Ing. Miroslav Turek Sigma Models in Curved Background (Sigma

České vysoké učenı́ technické v Praze
Fakulta jaderná a fyzikálně inženýrská
Katedra fyziky
Ing. Miroslav Turek
Sigma Models in Curved Background
(Sigma modely na zakřiveném pozadı́)
Doktorský studijnı́ program: Aplikace přı́rodnı́ch věd
Studijnı́ obor: Matematické inženýrstvı́
Teze disertačnı́ práce k zı́skánı́ akademického titulu “doktor”,
ve zkratce “Ph.D.”
Praha, Listopad 2012
Disertačnı́ práce byla vypracována v kombinované formě doktorského studia
na katedře fyziky Fakulty jaderné a fyzikálně inženýrské ČVUT v Praze.
Uchazeč:
Ing. Miroslav Turek
Katedra fyziky, FJFI ČVUT
Břehová 7, 115 19, Praha 1
Školitel:
Prof. RNDr. Ladislav Hlavatý, DrSc.
Katedra fyziky, FJFI ČVUT
Břehová 7, 115 19, Praha 1
Oponenti: Prof. RNDr. Jiřı́ Podolský, CSc., DSc.
Ústav teoretické fyziky, MFF UK
V Holešovičkách 2,180 00 Praha 8
Prof. Rikard von Unge, Ph.D.
Ústav teoretické fyziky a astrofyziky,
Kotlářská 267/2, Veveřı́, Brno
Teze byly rozeslány dne: .......................
Obhajoba disertačnı́ práce se koná ...................... v ............. hod. před komisı́
pro obhajobu disertačnı́ práce ve studijnı́m oboru Matematické inženýrstvı́
v zasedacı́ mı́stnosti č. ....... Fakulty jaderné a fyzikálně inženýrské ČVUT v
Praze.
S disertacı́ je možno se seznámit na děkanátě Fakulty jaderné a fyzikálně
inženýrské ČVUT v Praze, na oddělenı́ pro vědeckou a výzkumnou činnost,
Břehová 7, 115 19, Praha 1.
.................................................................
Prof. RNDr. Ladislav Hlavatý, DrSc.
předseda komise pro obhajobu disertačnı́ práce
ve studijnı́m oboru Matematické inženýrstvı́
FJFI ČVUT, Břehová 7, 115 19, Praha 1
Obsah
1 Introduction
4
2 Flat coordinates and dilaton fields for three-dimensional conformal sigma models
4
2.1 σ-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Riemannian coordinates . . . . . . . . . . . . . . . . . . . . . 5
2.3 Flat coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Classical solutions of sigma models in curved backgrounds
by the Poisson-Lie T-plurality
6
3.1 Drinfel’d double and Manin triple . . . . . . . . . . . . . . . . 6
3.2 String condition . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4 Nonabelian dualization of plane wave backgrounds
4.1 Homogenous plane wave metrics . . . . . . . . . . . . . . . . .
4.2 Lobatchevsky plane waves . . . . . . . . . . . . . . . . . . . .
7
7
8
5 Conclusions
8
A List of author’s papers
10
1
Introduction
Very important and well known feature of the string theory is connection
to the gravitation. One possible method to study the gravitation (and other
problems), from string theory point of view, is based on the concept of conformally invariant σ−models. These models are given by the metrics, the
torsion and the scalar dilaton field, which satisfy so-called β−equations.
1
0 = Rij − ∇i ∇j Φ − Himn Hjmn ,
4
0 = ∇k ΦHkij + ∇k Hkij ,
0 = R − 2∇k ∇k Φ − ∇k Φ∇k Φ −
(1)
1
Hkmn H kmn
12
To solve these equations is in general very dificult and therefore it is interesting to deal with these problems in lower dimensions.
2
Flat coordinates and dilaton fields for threedimensional conformal sigma models
The main goal of the work [1], was the constructions of the dilaton fields
of three dimensional conformal invariant σ−models. These models were obtained by Poisson-Lie T-plurality [2] and investigated in detail in [3]. The
mentioned models live on solvable Lie groups and are Poisson-Lie T-dual or
plural to models on the flat background with the constant dilaton field. The
conditions for construction the dilaton fields by means of Poisson-Lie transformation were also analyzed in [3] and solved for concrete examples. We
were investigated that these conditions, in terms of Riemannian cordinates,
can be satisfied for more general dilaton fields. The dual dilaton fields were
solved and written for arbitrary models.
2.1
σ-model
Generally, the σ−model can be defined [4] as a field theory on the d−dimensional
Minkowski spacetime M with values in the n−dimensional target manifold
T . Here d = 2. The σ−model action can be then written as follows1
Z
SF [φ] = dz + dz − ∂− φµ Fµν (φ)∂+ φν ,
(2)
1
we denote: ∂+ ≡
∂
∂z + ,
∂− ≡
∂
∂z − .
4
where Fµν is a covariant second order tensor field, z + , z − are light-cone coordinates and φµ : M → T, µ = 1, . . . n. If the symmetric part of the tensor
Fµν is nondegenerate then it may be interpreted as a metric on the target
manifold T . The metric is used for raising and lowering indices. For PoissonLie dualizable models it is necessary to consider tensors containing also an
antisymmetric part. The equations of motion of model (2) obtained as an
Euler-Lagrange equations by the variation of the σ−model action are in the
form
∂− ∂+ φµ + Γµνλ ∂− φν ∂+ φλ = 0,
(3)
where the components of the Levi-Civita connection Γµνλ are generally defined
as follows
1 −1 µα ∂Fαλ ∂Fνα ∂Fνλ
µ
+
−
,
(4)
Γνλ = (G )
2
∂φν
∂φλ
∂φα
where
1
Gµν (φ) = (Fµν (φ) + Fνµ (φ))
(5)
2
is the symmetric part of F (φ)µν . The antisymmetric part defines the torsion
potential B
1
Bµν (φ) = (Fµν (φ) − Fνµ (φ)).
2
2.2
(6)
Riemannian coordinates
There are two important types of coordinates on the manifold where the
σ−models live. The first one is called group coordinates and follows from
the Lie group structure of the manifold. The second type of coordinates
are those in which the metric of the model has a special simple form. They
are generally called Riemannian coordinates. Riemannian coordinates for flat
metric are called flat coordinates here. In these coordinates the flat metrics
become constant and the Christoffel symbols vanish so that the equations
of motion as well as the vanishing β−equations become very simple. This
was the main reason for finding transformations between group and Riemannian coordinates. In our investigation was obtained the explicit form of such
transformations for the certain class of models in the flat backgrounds. Then
was possible to write the general form dilaton fields, in terms of group coordinates, satisfying the β−equations (1) for these models. After that we found
the dual dilaton fields.
5
2.3
Flat coordinates
The method for finding the flat coordinates is based on the transformation
formula for the Levi-Civita connections Γλµν .
Γ0ijk =
∂x0i ∂xm ∂xn l
∂x0i ∂ 2 xl
+
Γ
.
∂xl ∂x0j ∂x0k mn ∂xl ∂x0j ∂x0k
(7)
This formula provides the system of partial differential equations for flat
coordinates ξ(y). The system is linear and moreover separated with respect
to the unknowns ξ i ’s. The possibility to solve it explicitly depends on the form
of the connection components Γλµν . The formula (7), in the flat coordinates
reads
i
∂ 2 xi
0l ∂x
=
Γ
.
(8)
jk
∂x0j ∂x0k
∂x0l
3
Classical solutions of sigma models in curved
backgrounds by the Poisson-Lie T-plurality
We dealt with the classical solutions of equations of motion of σ−models
in curved background by means of Poisson-Lie T-plurality. Poisson-Lie Tplurality, investigated in [2], provides the relation between larger class of the
models with different geometrical properties, than Poisson-Lie T-duality [5].
We considered two cases of three dimensional models and solved their equations of motion by plurality transformation. This solution can be interpreted
as a solution of equation of motion for the relativistic strings if an additional
string condition is required. These constraints have very simple form in flat
coordinates and are preserved by Poisson-Lie T-plurality transformations.
3.1
Drinfel’d double and Manin triple
It was shown in [6], that the basic concept for study the Poisson-Lie transformation is the Drinfel’d double. A Drinfel’d double D is defined as a connected
Lie group such its Lie algebra D is equiped with a symmetric, nondegenerate,
ad-invariant bilinear form h., .i and can be decomposed into a pair of subalgebras G and G̃, maximally isotropic with respect to h., .i and such that the
algebra D of the Drinfeld double as a vector space is the direct sum of these
two sub-algebras. The isotropic subspace of D means that the value of the
form on two arbitrary vectors belonging to the subspace vanishes and maximally isotropic means that the subspace cannot be enlarged while preserving
the property of isotropy. Any such decomposition of the double into a pair
of maximally isotropic subalgebras G +̇G̃ = D I shall refer to as the M anin
6
triple. This decomposition into the Manin triple defines the dual or plural
transformation.
3.2
String condition
It is well known that the solution of the σ−model equations of motion can
be interpreted as a solution of equation of motion for the relativistic strings
if an additional string condition is required.
∂a φµ Gµν (φ)∂b φν = ηab eω ,
where
η=
0 1
1 0
(9)
(10)
and ω is a function of x+ , x− .
4
Nonabelian dualization of plane wave backgrounds
We were interested in the plane-parallel wave metrics from the point of view
of their Poisson-Lie T-dualization. We considered the homogenous plane wave
metrics and Lobachevsky plane wave metrics as a background of nonlinear
dualizable σ−model on Lie groups. The dualizable metrics are constructed
by virtue of nonabelian dualization. The Lie algebra of the Drinfel’d double
is composed from the four-dimensional subalgebra of Killing vectors and
four-dimensional Abelian elgebra. Moreover, the four-dimensional subgroup
of isometries act freely and transitively on the Riemannian manifold where
the corresponding metric is defined. We have determined the metrics and
B-fields dual to plane waves. In the case of homogenous plane wave metrics
we also present the form of dilaton field that guarantees conformal invariance
of dual metric.
4.1
Homogenous plane wave metrics
Homogenous plane wave is generally defined by the metric of the following
form [7], [8]
ds2 = 2dudv − Aij (u)xi xj du2 + dx2 ,
(11)
where dx2 is the standard metrics on Euclidian space Ed and x ∈ Ed . We
have focused in our investigation on the special case of isotropic homogenous
7
plane wave metric Aij (u) = λ(u)δij
ds2 = 2dudv − λ(u)x2 du2 + dx2 .
For special choice of λ(u) =
vectors.
4.2
k
,
u2
(12)
k = const., the metric admits seven Killing
Lobatchevsky plane waves
Another type of metrics that have larger group of isometries are so called
Lobatchevsky plane wave [9],[10]. They are of general form
 −H(u,x,y)

− b21x2
0
0
b 2 x2
 − 21 2

0
0
0

b x
Gij (u, v, x, y) = 
.
(13)
1


0
0
− b2 x2
0
0
0
0
− b21x2
They satisfy Einstein equation with cosmological constant 3b2 iff
2∂
∂2
∂2
H(u,
x,
y)
−
+
H(u, x, y) = 0.
∂y 2
x∂x ∂x2
(14)
We shall investigate metric of the form (13) where that H(u, x, y) = xα . Then
the metric admits five Killing vectors.
5
Conclusions
Significant aspect of Poisson-Lie transformation is the fact, that it doesn’t
preserve the geometrical properties of backgrounds on which the models are
constructed. This feature is often used for solving the equations of motion
and determining other specifications of given models. We can therefore try
to solve the equations of motion of arbitrary models by means of the simpler
ones, especialy by the models in the flat backgrounds. We were also able to
used the Poisson-Lie transformation for solving the models related to the
gravitation. The concrete models of the plane paralel wave were investigated
in detail.
Reference
[1] L. Hlavatý, L. Šnobl: Poisson-Lie T-plurality of three-dimensional conformally invariant sigma models, J. High Energy Phys. 0405, 010 (2004),
[hep-th/0403164].
8
[2] R. von Unge: Poisson-Lie T-plurality, J. High Energy Phys. 0207, 014
(2002), [hep-th/00205245].
[3] L. Hlavatý, L. Šnobl: Poisson-Lie T-plurality of three-dimensional conformally invariant sigma models ——: Nondiagonal metrics and dilaton
puzzle, J. High Energy Phys. 0410, 045 (2004), [hep-th/0408126].
[4] C. Klimčı́k: Poisson-Lie T-duality, Nucl. Phys. B (Proc. Suppl.) 46, 116
(1996), [hep-th/9509095].
[5] L. Hlavatý: Classical solution of a sigma-model in curved background,
Phys. Lett. B 625, 285 (2005), [hep-th/0506188].
[6] C. Klimčı́k, P. Ševera: Dual Non-Abelian Duality and the Drinfeld Double, Phys. Lett. B 351, 455 (1995), [hep-th/9502122].
[7] G. Papadopoulos, J.G. Russo and A.A. Tseytlin: Solvable model of
strings in a time-dependent plane-wave background, Class. Quant. Grav.
20:969-1016 (2003), [hep-th/0211289]
[8] M. Blau, M. O‘Loughlin: Homogenous Plane Waves, Nucl. Phys. B654,
135-186 (2003), [hep-th/021135]
[9] J. Podolský: Interpretation of the Siklos solutions as exact gravitational
waves in the anti-de Sitter universe, Class. Quant. Grav. 15, (1998) 719.
[10] S.T.C. Siklos: Lobatchevski plane gravitational wave in Galaxies, Axisymmetric Systems and Relativity ed M A H MacCallum (Cambridge:
Cambridge University Press) 1985, p. 247.
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A
List of author’s papers
• Ladislav Hlavatý, Miroslav Turek: Flat coordinates and dilaton fields
for three-dimensional conformal sigma models, JHEP06 (2006) 003
• Ladislav Hlavatý, Jan Hýbl, Miroslav Turek: Classical solutions of sigma
models in curved backgrounds by the Poisson-Lie T-plurality, Int. J.
Mod. Phys. A22 (2007) 1039-1052,arXiv:0608069 [hep-th]
• Ladislav Hlavatý, Miroslav Turek: Nonabelian dualization of plane wave
backgrounds, J. Mod. Phys., accepted, arXiv:1201.5939 [hep-th].
10