### 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
where and represent coordinates on the coset space, i.e., they describe the sigma model map

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),