一维超膜的常曲率AdS_p+1真空与R^2重力模型的关系

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Eur. Phys. J. C (2019) 79 :633 Page 3 of 11 633

in curvature [35–38]. So, the transition to new geometric

variables uncovers the structure of non-linearities of funda-

mental branes with codim 1 and proves their similarity to

non-linearities of f (R) theories of gravity.

For c

= 0 action S

Di r

(4) contains only one coupling

constant κ

∼ T

3− p

2( p+1)

, where T

is the tension of p-branes.

The power law shows three different regimes of the behav-

ior of k

as a function of the tension and, consequently, the

energy corresponding to the cases p < 3, p > 3 and p = 3.

These regimes testify to the presence of the phases of asymp-

totic freedom and conﬁnement and their links with inﬂation

and collapse of the branes.

The phase with p = 3 is peculiar because the coupling

constant k

becomes dimensionless, so that S

Di r

p=3

is scale-

invariant. The latter forbids the Hilbert–Einstein (H–E) term

because it breaks the scale symmetry. To hold the desired

connection of 3-branes with our 4-dim. world, we have to

modify the potential of fundamental branes to activate the

H–E term [42].

Here we ﬁnd that the same problem interferes with 4-

dim. scale-invariant and renormalizable theories of gravity

quadratic in curvature (see [43–49] and refs. there). The latter

are converted to gravity theories containing the H–E term by

implementing the idea of spontaneously generated gravity.

This idea is illustrated by a simple example using the scale-

invariant action [45]

A =





|g|



R +

∇

ϕ∇

ϕg

μν

− V (ϕ, g)



(2)

Action (2) contains a scalar ϕ and a dimensionless constant

α. The potential V (ϕ, g) is assumed to have a deep minimum

at ϕ

= v which provides vacuum expectation value (vev) v

for ϕ. Expansion of the ﬁrst term in (2) around the minimum

generates the H–E term with the Newton constant G

≈

αv

So, the scale symmetry of (2) is spontaneously broken that

results in a 4-dim. gravity in the low energy limit. On the con-

trary, in the early universe, v as a function of the temperature,

is expected to vanish resulting in a scale-invariant R

action.

This model prevents the appearance of a cosmological con-

stant usually arising in such cases [50]. The replacement of

ϕ by a scalar

ψψ proposed by Adler does not improve the

situation [44].

Here we show that this problem is overcome by the substi-

tution of the tensor ﬁeld l

μν

for the scalar ϕ and the use of the

brane action S

Di r

p=3

(4), comprising potential (1), instead

of action (2). As a result, we ﬁnd a modiﬁed brane poten-

tial and the corresponding new models of gravity quadratic

in curvature. These models include the tensor l

μν

playing

Exact solutions for p-brane collapse were earlier obtained in [39–41].

the role of the Brans-Dicke scalars [51] commonly used in

extended models of gravity.

These results are based on the following observations.

First we note that the fundamental 3-brane potential (1) has

the extremal at l

μν

= 0. This means the absence of spon-

taneous breaking of the scaling, because we choose the vev

associated with l

μν

equal to the value of the diff-invariant

trace Spl ≡ l

μν

at the extremal. So, the vev turns out to

be zero at l

μν

= 0. Secondly, keeping in mind that Spl is

a fundamental invariant of embedded hypersurfaces, called

mean curvature, we resume that l

μν

= 0 corresponds to a

ﬂat h-ws. This makes us search for a deformation of poten-

tial (1) to a new one which has its extremal at l

μν

= l

oμν

= 0,

so that Spl

= μ, where the constant μ has the dimension

[μ]=[L

−1

]. From this we conclude that the required defor-

mation implies the transition from minimal h-ws 

mi n

constant mean curvature (cmc) h-ws 

cm c

This observa-

tion reduces the discussed problem to the construction of a

potential V

cm c

(l, g) which encodes hypersurfaces with the

constant mean curvature Spl = μ.

The required potential V

cm c

(l, g,μ) is constructed in

Sect. 3 extending the discussed mapping to ( p + 1)-dim.

hypersurfaces 

cm c

p+1

with codim 1 and constant mean curva-

ture. As a result, we ﬁnd action S

cm c

(23) encoding the H–E

action with a cosmological constant and equation of motion

(25) self-consistent with the cmc condition. We note that the

presence of the coupling constants T

and μ in the deformed

p-brane action controls the dynamical regimes produced by

the elastic and gravitational forces.

Studied in Sect. 4 is the Cauchy problem for the EOM

(25) which are the second order PDEs. We prove that the

Peterson-Codazzi (P-C) embedding conditions (14) and the

cmc conditions, spontaneously breaking the scale symmetry

for 3-branes, can be chosen as the initial data constraints

preserved by the PDEs (25). Then the Cauchy-Kowalevskaya

theorem of uniqueness ensures that the solution of Eq. (25)

describes the cmc hyper-worldsheet encoded by deformed

potential V

cm c

(l, g,μ)(22).

In Sect. 5 potential V

cm c

(l, g,μ) is converted to the

Lagrangian of R

gravity (58) including the H–E term in

addition to pure R

. Nowadays there is a great interest to

such Lagrangians extending the model [13] to scale-invariant

models (see e.g. [55–62]). The latter comprise scale-invariant

combinations built from R, R

, an interaction potential of

massless scalar ﬁeld(s), and a few of dimensionless param-

eters. The spontaneous breaking of the scale-symmetry fol-

lows from the solutions of the equations of motion includ-

ing a special scalar potential. Interest in such scale-invariant

models is due to the fact that they describe inﬂation and

The string model with constant mean curvature worldsheet embedded

into R

1,2

was studied in [52] by the extension of the Nambu-Goto action.

A similar dynamics is observed for string in curved space [53,54].

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