9.1 Nonlinear sigma models on finite-dimensional coset spaces
A nonlinear sigma model describes maps
from one Riemannian space
, equipped with
a metric
, to another Riemannian space, the “target space”
, with metric
. Let
be coordinates on
and
be coordinates on
. Then the standard action for this sigma model is
Solutions to the equations of motion resulting from this action will describe the maps
as functions of
.
A familiar example, of direct interest to the analysis below, is the case where
is one-dimensional,
parametrized by the coordinate
. Then the action for the sigma model reduces to
where
is
and ensures reparametrization invariance in the variable
. Extremization with
respect to
enforces the constraint
ensuring that solutions to this model are null geodesics on
. We have already encountered such a
sigma model before, namely as describing the free lightlike motion of the billiard ball in the
-dimensional scale-factor space. In that case
corresponds to the inverse “lapse-function”
and the metric
is a constant Lorentzian metric.
9.1.1 The Cartan involution and symmetric spaces
In what follows, we shall be concerned with sigma models on symmetric spaces
where
is a Lie
group with semi-simple real Lie algebra
and
its maximal compact subgroup with real Lie
algebra
, corresponding to the maximal compact subalgebra of
. Since elements of the coset are
obtained by factoring out
, this subgroup is referred to as the “local gauge symmetry group” (see
below). Our aim is to provide an algebraic construction of the metric on the coset and of the
Lagrangian.
We have investigated real forms in Section 6 and have found that the Cartan involution
induces a
Cartan decomposition of
into even and odd eigenspaces:
(direct sum of vector spaces), where
play central roles. The decomposition (9.4) is orthogonal, in the sense that
is the orthogonal
complement of
with respect to the invariant bilinear form
,
The commutator relations split in a way characteristic for symmetric spaces,
The subspace
is not a subalgebra. Elements of
transform in some representation of
, which
depends on the Lie algebra
. We stress that if the commutator
had also contained elements in
itself, this would not have been a symmetric space.
The coset space
is defined as the set of equivalence classes
of
defined by the
equivalence relation
i.e.,
Example: The coset space 
As an example to illustrate the Cartan involution we consider the coset space
. The
group
contains all
real matrices with determinant equal to one. The associated
Lie algebra
thus consists of real
traceless matrices. In this case the Cartan
involution is simply (minus) the ordinary matrix transpose
on the Lie algebra elements:
This implies that all antisymmetric traceless
matrices belong to
. The Cartan involution
is the differential at the identity of an involution
defined on the group itself, such that for real Lie
groups (real or complex matrix groups),
is just the inverse conjugate transpose. Defining
then gives in this example the group
. The Cartan decomposition of
thus
splits all elements into symmetric and antisymmetric matrices, i.e., for
we have
9.1.2 Nonlinear realisations
The group
naturally acts through (here, right) multiplication on the quotient space
as
This definition makes sense because if
, i.e.,
for some
, then
since
(left and right multiplications commute).
In order to describe a dynamical theory on the quotient space
, it is convenient to introduce as
dynamical variable the group element
and to construct the action for
in such a way that
the equivalence relation
corresponds to a gauge symmetry. The physical (gauge invariant) degrees of freedom are then parametrized
indeed by points of the coset space. We also want to impose Equation (9.13) as a rigid symmetry. Thus, the
action should be invariant under
One may develop the formalism without fixing the
-gauge symmetry, or one may instead fix the
gauge symmetry by choosing a specific coset representative
. When
is a maximal
compact subgroup of
there are no topological obstructions, and a standard choice, which is always
available, is to take
to be of upper triangular form as allowed by the Iwasawa decomposition. This is
usually called the Borel gauge and will be discussed in more detail later. In this case an arbitrary global
transformation,
will destroy the gauge choice because
will generically not be of upper triangular form. Then, a
compensating local
-transformation is needed that restores the original gauge choice. The total
transformation is thus
where
is again in the upper triangular gauge. Because now
depends nonlinearly on
, this is called a nonlinear realisation of
.
9.1.3 Three ways of writing the quadratic
-invariant action
Given the field
, we can form the Lie algebra valued one-form (Maurer–Cartan form)
Under the Cartan decomposition, this element splits according to Equation (9.4),
where
and
. We can use the Cartan involution
to write these explicitly as
projections onto the odd and even eigenspaces as follows:
If we define a generalized transpose
by
then
and
correspond to symmetric and antisymmetric elements, respectively,
Of course, in the special case when
and
, the generalized transpose
coincides with the ordinary matrix transpose
. The Lie algebra valued one-forms with
components
,
and
are invariant under rigid right multiplication,
.
Being an element of the Lie algebra of the gauge group,
can be interpreted as a gauge
connection for the local symmetry
. Under a local transformation
,
transforms as
which indeed is the characteristic transformation property of a gauge connection. On the other hand,
transforms covariantly,
because the element
is projected out due to the negative sign in the construction of
in Equation (9.20).
Using the bilinear form
we can now form a manifestly
-invariant expression by
simply “squaring”
, i.e., the
-dimensional action takes the form (cf. Equation (9.1))
We can rewrite this action in a number of ways. First, we note that since
can be
interpreted as a gauge connection we can form a “covariant derivative”
in a standard way as
which, by virtue of Equation (9.20), can alternatively be written as
We see now that the action can indeed be interpreted as a gauged nonlinear sigma model, in the sense that
the local invariance is obtained by minimally coupling the globally
-invariant expression
to the gauge field
through the “covariantization”
,
Thus, the action then takes the form
We can also form a generalized “metric”
that does not transform at all under the local
symmetry, but only transforms under rigid
-transformations. This is done, using the generalized
transpose (extended from the algebra to the group through the exponential map [93
]), in the following way,
which is clearly invariant under local transformations
for
, and transforms as follows under global transformations on
from the right,
A short calculation shows that the relation between
and
is given by
Since the factors of
drop out in the squared expression,
Equation (9.33) provides a third way to write the
-invariant action, completely in terms
of the generalized metric
,
(We call
a “generalized metric” because in the
-case, it does correspond to the
metric, the field
being the “vielbein”; see Section 9.3.2.)
All three forms of the action are manifestly gauge invariant under
. If desired, one can fix the
gauge, and thereby eliminating the redundant degrees of freedom.
9.1.4 Equations of motion and conserved currents
Let us now take a closer look at the equations of motion resulting from an arbitrary variation
of
the action in Equation (9.25). The Lie algebra element
can be decomposed according to
the Cartan decomposition,
The variation
along the gauge orbit will be automatically projected out by gauge invariance of the
action. Thus we can set
for simplicity. Let us then compute
. One easily gets
Since
is a Lie algebra valued scalar we can freely set
in the variation of the
action below, where
is a covariant derivative on
compatible with the Levi–Civita connection.
Using the symmetry and the invariance of the bilinear form one then finds
The equations of motion are therefore equivalent to
with
and simply express the covariant conservation of
.
It is also interesting to examine the dynamics in terms of the generalized metric
. The equations
of motion for
are
These equations ensure the conservation of the current
i.e.,
This is the conserved Noether current associated with the rigid
-invariance of the action.
9.1.5 Example:
(hyperbolic space)
Let us consider the example of the coset space
, which, although very simple, is
nevertheless quite illustrative. Recall from Section 6.2 that the Lie algebra
is constructed from
the Chevalley triple
,
with the following standard commutation relations
and matrix realisation
In the Borel gauge,
reads
where
and
represent coordinates on the coset space, i.e., they describe the sigma model map
An arbitrary transformation on
reads
which in infinitesimal form becomes
Let us then check how
transforms under the generators
. As expected, the Borel
generators
and
preserve the upper triangular structure
while the negative root generator
does not respect the form of
,
Thus, in this case we need a compensating transformation to restore the upper triangular form.
This transformation needs to cancel the factor
in the lower left corner of the matrix
and so it must necessarily depend on
. The transformation that does the job is
and we find
Finally, since the generalized transpose
in this case reduces to the ordinary matrix transpose, the
“generalized” metric becomes
The Killing form
corresponds to taking the trace in the adjoint representation of Equation (9.46)
and the action (9.35) therefore takes the form
9.1.6 Parametrization of 
The Borel gauge choice is always accessible when the group
is the maximal compact subgroup of
. In the noncompact case this is no longer true since one cannot invoke the Iwasawa decomposition (see,
e.g. [120] for a discussion of the subtleties involved when
is noncompact). This point will,
however, not be of concern to us in this paper. We shall now proceed to write down the sigma
model action in the Borel gauge for the coset space
, with
being the maximal
compact subgroup. Let
be a basis of the Cartan subalgebra
, and let
denote the set of positive roots. The Borel subalgebra of
can then be written as
where
is the positive root generator spanning the one-dimensional root space
associated to the
root
. The coset representative is then chosen to be
Because
is a finite Lie algebra, the sum over positive roots is finite and so this is a well-defined
construction.
From Equation (9.58) we may compute the Lie algebra valued one-form
explicitly. Let
us do this in some detail. First, we write the general expression in terms of
and
,
To compute the individual terms in this expression we need to make use of the Baker–Hausdorff formulas:
The first term in Equation (9.59) is easy to compute since all generators in the exponential commute. We
find
Secondly, we compute the corresponding expression for
. Here we must take into account all
commutators between the positive root generators
. Using the first of the Baker–Hausdorff
formulas above, the first terms in the series become
Each multi-commutator
corresponds to some new positive root generator, say
. However, since each term in the expansion (9.62) is a sum over all positive roots, the
specific generator
will get a contribution from all terms. We can therefore write the sum
in “closed form” with the coefficient in front of an arbitrary generator
taking the form
where
denotes the number corresponding to the last term in the Baker–Hausdorff expansion in which
the generator
appears. The explicit form of
must be computed individually for each root
.
The sum in Equation (9.62) can now be written as
To proceed, we must conjugate this expression with
in order to compute the full form of
Equation (9.59). This involves the use of the second Baker–Hausdorff formula in Equation (9.60) for each
term in the sum, Equation (9.64). Let
denote an arbitrary element of the Cartan subalgebra,
Then the commutators we need are of the form
where
denotes the value of the root
acting on the Cartan element
,
So, for each term in the sum in Equation (9.64) we obtain
We can now write down the complete expression for the element
,
Projection onto the coset
gives (see Equation (9.20) and Section 6.3)
where we have used that
and
.
Next we want to compute the explicit form of the action in Equation (9.25). Choosing the following
normalization for the root generators,
which implies
one finds the form of the
-invariant action in the parametrization of Equation (9.58),