On the Symmetric Space Sigma-Model Kinematics

Abstract

The solvable Lie algebra parametrization of the symmetric spaces is discussed. Based on the solvable Lie algebra gauge two equivalent formulations of the symmetric space sigma model are studied. Their correspondence is established by inspecting the normalization conditions and deriving the field transformation laws. Comment: 17 pages

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arXiv:0707.2150v1 [hep-th] 14 Jul 2007
On the Symmetric Space σ-model Kinematics
Nejat T. Yılmaz
Department of Mathematics and Computer Science,
C ¸ankaya University,
¨O˘ gretmenler Cad. No:14,
Balgat, Ankara, Turkey.
ntyilmaz@cankaya.edu.tr
06530,
February 1, 2008
Abstract
The solvable Lie algebra parametrization of the symmetric spaces
is discussed. Based on the solvable Lie algebra gauge two equivalent
formulations of the symmetric space sigma model are studied. Their
correspondence is established by inspecting the normalization condi-
tions and deriving the field transformation laws.
1Introduction
The symmetric space sigma model [1, 2, 3, 4, 5, 6, 7, 8] has a field content
of scalars which parametrize a homogeneous coset manifold G/K which is
a Riemannian globally symmetric space for all the G-invariant Riemannian
structures on it [9]. The theory is invariant under the parametrization pre-
serving global (rigid) G-action from the right and the local K-action from
the left. The scalars transform nonlinearly under these actions. In general
the global symmetry group G is a non-compact real form of a semi-simple
Lie group and K is its maximal compact subgroup. The formulation of the
symmetric space sigma model is based on the Cartan-Maurer form which
is induced by the local parametrization of the coset representatives of G/K
by the scalar fields of the theory. There are two equivalent formulations of
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the symmetric space sigma model, the more conventional one which is also
applicable to the cases where the scalar manifold is not a symmetric space
is based on the decomposition of the Cartan-Maurer form which reveals the
vielbein and the gauge connection identification of the scalar coset manifold
G/K [1, 2, 10]. This formulation divides the Cartan-Maurer form into two
parts; one is a coset generators valued one-form and the other takes values in
the algebra of the maximal compact subgroup K. The second formulation of
the symmetric space sigma model is based on the introduction of an internal
metric [3, 4]. When one uses the solvable Lie algebra parametrization [8, 11]
to generate the coset representatives the analysis of the theory has simpli-
fications in both of the formulations [5, 6, 7, 8]. In [8] the Cartan-Maurer
forms and the field equations of both of the formulations are studied when
the solvable Lie algebra gauge is assumed.
In this work following the construction of the solvable Lie algebra gauge
or the solvable Lie algebra parametrization which has an essential role in
our analysis we will study the lagrangians of both of the formulations in the
solvable Lie algebra gauge to find a correspondence between them. We will
inspect the normalization conditions then we will derive the transformation
laws between the field contents of both of the formulations by assuming a
normalization scheme. We will also mention about the relation between the
two formulations when the model is coupled to other fields.
In section two we will introduce the solvable Lie algebra gauge to pa-
rameterize the coset manifold G/K of the symmetric space sigma model. By
using the solvable Lie algebra parametrization we will derive the lagrangian of
the symmetric space sigma model explicitly for both of the above mentioned
formulations in section three. We will also compare the normalization condi-
tions and we will obtain the transformation laws between the field contents
of the two formulations by choosing a normalization convention.
2Symmetric Spaces and the Solvable Lie Al-
gebra Parametrization
The symmetric space sigma models are based on homogeneous manifolds
[12], the homogeneity is in the sense that there exists a transitive action of a
Lie group G on these manifolds. These homogeneous spaces are in the form
of a coset manifold G/K where G is in general a non-compact real form of
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any other semi-simple Lie group and K is a maximal compact subgroup of
G. When G is a compact real form its maximal compact subgroup K is G
itself thus the coset space G/G is a single point and the corresponding sigma
model is an empty set. The numerator group may as well be a split real form
(maximally non-compact) of a semi-simple Lie group.
The Lie algebra k0of the analytical subgroup K is a subalgebra of the
semi-simple Lie algebra g0which is the Lie algebra of G. The Lie algebra
k0is a maximal compactly imbedded Lie subalgebra of g0therefore it is an
element of a Cartan decomposition of g0[9]
g0= k0⊕ p0, (2.1)
which is a vector space direct sum of the Lie subalgebra k0 and a vector
subspace p0. Since G is a linear analytical Lie group the corresponding Lie
algebra g0is non-compact for both of the cases when G is a non-compact
or a split real form (although we consider it separately we should not forget
that it is a limiting non-compact case). The map u0: k + p −→ k − p, for
all k ∈ k0and p ∈ p0is the Cartan involution which generates (2.1). The
pair (g0,u0) is an orthogonal symmetric Lie algebra of the non-compact type
[9]. Thus we conclude that (G,K) is associated with (g0,u0) and it is of
non-compact type too. The Cartan decomposition (2.1) is the eigenspace
decomposition of u0where the elements of k0have +1 eigenvalues and the
elements of p0have −1 eigenvalues under the involution u0. We also know
that (G,K) is a Riemannian symmetric pair therefore the coset space G/K
has a unique analytical structure induced by the quotient topology of G.
The scalar manifold G/K is a Riemannian globally symmetric space for all
the G-invariant Riemannian structures on G/K.1The crucial consequence
of this identification is that the exponential map Exp : g0−→ G induces a
diffeomorphism
Exp : p0−→ G/K,(2.2)
from the Rdimp0manifold p0onto G/K since it maps the elements of p0onto
the representatives of the left cosets G/K [9]. This result will enable us to
define a parametrization of the scalar manifold G/K on which the sigma
model will be constructed in the next section.
Furthermore one may use the Iwasawa decomposition of g0which is built
1This is the origin of the name; symmetric space sigma model.
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on the Cartan decomposition (2.1)2and one may make use of the root space
decomposition basis of g0 to parametrize the scalar coset manifold. The
Iwasawa decomposition reads
g0= k0⊕ s0
= k0⊕ hp0⊕ n0,
(2.3)
where k0is the Lie algebra of K and the algebra direct sum s0= hp0⊕ n0
is a solvable Lie subalgebra of g0which is isomorphic to the vector space p0.
In (2.3) hp0is generated by r non-compact Cartan generators {Hi}. Also
the nilpotent Lie subalgebra n0 in (2.3) is generated by a subset {Eβ} of
the positive root generators of g0where β ∈ ∆+
non-compact roots with respect to a Cartan involution θ which is composed
of the conjugation induced by the Cartan decomposition in (2.3) and the
conjugation of g0 via its complexification [9]. The Cartan subalgebra h0
generates an abelian subgroup in G which is called the torus. Although we
call it torus it is not the ordinary torus topologically in fact it has the topology
(S1)m×Rnfor some m and n and if it is diagonalizable in R (such that m = 0)
then it is called an R-split torus. These definitions can be generalized for the
subalgebras of h0as well. The subspace of G which is generated by hp0is the
maximal R-split torus in G in the sense defined above and its dimension is
called the R-rank which we will denote by r. If r is maximal such that r = l
where l is the rank of G (l =dimh0), which also means that hp0= h0then
the Lie group G is said to be in split real form (maximally non-compact).
In this case hp0= h0 is generated by all the Cartan generators {Hi} and
∆+
of positive roots. Thus the solvable Lie subalgebra s0 coincides with the
Borel subalgebra which is generated by the entire Cartan and the positive
root generators of g0for the split real form case. If on the other hand r is
minimal such that r = 0 then G is a compact real form. All the other cases
in between are called non-compact semi-simple real forms.
For the non-compact real form G if we consider the Iwasawa decomposi-
tion and in (2.2) if we use the basis {Hi,Eβ} which generates the solvable
nc. The roots in ∆+
ncare the
nc= ∆+so that the generators {Eβ} of n0 correspond to the entire set
2In this respect the Iwasawa decomposition is not a Cartan decomposition but it intro-
duces a solvable Lie algebra of g0which is isomorphic to p0thus which generates another
parametrization of G/K via (2.2) [9].
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Lie subalgebra s0then we have the parametrization
Exp :
?
R{Hi,Eβ} −→ G/K. (2.4)
(2.4) is called the solvable Lie algebra parametrization or the solvable Lie
algebra gauge of the symmetric space G/K [11]. On the other hand when g
is a split real form (maximally non-compact) if we use the Borel subalgebra
basis which is made up of the entire set of the Cartan generators and the
positive root generators; then (2.4) is called the Borel parametrization or the
Borel gauge of G/K.
In summary in this section we have obtained a legitimate parametrization
of the symmetric space G/K by using the solvable Lie subalgebra s0of g0.
If we use the notation {Tm} for the basis vectors {Hj,Eβ| j = 1,...,r ; β ∈
∆+
then the map
ν(x) = eϕm(x)Tm,
nc} of s0and if {ϕm(x)} are C∞-maps over the D-dimensional spacetime
(2.5)
is an onto C∞-map from the D-dimensional spacetime to the Riemannian
globally symmetric space G/K. The gauge map, (2.5) which depends on
the scalar functions {ϕm(x)} is the building block in the construction of the
symmetric space sigma model.
3Normalization Conditions and the Duality
Transformations of the SSSM
In this section we will obtain the field transformations of the two equivalent
formulations of the symmetric space sigma model (SSSM) which are based
on the solvable Lie algebra parametrization introduced in the last section.
We will show that when one assumes a normalization convention relating
the matrix representations of the basis that is used in (2.5) of the two sep-
arate formulations one can find a correspondence between the sets of field
definitions of the two distinct constructions.
In order to construct the symmetric space sigma model we first consider
the set of G-valued maps ν(x). They transform onto each other as ν →
k(x)νg, ∀g ∈ G, k(x) ∈ K for some subgroup K of G. We will assume
that the map ν(x) corresponds to a parametrization of the coset G/K (for
convenience we will consider the left cosets). Thus as mentioned before we
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