, by using suitably constructed finite-field-dependent BRST transformation.

Quasi-symmetric invariant properties of Cantor metric spaces

For metric spaces, the doubling property, the uniform disconnectedness, and the uniform perfectness are known as quasi-symmetric invariant properties. The David–Semmes uniformization theorem states that if a compact metric space satisfies all the three properties, then it is quasi-symmetrically equivalent to the middle-third Cantor set. We say that a Cantor metric space is standard if it satisfies all the three properties; otherwise, it is exotic. In this paper, we conclude that for each of exotic type the class of all the conformal gauges of Cantor metric spaces exactly has continuum cardinality. As a byproduct of our study, we state that there exists a Cantor metric space with prescribed Hausdorff dimension and Assouad dimension.

Linear sigma EFT for nearly conformal gauge theories

We construct a generalized linear sigma model as an effective field theory (EFT) to describe nearly conformal gauge theories at low energies. The work is motivated by recent lattice studies of gauge theories near the conformal window, which have shown that the lightest flavor-singlet scalar state in the spectrum ($\sigma$) can be much lighter than the vector state ($\rho$) and nearly degenerate with the PNGBs ($\pi$) over a large range of quark masses. The EFT incorporates this feature. We highlight the crucial role played by the terms in the potential that explicitly break chiral symmetry. The explicit breaking can be large enough so that a limited set of additional terms in the potential can no longer be neglected, with the EFT still weakly coupled in this new range. The additional terms contribute importantly to the scalar and pion masses. In particular, they relax the inequality $M_{\sigma}^2 \ge 3 M_{\pi}^2$, allowing for consistency with current lattice data.

N=2 supersymmetric higher spin gauge theories and current multiplets in three dimensions

We describe several families of primary linear supermultiplets coupled to three-dimensional ${\cal N}=2$ conformal supergravity and use them to construct topological $BF$-type terms. We introduce conformal higher-spin gauge superfields and associate with them Chern-Simons-type actions that are constructed as an extension of the linearised action for ${\cal N}=2$ conformal supergravity. These actions possess gauge and super-Weyl invariance in any conformally flat superspace and involve a higher-spin generalisation of the linearised ${\cal N}=2$ super-Cotton tensor. For massless higher-spin supermultiplets in (1,1) anti-de Sitter (AdS) superspace, we propose two off-shell Lagrangian gauge formulations, which are related to each other by a dually transformation. Making use of these massless theories allows us to formulate consistent higher-spin supercurrent multiplets in (1,1) AdS superspace. Explicit examples of such supercurrent multiplets are provided for models of massive chiral supermultiplets. Off-shell formulations for massive higher-spin supermultiplets in (1,1) AdS superspace are proposed.

Construction of anti-de Sitter-like spacetimes using the metric conformal Einstein field equations: the vacuum case

We make use of the metric version of the conformal Einstein field equations to construct anti-de Sitter-like spacetimes by means of a suitably posed initial-boundary value problem. The evolution system associated to this initial-boundary value problem consists of a set of conformal wave equations for a number of conformal fields and the conformal metric. This formulation makes use of generalised wave coordinates and allows the free specification of the Ricci scalar of the conformal metric via a conformal gauge source function. We consider Dirichlet boundary conditions for the evolution equations at the conformal boundary and show that these boundary conditions can, in turn, be constructed from the 3D Lorentzian metric of the conformal boundary and a linear combination of the incoming and outgoing radiation as measured by certain components of the Weyl tensor. To show that a solution to the conformal evolution equations implies a solution to the Einstein field equations we also provide a discussion of the propagation of the constraints for this initial-boundary value problem. The existence of local solutions to the initial-boundary value problem in a neighbourhood of the corner where the initial hypersurface and the conformal boundary intersect is subject to compatibility conditions between the initial and boundary data. The construction described is amenable to numerical implementation and should allow the systematic exploration of boundary conditions.

Three-dimensional conformal geometry and prepotentials for four-dimensional fermionic higher-spin fields

We introduce prepotentials for fermionic higher-spin gauge fields in four space-time dimensions, generalizing earlier work on bosonic fields. To that end, we first develop tools for handling conformal fermionic higher-spin gauge fields in three dimensions. This is necessary because the prepotentials turn out to be three-dimensional fields that enjoy both “higher-spin diffeomorphism” and “higher-spin Weyl” gauge symmetries. We discuss a number of the key properties of the relevant Cotton tensors. The reformulation of the equations of motion as “twisted self-duality conditions” is then exhibited. We show next how the Hamiltonian constraints can be explicitly solved in terms of appropriate prepotentials and show that the action takes then the same remarkable form for all spins.

Galaxy two-point correlation function in general relativity

We perform theoretical and numerical studies of the full relativistic two-point galaxy correlation function, considering the linear-order scalar and tensor perturbation contributions and the wide-angle effects. Using the gauge-invariant relativistic description of galaxy clustering and accounting for the contributions at the observer position, we demonstrate that the complete theoretical expression is devoid of any long-mode contributions from scalar or tensor perturbations and it lacks the infrared divergences in agreement with the equivalence principle. By showing that the gravitational potential contribution to the correlation function converges in the infrared, our study justifies an IR cut-off (kIR ⩽ H0) in computing the gravitational potential contribution. Using the full gauge-invariant expression, we numerically compute the galaxy two-point correlation function and study the individual contributions in the conformal Newtonian gauge. We find that the terms at the observer position such as the coordinate lapses and the observer velocity (missing in the standard formalism) dominate over the other relativistic contributions in the conformal Newtonian gauge such as the source velocity, the gravitational potential, the integrated Sachs-Wolf effect, the Shapiro time-delay and the lensing convergence. Compared to the standard Newtonian theoretical predictions that consider only the density fluctuation and redshift-space distortions, the relativistic effects in galaxy clustering result in a few percent-level systematic errors beyond the scale of the baryonic acoustic oscillation (∼ 2% at 150 Mpc/h and redshift one). Our theoretical and numerical study provides a comprehensive understanding of the relativistic effects in the galaxy two-point correlation function, as it proves the validity of the theoretical prediction and accounts for effects that are often neglected in its numerical evaluation.

Topologically massive higher spin gauge theories

We elaborate on conformal higher-spin gauge theory in three-dimensional (3D) curved space. For any integer n > 2 we introduce a conformal spin- frac{n}{2} gauge field {h}_{(n)}={h}_{alpha_1dots {alpha}_n} (with n spinor indices) of dimension (2 − n/2) and argue that it possesses a Weyl primary descendant C(n) of dimension (1 + n/2). The latter proves to be divergenceless and gauge invariant in any conformally flat space. Primary fields C(3) and C(4) coincide with the linearised Cottino and Cotton tensors, respectively. Associated with C(n) is a Chern-Simons-type action that is both Weyl and gauge invariant in any conformally flat space. These actions, which for n = 3 and n = 4 coincide with the linearised actions for conformal gravitino and conformal gravity, respectively, are used to construct gauge-invariant models for massive higher-spin fields in Minkowski and anti-de Sitter space. In the former case, the higher-derivative equations of motion are shown to be equivalent to those first-order equations which describe the irreducible unitary massive spin- frac{n}{2} representations of the 3D Poincaré group. Finally, we develop mathcal{N}=1 supersymmetric extensions of the above results.

New Reflections on Gravitational Duality

In general terms duality consists of two descriptions of one physical system by using degrees of freedom of different nature. There are different kinds of dualities and they have been extremely useful to uncover the underlying strong coupling dynamics of gauge theories in various dimensions and those of the diverse string theories. Perhaps the oldest example exhibiting this property is Maxwell theory, which interchanges electric and magnetic fields. An extension of this duality involving the sources is also possible if the magnetic monopole is incorporated. At the present time a lot has been understood about duality in non-Abelian gauge theories as in the case of N=4 supersymmetric gauge theories in four dimensions or in the Seiberg-Witten duality for N=2 theories. Moreover, a duality that relates a gravitational theory (or a string theory) and a conformal gauge theory, as in the case of gauge/gravity correspondence, have been also studied with considerable detail. The case of duality between two gravitational theories is the so called gravitational duality. At the present time, this duality has not been exhaustively studied, however some advances have been reported in the literature. In the present paper we give a general overview of this subject. In particular we will focus on non-Abelian dualities, applied to various theories of gravity as developed by the authors, based in the Rocek-Verlinde duality procedure. Finally, as a new development in this direction, we study the gravitational duality in Hitchin's gravity in seven and six dimensions and their relation is also discussed.

Minisuperspace computation of the Mabuchi spectrum

It was shown recently that, beside the traditional Liouville action, other functionals appear in the gravitational action of two-dimensional quantum gravity in the conformal gauge, the most important one being the Mabuchi functional. In a letter we proposed a minisuperspace action for this theory and used it to perform its canonical quantization. We found that the Hamiltonian of the Mabuchi theory is equal to the one of the Liouville theory and thus that the spectrum and correlation functions match in this approximation. In this paper we provide motivations to support our conjecture.

Observational Viability of an Inflation Model with E-model Nonminimal Derivative Coupling

Abstract By starting with a two-fields model in which the fields and their derivatives are nonminimally coupled to gravity, and then by using a conformal gauge, we obtain a model in which the derivatives of the canonically normalized field are nonminimally coupled to gravity. By adopting some appropriate functions, we study two cases with constant and E-model nonminimal derivative coupling, while the potential in both cases is chosen to be E-model one. We show that contrary to the single-field α-attractor model, there is an attractor point in the large N and small α limits in our setup, and for both mentioned cases there is an attractor line in these limits that the r−n s trajectories tend to. By studying the linear and nonlinear perturbations in this setup, and by comparing the numerical results with Planck2015 observational data, we obtain some constraints on the free parameter α. We show that by considering the E-model potential and coupling function, the model is observationally viable for all values of M (mass scale of the model). We use the observational constraints on the tensor-to-scalar ratio and the consistency relation to obtain some constraints on the sound speed of the perturbations in this model. As a result, we show that in a nonminimal derivative α-attractor model, it is possible to have small sound speed and therefore large non-Gaussianity.

Uhlenbeck\u2019s Decomposition in Sobolev and Morrey\u2013Sobolev Spaces

We present a self-contained proof of Rivière’s theorem on the existence of Uhlenbeck’s decomposition for Omega in L^p(mathbb {B}^n,so(m)otimes Lambda ^1mathbb {R}^n) for pin (1,n), with Sobolev type estimates in the case p in [n/2,n) and Morrey–Sobolev type estimates in the case pin (1,n/2). We also prove an analogous theorem in the case when Omega in L^p( mathbb {B}^n, TCO_{+}(m) otimes Lambda ^1mathbb {R}^n), which corresponds to Uhlenbeck’s decomposition with conformal gauge group.

Constrained dynamics of two interacting relativistic particles in the Faddeev-Jackiw symplectic framework

The Faddeev-Jackiw symplectic formalism for constrained systems is applied to analyze the dynamical content of a model describing two massive relativistic particles with interaction, which can also be interpreted as a bigravity model in one dimension. We systematically investigate the nature of the physical constraints, for which we also determine the zero-modes structure of the corresponding symplectic matrix. After identifying the whole set of constraints, we find out the transformation laws for all the set of dynamical variables corresponding to gauge symmetries, encoded in the remaining zero-modes. In addition, we use an appropriate gauge-fixing procedure, the conformal gauge, to compute the quantization brackets (Faddeev-Jackiw brackets) and also obtain the number of physical degree of freedom. Finally, we argue that this symplectic approach can be helpful for assessing physical constraints and understanding the gauge structure of theories of interacting spin-2 fields.

Weyl degree of freedom in the Nambu-Goto string through field transformation

We show how Weyl degrees of freedom can be introduced in the Nambu-Goto string in the path-integral formulation using the reparametrization invariant measure. We first identify Weyl degrees in conformal gauge using BFV formulation. Further we change the Nambu-Goto string action to the Polyakov action. The generating functional in light-cone gauge is then obtained from the generating functional corresponding to the Polyakov action in conformal gauge by using suitably constructed finite-field–dependent BRST transformation.

Multiloop amplitudes of light-cone gauge superstring field theory: odd spin structure contributions

We study the odd spin structure contributions to the multiloop amplitudes of light-cone gauge superstring field theory. We show that they coincide with the amplitudes in the conformal gauge with two of the vertex operators chosen to be in the pictures different from the standard choice, namely (−1, −1) picture in the type II case and −1 picture in the heterotic case. We also show that the contact term divergences can be regularized in the same way as in the amplitudes for the even structures and we get the amplitudes which coincide with those obtained from the first-quantized approach.

Correlators between Wilson loop and chiral operators in mathcal{N}=2 conformal gauge theories

We consider conformal mathcal{N}=2 super Yang-Mills theories with gauge group SU(N) and Nf = 2N fundamental hypermultiplets in presence of a circular 1/2-BPS Wilson loop. It is natural to conjecture that the matrix model which describes the expectation value of this system also encodes the one-point functions of chiral scalar operators in presence of the Wilson loop. We obtain evidence of this conjecture by successfully comparing, at finite N and at the two-loop order, the one-point functions computed in field theory with the vacuum expectation values of the corresponding normal-ordered operators in the matrix model. For the part of these expressions with transcendentality ζ(3), we also obtain results in the large-N limit that are exact in the ’t Hooft coupling λ.

Inhomogeneous tensionless superstrings

We construct a novel tensionless limit of Superstring theory that realises the Inhomogeneous Super Galilean Conformal Algebra (SGCAI ) as the residual symmetry in the analogue of the conformal gauge, as opposed to previous constructions of the tensionless superstring, where a smaller symmetry algebra called the Homogeneous SGCA emerged as the residual gauge symmetry on the worldsheet. We obtain various features of the new tensionless theory intrinsically as well as from a systematic limit of the corresponding features of the tensile theory. We discuss why it is desirable and also natural to work with this new tensionless limit and the larger algebra.

Perturbative four-point functions from the analytic conformal bootstrap

We apply the analytic conformal bootstrap method to study weakly coupled conformal gauge theories in four dimensions. We employ twist conformal blocks to find the most general form of the one-loop four-point correlation function of identical scalar operators, without any reference to Feynman calculations. The method relies only on symmetries of the model. In particular, it does not require introducing any regularisation and it is free from the redundancies usually associated with the Feynman approach. By supplementing the general solution with known data for a small number of operators, we recover explicit forms of one-loop correlation functions of four Konishi operators as well as of four half-BPS operators {{mathcal{O}}_{20}}_{prime } in mathcal{N} = 4 super Yang-Mills.