arXiv subjects

Ryuichi Nakayama

Publications and source records attributed to Ryuichi Nakayama.

At least 19 recordsLinked to original sources

arXiv2022

Covariant BRST Quantization of Unimodular Gravity II -- Formulation with a vector antighost --

In our previous paper, we have presented a covariant BRST quantization of unimodular gravity which may account for the smallness of the cosmological constant, and have shown that the physical degrees of freedom in the theory are the same as general relativity. The formulation has been given by using rank-2 antisymmetric tensor fields for both ghosts and antighosts. Here we give an alternative formulation using a vector field for the antighost but keeping the same structure for the ghosts. This gives a significantly simpler covariant quantization with less ghosts and no tripole modes in the ghost sector. We show that this also gives only two physical transverse modes as in general relativity.

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arXiv2022

Covariant BRST Quantization of Unimodular Gravity I -- Formulation with antisymmetric tensor ghosts --

Unimodular gravity (UG) is an interesting theory that may explain why the cosmological constant is extremely small, in contrast to general relativity (GR). The theory has only the transverse diffeomorphism invariance and this causes lots of debate as to the equivalence of UG to GR in the covariant quantization. We study the covariant BRST quantization of UG by gauge fixing only the transverse diffeomorphism and show that the remaining physical degrees of freedom are two, the same number as GR. This is achieved by using antisymmetric tensor ghost fields which automatically satisfy the transverse condition without nonlocal projection operator. The theory exhibits the ghosts for ghosts phenomenon, which requires further gauge fixing and introduction of more ghosts. We identify the BRST quartet structure among the various fields and single out the remaining physical degrees of freedom.

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arXiv2021

BRST Quantization of General Relativity in Unimodular Gauge and Unimodular Gravity

Unimodular gravity (UG) is an important theory which may explain the smallness of the cosmological constant. To get insight into the covariant quantization of UG, we discuss the BRST quantization of General Relativity (GR) with a cosmological constant in the unimodular gauge. We develop a novel way to gauge fix the transverse diffeomorphism (TDiff) and then further to fulfill the unimodular gauge. This process requires the introduction of an additional pair of BRST doublets which decouple from the physical sector together with the other three pairs of BRST doublets for the TDiff. We show that the physical spectrum is the same as GR in the usual covariant gauge fixing. We then study a theory derived by making "Fourier transform" of GR in the unimodular gauge with respect to the cosmological constant as a candidate of "quantum UG." We clarify the difference from GR and point out problems in this theory.

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arXiv2021

On Unimodular Gauge of Quantum Gravity

BRST-invariant action of general relativity in the unimodular gauge proposed by Baulieu is studied without using perturbative expansions. The expression for the path integral in the unimodular gauge is reduced to a form in which a functional measure is defined by a norm invariant under Transverse Diffeomorphism. It is shown that general relativity in the unimodular gauge with this action and the quantum unimodular gravity are equivalent. It is also shown that Vacuum Expectation Values (VEVs) of Diff invariant operators in the unimodular gauge and other gauge such as the harmonic gauge take distinct values. A path integral for harmonic gauge is found to be gauge equivalent to a superposition of that for unimodular gauge obtained by performing constant Weyl transformation of the metric, after a non-dynamical cosmological term is introduced into the action of the unimodular gauge.

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arXiv2020

Interior of the Horizon of BTZ Black Hole

A quantum scalar field inside the horizon of a non-rotating BTZ black hole is studied. Not only the near-horizon modes but also the normal modes deep inside the horizon are obtained. It is shown that the matching condition for the normal modes of a scalar field at the horizon does not uniquely determine the normal-mode expansion of a scalar field inside the horizon. By choosing a certain appropriate prescription for removing this ambiguity an integral form of a new scalar propagator for points on both sides of the horizon are obtained. A similar problem may arise in higher-dimensional black holes.

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arXiv2019

Holographic Duality for 3D Spin-3 Gravity Coupled to Scalar Field

The 3d spin-3 gravity theory is holographically dual to a 2d ${\cal W}_3$-extended CFT. In a large-c limit the symmetry algebra of the CFT reduces to $SU(1,2) \times SU(1,2)$. On the ground of symmetry the dual bulk space-time will be given by an 8d group manifold $SU(1,2)$. Hence we need to introduce five extra coordinates in addition to three ordinary ones. The 3d space-time is a 3d hyper-surface $Σ$ embedded at constant values of the extra variables. Operators in the CFT at the boundary of $Σ$ are expressed in terms of ${\cal W}$ descendants of the operators at the boundary of $Σ_0$, where the extra variables vanish. In this paper it is shown that AdS/CFT correspondence for a scalar field coupled to 3d spin-3 gravity is realized in this auxiliary 8d space. A bulk-to-boundary propagator of a scalar field is found and a generating functional of boundary two-point functions of scalar ${\cal W}$-descendant operators is obtained by using the classical action for the scalar field. Classically, the scalar field must satisfy both Klein-Gordon equation and a third-order differential equation, which are related to the quadratic and cubic Casimir operators of $su(1,2)$. It is found that the coefficient function of the derivatives of the scalar field in the latter equation is the spin-3 gauge field, when restricted to the hypersurface. An action integral in the 8d auxiliary space for the 3d spin-3 gravity coupled to a scalar field is presented. An 8d local frame is introduced and the equations of motion for the 8d connections $A_μ$, $\overline{A}_μ$ are solved. By restricting those solutions onto $Σ$, flat connections in 3d $SL(3,\mathbb{R}) \times SL(3,\mathbb{R})$ Chern-Simons theory are obtained and new 3d black hole solutions with and without spin-3 charge are found by this method.

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arXiv2018

A Bulk Localized State and New Holographic Renormalization Group Flow in 3D Spin-3 Gravity

We construct a localized state of a scalar field in 3D spin-3 gravity. 3D spin-3 gravity is thought to be holographically dual to W$_3$ extended CFT on a boundary at infinity. It is known that while W$_3$ algebra is a non-linear algebra, in the limit of large central charge $c$ a linear finite-dimensional subalgebra generated by $W_n \, (n=0,\pm1,\pm2)$ and $L_n (n= 0,\pm1)$ is singled out. The localized state is constructed in terms of these generators. To write down an equation of motion for a scalar field which is satisfied by this localized state it is necessary to introduce new variables for an internal space $α^{\pm}$, $β^{\pm}$, $γ$, in addition to ordinary coordinates $x^{\pm}$ and $y$. The higher-dimensional space, which combines the bulk spacetime with the `internal space', which is an analog of superspace in supersymmetric theory, is introduced. The `physical bulk spacetime' is a 3D hypersurface with constant $α^{\pm}$, $β^{\pm}$ and $γ$ embedded in this space. We will work in Poincaré coordinates of AdS space and consider W-quasi-primary operators $Φ_{h}(x^+)$ with a conformal weight $h$ in the boundary and study two and three point functions of W-quasi-primary operators transformed as $e^{ix^+L^h_{-1}} e^{β^+W^h_{-1}} Φ_{h}(0) e^{-β^+W^h_{-1}}e^{-ix^+L^h_{-1}}$. Here $L^h_n$ and $W^h_n$ are sl(3,R) generators in the hyperbolic basis for Poincaré coordinates. It is shown that in the $β^+ \rightarrow \infty$ limit, the conformal weight changes to a new value $h'=h/2$. This may be regarded as a Renormalization Group (RG) flow. It is argued that this RG flow will be triggered by terms $ΔS \propto β^+ W^h_{-1}+β^- \overline{W}^h_{-1}$ added to the action.

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arXiv2017

Study of the AdS$_2$/CFT$_1$ Correspondence with the Contribution from the Weyl Anomaly

In this paper we will consider the Almheiri-Polchinski model of the AdS$_2$ back reaction coupled with Liouville field, which is necessary for quantum consistency. In this model, the Liouville field is determined classically by a bulk conformal transformation. The boundary time is also reparametrized by this transformation. It is shown that the on-shell action on the boundary for the gravity sector is given by a bulk integral containing the Liouville field. This integral stems from Weyl anomaly and is SL(2,R) invariant. A prescription is given for computing correlation functions of the operators dual to massless scalars. The generating function of the correlation functions of these operators is given by a sum of matter action and the bulk integral containing the Liouville field. The latter integral leads to extra contributions to $n(\geq 6)$ point functions.

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arXiv2016

Kerr/Fluid Duality and Singularity of Solutions to the Fluid Equation

An equation for a viscous incompressible fluid on a spheroidal surface which is dual to the perturbation around the near-near horizon extreme Kerr (n-NHEK) black hole is derived. It is also shown that an expansion scalar $θ$ of a congruence of null geodesics on the null horizon of the perturbed n-NHEK spacetime, which is dual to a viscous incompressible fluid, is not positive semi-definite, even if initial conditions on the velocity are smooth. Unless initial conditions are elaborated, caustics of null congruence will occur on the horizon in the future. A similar result is obtained for a perturbed Schwarzschild black hole spacetime which is dual to a viscous incompressible fluid on $S^2$. An initial condition that $θ$ be positive semi-definite at any point on $S^2$ is a necessary condition for the existence of smooth solutions to incompressible Navier-Stokes (NS) equation on $S^2$.

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arXiv2015

$BMS_4$ Surface-Charge Algebra via Hamiltonian Framework

Surface-charge algebra associated with BMS$_4$ symmetry on the null infinity of asymptotically flat spacetime is studied via the Hamiltonian framework. A coordinate system, where boundaries of constant-time hypersurfaces cross the null infinity, is adopted. The equation itself which determines the variation of the surface charges turns out the same as that previously obtained via the covariant framework by Barnich and Troessaert, and is non-integrable for general radiation field $C_{AB}$. However, if $C_{AB}$ is independent of retarded time $u$, the variation equation is integrable and the conserved surface charges generate BMS$_4$ algebra without central extension.

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arXiv2014

Quantization of a Scalar Field in Two Poincaré Patches of Anti-de Sitter Space and AdS/CFT

Two sets of modes of a massive free scalar field are quantized in a pair of Poincaré patches of Lorentzian anti-de Sitter (AdS) space, AdS$_{d+1}$ ($d \geq 2$). It is shown that in Poincaré coordinates $(r,t,\vec{x})$, the two boundaries at $r=\pm \infty$ are connected. When the scalar mass $m$ satisfies a condition $0 < ν=\sqrt{(d^2/4)+(m\ell)^2} <1$, there exist two sets of mode solutions to Klein-Gordon equation, with distinct fall-off behaviors at the boundary. By using the fact that the boundaries at $r=\pm \infty$ are connected, a conserved Klein-Gordon norm can be defined for these two sets of scalar modes, and these modes are canonically quantized. Energy is also conserved. A prescription within the approximation of semi-classical gravity is presented for computing two- and three-point functions of the operators in the boundary CFT, which correspond to the two fall-off behaviours of scalar field solutions.

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arXiv2013

AdS/CFT for 3D Higher-Spin Gravity Coupled to Matter Fields

New holographic prescription for the model of 3d higher-spin gravity coupled to real matter fields $B_{μν}$ and $C$, which was introduced in ArXiv:1304.7941[hep-th], is formulated. By using a local symmetry, two of the components of $B_{μν}$ are eliminated, and gauge-fixing conditions are imposed such that the non-vanishing component, $B_{ϕρ}$, satisfies a covariantly-constancy condition in the background of Chern-Simons gauge fields $A_μ$, $\bar{A}_μ$. In this model, solutions to the classical equations of motion for $A_μ$ and $\bar{A}_μ$ are non-flat due to the interactions with matter fields. The solutions for the gauge fields can, however, be split into two parts, flat gauge fields ${\cal A}_μ$, $\bar{\cal A}_μ$, and those terms that depend on the matter fields. The equations for the matter fields then coincide with covariantly-constancy equations in the flat backgrounds ${\cal A}_μ$ and $\bar{\cal A}_μ$, which are exactly the same as those in linearized 3d Vasiliev gravity. The two- and three-point correlation functions of operators in the boundary CFT are computed by using an on-shell action, $ \text{tr}\, (B_{ϕρ} \, C)$. This term does not depend on coordinates due to the matter equations of motion, and it is not necessary to take the near-boundary limit $ρ\rightarrow \infty$. Analysis is presented for SL(3,R) $\times$ SL(3,R) as well as $HS[\frac{1}{2}] \times HS[\frac{1}{2}]$ higher-spin gravity. In the latter model, scalar operators with scaling dimensions $Δ_+=\frac{3}{2}$ and $Δ_-=\frac{1}{2}$ appear in a single quantization.

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arXiv2013

Metric-Like Formalism for Matter Fields Coupled to 3D Higher Spin Gravity

Action integral for a matter system composed of 0- and 2-forms, $C$ and $B_{μν}$, topologically coupled to 3D spin-3 gravity is considered first in the frame-like formalism. The field $C$ satisfies an eq of motion, $\partial_μ \, C+A_μ \, C-C \, \bar{A}_μ=0$. With a suitable gauge fixing of a new local symmetry and diffeomorphism, only one component of $B_{μν}$, say $B_{ϕr}$, remains non-vanishing and satisfies an equation similar to that for $C$ with $A_μ$ and $\bar{A}_μ$ interchanged. The spin connection is eliminated by solving the eq of motion for the total action, and in the resulting metric-like formalism, $(BC)^2$ interaction terms are induced because of the torsion. The world-volume components of the matter field, $C^0$, $C^μ$ and $C^{(μν)}$, are introduced by contracting the local-frame index of $C$ with those of the inverse vielbeins, $E_a^μ$ and $E_a^{(μν)}$, which were defined by the present authors in ArXiv:1209.0894 [hep-th]. The metric-like fields, as well as the new connections and the generalized curvature tensors, introduced in the above mentioned paper, are explicitly expressed in terms of the metric $g_{μν}$ and the spin-3 field $ϕ_{μνλ}$ by means of the $ϕ$-expansion. The action integral for the pure spin-3 gravity in the metric-like formalism up to ${\cal O}(ϕ^2)$, obtained before in the literature, is re-derived. Then the matter action is re-expressed in terms of $g_{μν}$, $ϕ_{μνρ}$ and the covariant derivatives for spin-3 geometry. Spin-3 gauge transformation is extended to the matter fields.

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arXiv2012

Second-Order Formalism for 3D Spin-3 Gravity

A second-order formalism for the theory of 3D spin-3 gravity is considered. Such a formalism is obtained by solving the torsion-free condition for the spin connection ω^a_μ, and substituting the result into the action integral. In the first-order formalism of the spin-3 gravity defined in terms of SL(3,R) X SL(3,R) Chern-Simons (CS) theory, however, the generalized torsion-free condition cannot be easily solved for the spin connection, because the vielbein e^a_μ itself is not invertible. To circumvent this problem, extra vielbein-like fields e^a_{μν} are introduced as a functional of e^a_μ. New set of affine-like connections Γ_{μM}^N are defined in terms of the metric-like fields, and a generalization of the Riemann curvature tensor is also presented. In terms of this generalized Riemann tensor the action integral in the second-order formalism is expressed. The transformation rules of the metric and the spin-3 gauge field under the generalized diffeomorphims are obtained explicitly. As in Einstein gravity, the new affine-like connections are related to the spin connection by a certain gauge transformation, and a gravitational CS term expressed in terms of the new connections is also presented.

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arXiv2011

The World-Line Quantum Mechanics Model at Finite Temperature which is Dual to the Static Patch Observer in de Sitter Space

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 Sitter space is also considered and then a large-N matrix model is obtained.

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arXiv2011

The Holographic Fluid on the Sphere Dual to the Schwarzschild Black Hole

We consider deformation of the d+2 dimensional asymptotically flat Schwarzschild black hole spacetime with the induced metric on a d-sphere at $r=r_c$ held fixed. This is done without taking the near horizon limit. The deformation is determined so that the $Λ=0$ vacuum Einstein equation is satisfied and the metric is regular on the horizon. In this paper the velocity of a dual fluid $v^i$ is assumed to be a Killing field and small, and the deformed metric is obtained up to $O(v^2)$. At this order of hydrodynamic expansion the dual fluid is an ideal one. The structure of the metric is fairly different from the near horizon result of Bredberg and Strominger in arXiv:1106.3084.

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arXiv2006

Fuzzy Torus and q-Deformed Lie Algebra

It will be shown that the defining relations for fuzzy torus and deformed (squashed) sphere proposed by J. Arnlind, et al (hep-th/0602290) (ABHHS) can be rewriten as a new algebra which contains q-deformed commutators. The quantum parameter q (|q|=1) is a function of \hbar. It is shown that the q -> 1 limit of the algebra with the parameter μ<0 describes fuzzy S^2 and that the squashed S^2 with q \neq 1 and μ<0 can be regarded as a new kind of quantum S^2. Throughout the paper the value of the invariant of the algebra, which defines the constraint for the surfaces, is not restricted to be 1. This allows the parameter q to be treated as independent of N (the dimension of the representation) and μ. It was shown by ABHHS that there are two types of representations for the algebra, ``string solution'' and ``loop solution''. The ``loop solution'' exists only for q a root of unity (q^N=1) and contains undetermined parameters. The 'string solution' exists for generic values of q (q^N \neq 1). In this paper we will explicitly construct the representation of the q-deformed algebra for generic values of q (q^N \neq 1) and it is shown that the allowed range of the value of q+q^{-1} must be restricted for each fixed N.

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arXiv2004

Matrix Configurations for Spherical 4-branes and Non-commutative Structures on S^4

We present a Matrix theory action and Matrix configurations for spherical 4-branes. The dimension of the representations is given by N=2(2j+1) (j=1/2,1,3/2,...). The algebra which defines these configurations is not invariant under SO(5) rotations but under SO(3) \otimes SO(2). We also construct a non-commutative product for field theories on S^4 in terms of that on S^2. An explicit formula of the non-commutative product which corresponds to the N=4 dim representation of the non-commutative S^4 algebra is worked out. Because we use S^2 \otimes S^2 parametrization of S^4, our S^4 is doubled and the non-commutative product and functions on S^4 are indeterminate on a great circle (S^1) on S^4. We will however, show that despite this mild singularity it is possible to write down a finite action integral of the non-commutative field thoery on S^4. NS-NS B field background on S^4 which is associated with our Matrix S^4 configurations is also constructed.

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