arXiv:0704.0752v3  [hep-th]  18 Apr 2008
Actions for the Bosonic String with the Curved
Worldsheet
Davoud Kamani
Faculty of Physics, Amirkabir University of Technology (Tehran Polytechnic)
P.O.Box: 15875-4413, Tehran, Iran
e-mail: kamani@cic.aut.ac.ir
Abstract
At first we introduce an action for the string, which leads to a worldsheet that
always is curved. For this action we study the Poincaré symmetry and the associated
conserved currents. Then, a generalization of the above action, which contains an
arbitrary function of the two-dimensional scalar curvature, will be introduced. An
extra scalar field enables us to modify these actions to Weyl invariant models.
PACS: 11.25.-w; 11.30.Cp
Keywords: Curved worldsheet; 2d scalar curvature; Poincaré symmetry.
http://arxiv.org/abs/0704.0752v3
1 Introduction
The two-dimensional models have widely been used in the context of the two-dimensional
gravity (e.g. see [1, 2, 3, 4] and references therein) and string theory. From the 2d-gravity
point of view, higher-dimensional gravity models, by dimensional reduction reduce to the
2d-gravity [1, 2, 3]. From the string theory point of view, the (1+1)-dimensional actions
are fundamental tools of the theory. However, 2d-gravity and 2d- string theory are closely
related to each other.
The known sigma models for string, in the presence of the dilaton field Φ(X), contain
the two-dimensional scalar curvature R(hab),
hRΦ(X). (1)
In two dimensions the combination
hR is total derivative. Thus, in the absence of the
dilaton field, this action is a topological invariant that gives no dynamics to the worldsheet
metric hab.
In fact, in the action (1), the dilaton is not the only choice. For example, replacing the
dilaton field with the scalar curvature R, leads to the R2-gravity [1, 4, 5]. In particular
the Polyakov action is replaced by a special combination of the worldsheet fields, which
include an overall factor R−1. Removing the dilaton and replacing it with another quantities
motivated us to study a class of two-dimensional actions. They are useful in the context of
the non-critical strings with curved worldsheet, and the 2-dimensional gravity.
Instead of the dilaton field, we introduce some combinations of hab, R and the induced
metric on the worldsheet, i.e. γab, which give dynamics to hab. These non-linear combinations
can contain an arbitrary function f(R) of the scalar curvature R. We observe that these
dynamics lead to the constraint equation for hab, extracted from the Polyakov action.
For the flat spacetime, these models have the Poincaré symmetry. In addition, they are
reparametrization invariant. However, for any function f(R), they do not have the Weyl
symmetry. Therefore, the string worldsheet at most is conformally flat. By introducing an
extra scalar field in these actions, they also find the Weyl symmetry. Note that a Weyl
non-invariant string theory has noncritical dimension, e.g. see [6].
This paper is organized as follows. In section 2, we introduce a new action for the string
in which the corresponding worldsheet always is curved. In section 3, the Poincaré symmetry
of this string model will be studied. In section 4, the generalized form of the above action
will be introduced and it will be analyzed.
2 Curved worldsheet in the curved spacetime
We consider the following action for the string, which propagates in the curved spacetime
S = −T
habγab
, (2)
where h = − det hab, and T is a dimensionless constant. In addition, R denotes the two-
dimensional scalar curvature which is made from hab. The string coordinates are {Xµ(σ, τ)}.
The induced metric on the worldsheet, i.e. γab, is also given by
γab = gµν(X)∂aX
µ(σ, τ)∂bX
ν(σ, τ), (3)
where gµν(X) is the spacetime metric.
In two dimensions, the symmetries of the curvature tensor imply the identity
Rab −
habR = 0. (4)
Therefore, the variation of the action (2) leads to the following equation of motion for hab,
Rab −
γab = 0. (5)
This implies that the energy-momentum tensor, extracted from the action (2), vanishes.
Contraction of this equation by hab gives R = 1
habγab. Introducing this equation and
the equation (5) into (4) leads to
(Polyakov)
ab ≡ γab −
hab(h
a′b′γa′b′) = 0. (6)
This is the constraint equation, extracted from the Polyakov action. Note that the energy-
momentum tensor, due to the action (2), is proportional to the left-hand-side of the equation
(5). Thus, it is different from (6).
The equation of motion of the string coordinate Xµ(σ, τ) also is
hRhab∂bX
hRhabΓ
νλ∂aX
λ = 0. (7)
Presence of the scalar curvature R distinguishes this equation from its analog, extracted
from the Polyakov action.
Now consider those solutions of the equations of motion (5) and (7), which admit constant
scalar curvature R. For these solutions, the equation (7) reduces to the equation of motion
of the string coordinates, extracted from the Polyakov action with the curved background.
However, for general solutions the scalar curvature R depends on the worldsheet coordinates
σ and τ , and hence this coincidence does not occur.
2.1 The model in the conformal gauge
Under reparametrization of σ and τ , the action (2) is invariant. That is, in two dimensions
the general coordinate transformations σ → σ′(σ, τ) and τ → τ ′(σ, τ), depend on two free
functions, namely the new coordinates σ′ and τ ′. By means of such transformations any
two of the three independent components of hab can be eliminated. A standard choice is a
parametrization of the worldsheet such that
hab = e
φ(σ,τ)ηab, (8)
where ηab = diag(−1, 1), and eφ(σ,τ) is an unknown conformal factor. The choice (8) is called
the conformal gauge. Since the action (2) does not have the Weyl symmetry (a local rescaling
of the worldsheet metric hab) we cannot choose the gauge hab = ηab.
The scalar curvature corresponding to the metric (8) is
R = −e−φ∂2φ, (9)
where ∂2 = ηab∂a∂b. Thus, the action (2) reduces to
S ′ = −T
d2σe−φ∂2φ
ηabγab
. (10)
According to the gauge (8), this action describes a conformally flat worldsheet.
3 Poincaré symmetry of the model
In this section we consider flat Minkowski space, i.e. gµν(X) = ηµν . Therefore, the equations
of motion are simplified to
Rab −
ηµν∂aX
ν = 0, (11)
hRhab∂bX
µ) = 0. (12)
The Poincaré symmetry reflects the symmetry of the background in which the string is
propagating. It is described by the transformations
δXµ = aµνX
ν + bµ,
δhab = 0, (13)
where aµν and b
µ are independent of the worldsheet coordinates σ and τ , and aµν = ηµλa
is antisymmetric. Thus, from the worldsheet point of view, these transformations are global
symmetries. Under these transformations the action (2) is invariant.
3.1 The conserved currents
The Poincaré invariance of the action (2) is associated to the following Noether currents
J µνa = T
hRhab(Xµ∂bX
ν −Xν∂bXµ),
Pµa = T
hRhab∂bX
µ, (14)
where the current Pµa is corresponding to the translation invariance and J µνa is the current
associated to the Lorentz symmetry. According to the equation of motion (12) these are
conserved currents
∂aJ µνa = 0,
∂aPµa = 0. (15)
3.2 The covariantly conserved currents
It is possible to construct two other currents from (14), in which they be covariantly con-
served. For this, there is the useful formula
∇aKa =
a), (16)
where Ka is a worldsheet vector. Therefore, we define the currents Jµνa and P µa as in the
following
Jµνa =
J µνa,
P µa =
Pµa. (17)
According to the equations (15) and (16), these are covariantly conserved currents, i.e.,
∇aJµνa = ∇aP µa = 0. (18)
The currents (17) can also be written as
Jµνa =
R(Xµ∂aX
ν −Xν∂aXµ),
P µa =
µ. (19)
Since there is ∇ahbc = 0, the conservation laws (18) also imply the covariantly conservation
of the currents (19).
4 Generalization of the model
The generalized form of the action (2) is
I = −T
f(R)−
, (20)
where f(R) is an arbitrary differentiable function of the scalar curvature R. The set
{Xµ(σ, τ)} describes a string worldsheet in the spacetime. These string coordinates ap-
peared in the induced metric γab through the equation (3). Thus, (20) is a model for the
string action.
The equation of motion of Xµ is as previous, i.e. (7). Vanishing the variation of this
action with respect to the worldsheet metric hab, gives the equation of motion of hab,
df(R)
γab = 0. (21)
The trace of this equation is
df(R)
habγab = 0. (22)
Combining the equations (4), (21) and (22) again leads to the equation (6).
As an example, consider the function f(R) = α lnR + β. Thus, the field equation (21)
implies that the intrinsic metric hab becomes proportional to the induced metric γab, that is
hab =
Since the Poincaré transformations contain δhab = 0, the generalized action (20) for
the flat background metric gµν = ηµν , also has the Poincaré invariance. This leads to the
previous conserved currents, i.e. (14) and (19).
4.1 Weyl invariance in the presence of a new scalar field
The action (20) under the reparametrization transformations is symmetric. The Weyl trans-
formation is also defined by
hab −→ h′ab = eρ(σ,τ)hab. (23)
Thus, the scalar curvature transforms as
R −→ R′ = e−ρ(R−∇2ρ), (24)
where ∇2ρ = 1√
hhab∂bρ). The equations (23) and (24) imply that the action (20), for
any function f(R), is Weyl non-invariant.
Introducing (23) and (24) into the action (20) gives a new action which contains the field
ρ(σ, τ),
I ′ = −T
h(R−∇2ρ)
f [e−ρ(R−∇2ρ)]− 1
e−ρhabγab
. (25)
We can ignore the origin of this action. In other words, it is another model for string.
However, under the Weyl transformations
hab −→ eu(σ,τ)hab,
ρ −→ ρ− u, (26)
the action I ′, for any function f , is symmetric. Note that according to the definition of ∇2
there is the transformation ∇2 → e−u∇2.
5 Conclusions
We considered some string actions which give dynamics to the worldsheet metric hab. Due
to the absence of the Weyl invariance, these models admit at most conformally flat (but not
flat) worldsheet. We observed that the constraint equation on the metric, extracted from
the Polyakov action, is a special result of the field equations of our string models. Obtaining
this constraint equation admits us to introduce an arbitrary function of the scalar curvature
to the action. For the case f(R) = α lnR + β, the metric hab becomes proportional to the
induced metric of the worldsheet.
By introducing a new degree of freedom we obtained a string action, in which for any
function f is Weyl invariant.
Our string models with arbitrary f(R), in the flat background have the Poincaré sym-
metry. The associated conserved currents are proportional to the scalar curvature R. We
also constructed the covariantly conserved currents from the Poincaré currents.
References
[1] H.J. Schmidt, Int. J. Mod. Phys. D7 (1998) 215, gr-qc/9712034.
[2] D. Park and Y. Kiem, Phys. Rev. D53 (1996) 5513; Phys. Rev. D53 (1996) 747.
[3] A. Achucarro and M. Ortiz, Phys. Rev. D48 (1993) 3600.
[4] D. Grumiller, W. Kummer and D.V. Vassilevich, Phys. Rept. 369 (2002) 327-429, hep-
th/0204253.
[5] M.O. Katanaev and I.V. Volovich, Phys. Lett. B175 (1986) 413, hep-th/0209014.
[6] F. David, Mod. Phys. Lett. A3 (1988) 1651; J. Distler, H. Kawai, Nucl. Phys. B321
(1989) 509; A. A. Tseytlin, Int. Jour. Mod. Phys. A4 (1989) 1257; J. Polchinski, Nucl.
Phys. B324 (1989) 123.

  At first we introduce an action for the string, which leads to a worldsheet
that always is curved. For this action we study the Poincar\'e symmetry and the
associated conserved currents. Then, a generalization of the above action,
which contains an arbitrary function of the two-dimensional scalar curvature,
will be introduced. An extra scalar field enables us to modify these actions to
Weyl invariant models.

Introduction
The two-dimensional models have widely been used in the context of the two-dimensional
gravity (e.g. see [1, 2, 3, 4] and references therein) and string theory. From the 2d-gravity
point of view, higher-dimensional gravity models, by dimensional reduction reduce to the
2d-gravity [1, 2, 3]. From the string theory point of view, the (1+1)-dimensional actions
are fundamental tools of the theory. However, 2d-gravity and 2d- string theory are closely
related to each other.
The known sigma models for string, in the presence of the dilaton field Φ(X), contain
the two-dimensional scalar curvature R(hab),
hRΦ(X). (1)
In two dimensions the combination
hR is total derivative. Thus, in the absence of the
dilaton field, this action is a topological invariant that gives no dynamics to the worldsheet
metric hab.
In fact, in the action (1), the dilaton is not the only choice. For example, replacing the
dilaton field with the scalar curvature R, leads to the R2-gravity [1, 4, 5]. In particular
the Polyakov action is replaced by a special combination of the worldsheet fields, which
include an overall factor R−1. Removing the dilaton and replacing it with another quantities
motivated us to study a class of two-dimensional actions. They are useful in the context of
the non-critical strings with curved worldsheet, and the 2-dimensional gravity.
Instead of the dilaton field, we introduce some combinations of hab, R and the induced
metric on the worldsheet, i.e. γab, which give dynamics to hab. These non-linear combinations
can contain an arbitrary function f(R) of the scalar curvature R. We observe that these
dynamics lead to the constraint equation for hab, extracted from the Polyakov action.
For the flat spacetime, these models have the Poincaré symmetry. In addition, they are
reparametrization invariant. However, for any function f(R), they do not have the Weyl
symmetry. Therefore, the string worldsheet at most is conformally flat. By introducing an
extra scalar field in these actions, they also find the Weyl symmetry. Note that a Weyl
non-invariant string theory has noncritical dimension, e.g. see [6].
This paper is organized as follows. In section 2, we introduce a new action for the string
in which the corresponding worldsheet always is curved. In section 3, the Poincaré symmetry
of this string model will be studied. In section 4, the generalized form of the above action
will be introduced and it will be analyzed.
2 Curved worldsheet in the curved spacetime
We consider the following action for the string, which propagates in the curved spacetime
S = −T
habγab
, (2)
where h = − det hab, and T is a dimensionless constant. In addition, R denotes the two-
dimensional scalar curvature which is made from hab. The string coordinates are {Xµ(σ, τ)}.
The induced metric on the worldsheet, i.e. γab, is also given by
γab = gµν(X)∂aX
µ(σ, τ)∂bX
ν(σ, τ), (3)
where gµν(X) is the spacetime metric.
In two dimensions, the symmetries of the curvature tensor imply the identity
Rab −
habR = 0. (4)
Therefore, the variation of the action (2) leads to the following equation of motion for hab,
Rab −
γab = 0. (5)
This implies that the energy-momentum tensor, extracted from the action (2), vanishes.
Contraction of this equation by hab gives R = 1
habγab. Introducing this equation and
the equation (5) into (4) leads to
(Polyakov)
ab ≡ γab −
hab(h
a′b′γa′b′) = 0. (6)
This is the constraint equation, extracted from the Polyakov action. Note that the energy-
momentum tensor, due to the action (2), is proportional to the left-hand-side of the equation
(5). Thus, it is different from (6).
The equation of motion of the string coordinate Xµ(σ, τ) also is
hRhab∂bX
hRhabΓ
νλ∂aX
λ = 0. (7)
Presence of the scalar curvature R distinguishes this equation from its analog, extracted
from the Polyakov action.
Now consider those solutions of the equations of motion (5) and (7), which admit constant
scalar curvature R. For these solutions, the equation (7) reduces to the equation of motion
of the string coordinates, extracted from the Polyakov action with the curved background.
However, for general solutions the scalar curvature R depends on the worldsheet coordinates
σ and τ , and hence this coincidence does not occur.
2.1 The model in the conformal gauge
Under reparametrization of σ and τ , the action (2) is invariant. That is, in two dimensions
the general coordinate transformations σ → σ′(σ, τ) and τ → τ ′(σ, τ), depend on two free
functions, namely the new coordinates σ′ and τ ′. By means of such transformations any
two of the three independent components of hab can be eliminated. A standard choice is a
parametrization of the worldsheet such that
hab = e
φ(σ,τ)ηab, (8)
where ηab = diag(−1, 1), and eφ(σ,τ) is an unknown conformal factor. The choice (8) is called
the conformal gauge. Since the action (2) does not have the Weyl symmetry (a local rescaling
of the worldsheet metric hab) we cannot choose the gauge hab = ηab.
The scalar curvature corresponding to the metric (8) is
R = −e−φ∂2φ, (9)
where ∂2 = ηab∂a∂b. Thus, the action (2) reduces to
S ′ = −T
d2σe−φ∂2φ
ηabγab
. (10)
According to the gauge (8), this action describes a conformally flat worldsheet.
3 Poincaré symmetry of the model
In this section we consider flat Minkowski space, i.e. gµν(X) = ηµν . Therefore, the equations
of motion are simplified to
Rab −
ηµν∂aX
ν = 0, (11)
hRhab∂bX
µ) = 0. (12)
The Poincaré symmetry reflects the symmetry of the background in which the string is
propagating. It is described by the transformations
δXµ = aµνX
ν + bµ,
δhab = 0, (13)
where aµν and b
µ are independent of the worldsheet coordinates σ and τ , and aµν = ηµλa
is antisymmetric. Thus, from the worldsheet point of view, these transformations are global
symmetries. Under these transformations the action (2) is invariant.
3.1 The conserved currents
The Poincaré invariance of the action (2) is associated to the following Noether currents
J µνa = T
hRhab(Xµ∂bX
ν −Xν∂bXµ),
Pµa = T
hRhab∂bX
µ, (14)
where the current Pµa is corresponding to the translation invariance and J µνa is the current
associated to the Lorentz symmetry. According to the equation of motion (12) these are
conserved currents
∂aJ µνa = 0,
∂aPµa = 0. (15)
3.2 The covariantly conserved currents
It is possible to construct two other currents from (14), in which they be covariantly con-
served. For this, there is the useful formula
∇aKa =
a), (16)
where Ka is a worldsheet vector. Therefore, we define the currents Jµνa and P µa as in the
following
Jµνa =
J µνa,
P µa =
Pµa. (17)
According to the equations (15) and (16), these are covariantly conserved currents, i.e.,
∇aJµνa = ∇aP µa = 0. (18)
The currents (17) can also be written as
Jµνa =
R(Xµ∂aX
ν −Xν∂aXµ),
P µa =
µ. (19)
Since there is ∇ahbc = 0, the conservation laws (18) also imply the covariantly conservation
of the currents (19).
4 Generalization of the model
The generalized form of the action (2) is
I = −T
f(R)−
, (20)
where f(R) is an arbitrary differentiable function of the scalar curvature R. The set
{Xµ(σ, τ)} describes a string worldsheet in the spacetime. These string coordinates ap-
peared in the induced metric γab through the equation (3). Thus, (20) is a model for the
string action.
The equation of motion of Xµ is as previous, i.e. (7). Vanishing the variation of this
action with respect to the worldsheet metric hab, gives the equation of motion of hab,
df(R)
γab = 0. (21)
The trace of this equation is
df(R)
habγab = 0. (22)
Combining the equations (4), (21) and (22) again leads to the equation (6).
As an example, consider the function f(R) = α lnR + β. Thus, the field equation (21)
implies that the intrinsic metric hab becomes proportional to the induced metric γab, that is
hab =
Since the Poincaré transformations contain δhab = 0, the generalized action (20) for
the flat background metric gµν = ηµν , also has the Poincaré invariance. This leads to the
previous conserved currents, i.e. (14) and (19).
4.1 Weyl invariance in the presence of a new scalar field
The action (20) under the reparametrization transformations is symmetric. The Weyl trans-
formation is also defined by
hab −→ h′ab = eρ(σ,τ)hab. (23)
Thus, the scalar curvature transforms as
R −→ R′ = e−ρ(R−∇2ρ), (24)
where ∇2ρ = 1√
hhab∂bρ). The equations (23) and (24) imply that the action (20), for
any function f(R), is Weyl non-invariant.
Introducing (23) and (24) into the action (20) gives a new action which contains the field
ρ(σ, τ),
I ′ = −T
h(R−∇2ρ)
f [e−ρ(R−∇2ρ)]− 1
e−ρhabγab
. (25)
We can ignore the origin of this action. In other words, it is another model for string.
However, under the Weyl transformations
hab −→ eu(σ,τ)hab,
ρ −→ ρ− u, (26)
the action I ′, for any function f , is symmetric. Note that according to the definition of ∇2
there is the transformation ∇2 → e−u∇2.
5 Conclusions
We considered some string actions which give dynamics to the worldsheet metric hab. Due
to the absence of the Weyl invariance, these models admit at most conformally flat (but not
flat) worldsheet. We observed that the constraint equation on the metric, extracted from
the Polyakov action, is a special result of the field equations of our string models. Obtaining
this constraint equation admits us to introduce an arbitrary function of the scalar curvature
to the action. For the case f(R) = α lnR + β, the metric hab becomes proportional to the
induced metric of the worldsheet.
By introducing a new degree of freedom we obtained a string action, in which for any
function f is Weyl invariant.
Our string models with arbitrary f(R), in the flat background have the Poincaré sym-
metry. The associated conserved currents are proportional to the scalar curvature R. We
also constructed the covariantly conserved currents from the Poincaré currents.
References
[1] H.J. Schmidt, Int. J. Mod. Phys. D7 (1998) 215, gr-qc/9712034.
[2] D. Park and Y. Kiem, Phys. Rev. D53 (1996) 5513; Phys. Rev. D53 (1996) 747.
[3] A. Achucarro and M. Ortiz, Phys. Rev. D48 (1993) 3600.
[4] D. Grumiller, W. Kummer and D.V. Vassilevich, Phys. Rept. 369 (2002) 327-429, hep-
th/0204253.
[5] M.O. Katanaev and I.V. Volovich, Phys. Lett. B175 (1986) 413, hep-th/0209014.
[6] F. David, Mod. Phys. Lett. A3 (1988) 1651; J. Distler, H. Kawai, Nucl. Phys. B321
(1989) 509; A. A. Tseytlin, Int. Jour. Mod. Phys. A4 (1989) 1257; J. Polchinski, Nucl.
Phys. B324 (1989) 123.
