There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

We study certain bi-scalar-tensor theories emanating from conformal symmetry requirements of Horndeski’s four-dimensional action. The former scalar is a Galileon with shift symmetry whereas the latter scalar is adjusted to have a higher order conformal coupling. Employing techniques from local Weyl geometry certain Galileon higher order terms are thus constructed to be conformally invariant. The combined shift and partial conformal symmetry of the action, allow us to construct exact black hole solutions. The black holes initially found are of planar horizon geometry embedded in anti de Sitter space and can accommodate electric charge. The conformally coupled scalar comes with an additional independent charge and it is well-defined on the horizon whereas additional regularity of the Galileon field is achieved allowing for time dependence. Guided by our results in adS space-time we then consider a higher order version of the BBMB action and construct asymptotically flat, regular, hairy black holes. The addition of the Galileon field is seen to cure the BBMB scalar horizon singularity while allowing for the presence of primary scalar hair seen as an independent integration constant along-side the mass of the black hole.

/pdf/black-holes-in-bi-scalar-extensions-of-horndeski-theories-3klzz9uzm1.pdf

Black holes in bi-scalar extensions of Horndeski theories

We observe that the partition function of the set of all free massless higher spins s = 0, 1, 2, 3,... in flat space is equal to one: the ghost determinants cancel against the 'physical' ones or, equivalently, the (regularized) total number of degrees of freedom vanishes. This reflects large underlying gauge symmetry and suggests analogy with supersymmetric or topological theory. The Z = 1 property extends also to the AdS background, i.e. the 1-loop vacuum partition function of Vasiliev theory is equal to 1 (assuming a particular regularization of the sum over spins); this was noticed earlier as a consistency requirement for the vectorial AdS/CFT duality. We find that Z = 1 is true also in the conformal higher spin theory (with higher-derivative kinetic terms) expanded near flat or conformally flat S4 background. We also consider the partition function of free conformal theory of symmetric traceless rank s tensor field which has 2-derivative kinetic term but only scalar gauge invariance in flat 4d space. This non-unitary theory has Weyl-invariant action in curved background and it corresponds to 'partially massless' field in AdS5. We discuss in detail the special case of s = 2 (or 'conformal graviton'), compute the corresponding conformal anomaly coefficients and compare them with previously found expressions for generic representations of conformal group in 4 dimensions.

On higher spin partition functions

Studies in History and Philosophy of Science : 抄録雑誌の概要

Relations between the various formulations of nonlinear p-form electrodynamics with conformal-invariant weak-field and strong-field limits are clarified, with a focus on duality invariant (2n − 1)-form electrodynamics and chiral 2n-form electrodynamics in Minkowski spacetime of dimension D = 4n and D = 4n + 2, respectively. We exhibit a new family of chiral 2-form electrodynamics in D = 6 for which these limits exhaust the possibilities for conformal invariance; the weak-field limit is related by dimensional reduction to the recently discovered ModMax generalisation of Maxwell’s equations. For n > 1 we show that the chiral ‘strong-field’ 2n-form electrodynamics is related by dimensional reduction to a new Sl(2; ℝ)-duality invariant theory of (2n − 1)-form electrodynamics.

/pdf/on-p-form-gauge-theories-and-their-conformal-limits-4pj5j846ga.pdf

On p-form gauge theories and their conformal limits

We present a simple, systematic and practical method to construct conformally invariant equations in arbitrary Riemann spaces. 
This method that we call "Weyl-to-Riemann" is based on two features of Weyl geometry. i) A Weyl space is defined by the metric tensor and the Weyl vector $W$, it becomes equivalent to a Riemann space when $W$ is gradient. ii) Any homogeneous differential equation written in a Weyl space by means of the Weyl connection is conformally invariant. The Weyl-to-Riemann method selects those equations whose conformal invariance is preserved when reducing to a Riemann space. Applications to scalar, vector and spin-2 fields are presented, which demonstrates the efficiency of the present method. In particular, a new conformally invariant spin-2 field equation is exhibited. This equation extends Grishchuk-Yudin's equation and fixes its limitations since it does not require the Lorenz gauge. Moreover this equation reduces to the Drew-Gegenberg and Deser-Nepomechie equations in respectively Minkowski and de Sitter spaces.

Constructing conformally invariant equations by using Weyl geometry

The present work deals with two different but subtilely related kinds of conformal mappings: Weyl rescaling in d > 2 dimensional spaces and SO(2,d) transformations. We express how the difference between the two can be compensated by diffeomorphic transformations. This is well known in the framework of string theory but in the particular case of d = 2 spaces. Indeed, the Polyakov formalism describes world sheets in terms of two-dimensional conformal field theory (CFT). On the other hand, Zumino had shown that a classical four-dimensional Weyl-invariant field theory restricted to live in Minkowski space leads to an SO(2,4)-invariant field theory. We extend Zumino's result to relate Weyl and SO(2,d) symmetries in arbitrary conformally flat spaces (CFS). This allows us to assert that a classical SO(2,d)-invariant field does not distinguish, at least locally, between two different d-dimensional CFSs.

http://iopscience.iop.org/article/10.1209/0295-5075/101/31002/pdf

Conformal invariance: From Weyl to SO(2,d)

This paper is an extension for spinor fields the recently developed Dynamical Bridge formalism which relates a linear dynamics in a curved space to a nonlinear dynamics in Minkowski space. This leads to a new geometrical mechanism to generate a chiral symmetry breaking without mass, providing an alternative explanation for the absence of right-handed neutrinos. We analyze a spinor field obeying the Dirac equation in a curved space which is constructed by its own current. This way, both chiralities of the spinor field satisfy the same dynamics in the curved space. Afterward, the dynamical equation is re-expressed in terms of the flat Minkowski space and then each chiral component behaves differently. The left-handed part of the spinor field satisfies the Dirac equation while the right-handed part is trapped by a Nambu–Jona-Lasinio type potential.

http://inspirehep.net/record/1299667/files/arXiv:1406.2014.pdf

Chiral symmetry breaking as a geometrical process

This article expands for spinor fields the recently developed Dynamical Bridge formalism which relates a linear dynamics in a curved space to a nonlinear dynamics in Minkowski space. Astonishingly, this leads to a new geometrical mechanism to generate a chiral symmetry breaking without mass, providing an alternative explanation for the undetected right-handed neutrinos. We consider a spinor field obeying the Dirac equation in an effective curved space constructed by its own currents. This way, both chiralities of the spinor field satisfy the same dynamics in the curved space. Subsequently, the dynamical equation is re-expressed in terms of the flat Minkowski space and then each chiral component behaves differently. The left-handed part of the spinor field satisfies the Dirac equation while the right-handed part is trapped by a Nambu-Jona-Lasinio (NJL) type potential.

The present work deals with two different but subtilely related kinds of conformal mappings: Weyl rescaling in $d>2$ dimensional spaces and SO(2,d) transformations. We express how the difference between the two can be compensated by diffeomorphic transformations. This is well known in the framework of String Theory but in the particular case of $d=2$ spaces. Indeed, the Polyakov formalism describes world-sheets in terms of two-dimensional conformal field theory. On the other hand, B. Zumino had shown that a classical four-dimensional Weyl-invariant field theory restricted to live in Minkowski space leads to an SO(2,4)-invariant field theory. We extend Zumino's result to relate Weyl and SO(2,d) symmetries in arbitrary conformally flat spaces (CFS). This allows us to assert that a classical $SO(2,d)$-invariant field does not distinguish, at least locally, between two different $d$-dimensional CFSs.

Sofiane Faci

Papers

Constructing conformally invariant equations by using Weyl geometry

Conformal invariance: From Weyl to SO(2,d)

Chiral symmetry breaking as a geometrical process

Chiral symmetry breaking as a geometrical process

Conformal invariance: from Weyl to SO(2,d)