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9.1 Nonlinear sigma models on finite-dimensional coset spaces

A nonlinear sigma model describes maps ξ from one Riemannian space X, equipped with a metric γ, to another Riemannian space, the “target space” M, with metric g. Let xm (m = 1,⋅⋅⋅ ,p = dim X ) be coordinates on X and ξα (α = 1,⋅⋅⋅ ,q = dim M ) be coordinates on M. Then the standard action for this sigma model is
∫ √ -- S = dpx γ γmn (x) ∂mξα (x )∂nξβ(x) gαβ (ξ(x)) . (9.1 ) X
Solutions to the equations of motion resulting from this action will describe the maps ξ α as functions of xm.

A familiar example, of direct interest to the analysis below, is the case where X is one-dimensional, parametrized by the coordinate t. Then the action for the sigma model reduces to

∫ α β Sgeodesic = dtA d-ξ-(t) dξ-(t)gαβ (ξ(t)), (9.2 ) dt dt
where A is γ11√γ- and ensures reparametrization invariance in the variable t. Extremization with respect to A enforces the constraint
dξα(t)dξβ (t) ------------gαβ (ξ(t)) = 0, (9.3 ) dt dt
ensuring that solutions to this model are null geodesics on M. We have already encountered such a sigma model before, namely as describing the free lightlike motion of the billiard ball in the (dim M − 1 )-dimensional scale-factor space. In that case A corresponds to the inverse “lapse-function” N − 1 and the metric g αβ 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 π’’βˆ•π’¦ (𝒒 ) where 𝒒 is a Lie group with semi-simple real Lie algebra 𝔀 and 𝒦 (𝒒) its maximal compact subgroup with real Lie algebra 𝔨, corresponding to the maximal compact subalgebra of 𝔀. Since elements of the coset are obtained by factoring out 𝒦 (𝒒), 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 θ induces a Cartan decomposition of 𝔀 into even and odd eigenspaces:

𝔀 = 𝔨 ⊕ 𝔭 (9.4 )
(direct sum of vector spaces), where
𝔨 = {x ∈ 𝔀|θ (x ) = x}, (9.5 ) 𝔭 = {y ∈ 𝔀|θ(y ) = − y}
play central roles. The decomposition (9.4View Equation) is orthogonal, in the sense that 𝔭 is the orthogonal complement of 𝔨 with respect to the invariant bilinear form (⋅|⋅) ≡ B (⋅,⋅),
𝔭 = {y ∈ 𝔀| ∀x ∈ 𝔨 : (y |x ) = 0 }. (9.6 )
The commutator relations split in a way characteristic for symmetric spaces,
[𝔨,𝔨] ⊂ 𝔨, [𝔨,𝔭] ⊂ 𝔭, [𝔭,𝔭] ⊂ 𝔨. (9.7 )
The subspace 𝔭 is not a subalgebra. Elements of 𝔭 transform in some representation of 𝔨, which depends on the Lie algebra 𝔀. We stress that if the commutator [𝔭,𝔭] had also contained elements in 𝔭 itself, this would not have been a symmetric space.

The coset space π’’βˆ•π’¦ (𝒒) is defined as the set of equivalence classes [g ] of 𝒒 defined by the equivalence relation

g ∼ g′ iff gg′−1 ∈ 𝒦 (𝒒) and g,g′ ∈ 𝒒, (9.8 )
i.e.,
[g ] = {kg |∀k ∈ 𝒦 (𝒒 )}. (9.9 )

Example: The coset space SL (n, ℝ )βˆ•SO (n)

As an example to illustrate the Cartan involution we consider the coset space SL (n,ℝ )βˆ•SO (n). The group SL (n, ℝ) contains all n × n real matrices with determinant equal to one. The associated Lie algebra 𝔰𝔩(n, ℝ) thus consists of real n × n traceless matrices. In this case the Cartan involution is simply (minus) the ordinary matrix transpose T () on the Lie algebra elements:

τ : a ↦−→ − aT a ∈ 𝔰𝔩(n,ℝ ). (9.10 )
This implies that all antisymmetric traceless n × n matrices belong to 𝔨 = 𝔰𝔬(n ). The Cartan involution θ is the differential at the identity of an involution Θ defined on the group itself, such that for real Lie groups (real or complex matrix groups), θ is just the inverse conjugate transpose. Defining
𝒦 (𝒒) = {g ∈ 𝒒 |Θg = g} (9.11 )
then gives in this example the group 𝒦 (𝒒 ) = SO (n). The Cartan decomposition of 𝔰𝔩(n,ℝ ) thus splits all elements into symmetric and antisymmetric matrices, i.e., for a ∈ 𝔰𝔩(n,ℝ ) we have
a − aT ∈ 𝔰𝔬(n), a + aT ∈ 𝔭. (9.12 )

9.1.2 Nonlinear realisations

The group 𝒒 naturally acts through (here, right) multiplication on the quotient space 𝒒 βˆ•π’¦ (𝒒)34 as

[h ] ↦→ [hg]. (9.13 )
This definition makes sense because if h ∼ h ′, i.e., h′ = kh for some k ∈ 𝒦 (𝒒), then h′g ∼ hg since ′ h g = (kh )g = k(hg) (left and right multiplications commute).

In order to describe a dynamical theory on the quotient space π’’βˆ•π’¦ (𝒒), it is convenient to introduce as dynamical variable the group element V(x) ∈ 𝒒 and to construct the action for V (x) in such a way that the equivalence relation

∀k (x ) ∈ 𝒦 (𝒒 ) : V (x) ∼ k(x)V (x) (9.14 )
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 (9.13View Equation) as a rigid symmetry. Thus, the action should be invariant under
V (x) ↦−→ k(x)V (x )g, k(x ) ∈ 𝒦 (𝒒), g ∈ 𝒒. (9.15 )

One may develop the formalism without fixing the 𝒦(𝒒 )-gauge symmetry, or one may instead fix the gauge symmetry by choosing a specific coset representative V(x) ∈ π’’βˆ• 𝒦(𝒒 ). When 𝒦 (𝒒 ) is a maximal compact subgroup of 𝒒 there are no topological obstructions, and a standard choice, which is always available, is to take V(x ) 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,

V (x) ↦−→ V (x)′ = V (x)g, g ∈ 𝒒, (9.16 )
will destroy the gauge choice because V ′(x ) will generically not be of upper triangular form. Then, a compensating local 𝒦 (𝒒 )-transformation is needed that restores the original gauge choice. The total transformation is thus
′′ V(x ) ↦− → V (x ) = k(V (x),g)V (x)g, k(V (x),g) ∈ 𝒦(𝒒 ), g ∈ 𝒒, (9.17 )
where V′′(x ) is again in the upper triangular gauge. Because now k(V (x),g) depends nonlinearly on V (x), this is called a nonlinear realisation of 𝒒.

9.1.3 Three ways of writing the quadratic π“š (𝓖 )local × π“–rigid-invariant action

Given the field V(x), we can form the Lie algebra valued one-form (Maurer–Cartan form)

dV (x)V (x)−1 = dxμ ∂μV (x)V(x )− 1. (9.18 )
Under the Cartan decomposition, this element splits according to Equation (9.4View Equation),
∂ V (x)V (x)−1 = Q (x) + P (x), (9.19 ) μ μ μ
where Qμ (x ) ∈ 𝔨 and Pμ(x) ∈ 𝔭. We can use the Cartan involution θ to write these explicitly as projections onto the odd and even eigenspaces as follows:
1-[ −1 ( −1)] Q μ(x) = 2 ∂μV (x)V (x) + θ ∂μV (x)V (x) ∈ 𝔨, 1 [ ( )] (9.20 ) Pμ(x) = -- ∂μV (x)V (x)−1 − θ ∂μV (x)V (x)−1 ∈ 𝔭. 2
If we define a generalized transpose 𝒯 by
()𝒯 ≡ − θ(), (9.21 )
then P μ(x) and Q μ(x) correspond to symmetric and antisymmetric elements, respectively,
𝒯 𝒯 Pμ(x) = P μ(x), Q μ(x) = − Q μ(x). (9.22 )
Of course, in the special case when 𝔀 = 𝔰𝔩(n,ℝ ) and 𝔨 = 𝔰𝔬(n), the generalized transpose ( )𝒯 coincides with the ordinary matrix transpose ( )T. The Lie algebra valued one-forms with components −1 ∂μV (x )V(x), Q μ(x) and P μ(x) are invariant under rigid right multiplication, V (x) ↦→ V (x)g.

Being an element of the Lie algebra of the gauge group, Q μ(x) can be interpreted as a gauge connection for the local symmetry 𝒦(𝒒 ). Under a local transformation k (x ) ∈ 𝒦 (𝒒 ), Q μ(x) transforms as

𝒦 (𝒒) : Q μ(x) ↦−→ k(x)Q μ(x )k(x)−1 + ∂μk(x)k (x )−1, (9.23 )
which indeed is the characteristic transformation property of a gauge connection. On the other hand, P μ(x) transforms covariantly,
𝒦 (𝒒 ) : P (x) ↦−→ k (x )P (x)k(x )− 1, (9.24 ) μ μ
because the element ∂μk(x)k(x)− 1 is projected out due to the negative sign in the construction of P μ(x) in Equation (9.20View Equation).

Using the bilinear form (⋅|⋅) we can now form a manifestly 𝒦 (𝒒 ) × π’’ local rigid-invariant expression by simply “squaring” Pμ(x), i.e., the p-dimensional action takes the form (cf. Equation (9.1View Equation))

∫ p √ -- μν ( || ) Sπ’’βˆ•π’¦ (𝒒) = d x γγ P μ(x) Pν(x) . (9.25 ) X

We can rewrite this action in a number of ways. First, we note that since Q μ(x ) can be interpreted as a gauge connection we can form a “covariant derivative” D μ in a standard way as

D μV (x ) ≡ ∂μV (x ) − Q μ(x)V (x), (9.26 )
which, by virtue of Equation (9.20View Equation), can alternatively be written as
D μV (x) = Pμ(x )V (x). (9.27 )
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 𝒒-invariant expression −1 μ − 1 (∂μV (x)V (x) |∂ V (x )V(x) ) to the gauge field Q μ(x) through the “covariantization” ∂ μ → D μ,
( | ) ( | ) ( | ) ∂μV (x)V (x)−1|∂μV (x)V (x )−1 −→ D μV (x)V(x )− 1|D μV (x)V (x )−1 = Pμ(x )|P ν(x) .(9.28 )
Thus, the action then takes the form
∫ p √-- μν ( −1|| −1) S π’’βˆ•π’¦ (𝒒) = d x γ γ D μV (x)V (x ) DνV (x)V (x ) . (9.29 ) X

We can also form a generalized “metric” M (x) that does not transform at all under the local symmetry, but only transforms under rigid 𝒒-transformations. This is done, using the generalized transpose (extended from the algebra to the group through the exponential map [93Jump To The Next Citation Point]), in the following way,

M (x) ≡ V (x)𝒯V (x), (9.30 )
which is clearly invariant under local transformations
( ) 𝒦 (𝒒) : M (x) ↦−→ (k(x)V (x))𝒯 (k (x )V(x)) = V (x)𝒯 k (x )𝒯k(x) V (x) = M (x) (9.31 )
for k(x) ∈ 𝒦 (𝒒 ), and transforms as follows under global transformations on V (x) from the right,
𝒯 𝒒 : M (x ) ↦− → g M (x)g, g ∈ 𝒒. (9.32 )
A short calculation shows that the relation between M (x) ∈ 𝒒 and P(x) ∈ 𝔭 is given by
1- − 1 1-( 𝒯 )− 1 𝒯 ( 𝒯 )−1 𝒯 2M (x ) ∂μM (x) = 2 V (x ) V(x ) ∂μV (x) V (x) + V(x) V (x) V(x ) ∂μV(x ) 1 [( ) ] = --V(x)−1 ∂μV (x)V(x )− 1 𝒯 + ∂μV (x )V(x)−1 V (x ) 2 = V (x )−1Pμ(x )V (x). (9.33 )
Since the factors of V(x ) drop out in the squared expression,
( −1 || −1 μ ) ( || μ ) V (x) P μ(x)V(x )V (x) P (x )V(x) = P μ(x) P (x ) , (9.34 )
Equation (9.33View Equation) provides a third way to write the 𝒦 (𝒒 ) × π’’ local rigid-invariant action, completely in terms of the generalized metric M (x),
1 ∫ √ -- ( | ) Sπ’’βˆ•π’¦ (𝒒) = -- dpx γ γμν M (x)− 1∂ μM (x)|M (x )−1∂νM (x) . (9.35 ) 4 X
(We call M a “generalized metric” because in the GL (n,ℝ )βˆ•SO (n)-case, it does correspond to the metric, the field V being the “vielbein”; see Section 9.3.2.)

All three forms of the action are manifestly gauge invariant under 𝒦 (𝒒 ) local. 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 δV (x) of the action in Equation (9.25View Equation). The Lie algebra element δV (x )V(x)−1 ∈ 𝔀 can be decomposed according to the Cartan decomposition,

−1 δV (x)V (x ) = Σ(x) + Λ (x), Σ (x) ∈ 𝔨, Λ(x ) ∈ 𝔭. (9.36 )
The variation Σ (x) along the gauge orbit will be automatically projected out by gauge invariance of the action. Thus we can set Σ (x) = 0 for simplicity. Let us then compute δP μ(x). One easily gets
δPμ(x ) = ∂ μΛ(x) − [Qμ(x ),Λ (x)]. (9.37 )
Since Λ (x ) is a Lie algebra valued scalar we can freely set ∂ μΛ(x) → ∇ μΛ(x ) in the variation of the action below, where ∇ μ is a covariant derivative on X compatible with the Levi–Civita connection. Using the symmetry and the invariance of the bilinear form one then finds
∫ √ -- [( | )] δSπ’’βˆ•π’¦ (𝒒) = dpx γγ μν2 − ∇ νPμ(x ) + [Q ν(x ),Pμ(x)]|Λ(x) . (9.38 ) X
The equations of motion are therefore equivalent to
D μPμ (x) = 0, (9.39 )
with
D μP ν(x) = ∇ μPν(x) − [Qμ(x ),P ν(x)], (9.40 )
and simply express the covariant conservation of P μ(x).

It is also interesting to examine the dynamics in terms of the generalized metric M (x ). The equations of motion for M (x) are

( ) 1∇ μ M (x)−1∂μM (x ) = 0. (9.41 ) 2
These equations ensure the conservation of the current
1- −1 −1 π’₯μ ≡ 2 M (x) ∂μM (x) = V (x) P μ(x )V(x), (9.42 )
i.e.,
μ ∇ π’₯μ = 0. (9.43 )
This is the conserved Noether current associated with the rigid 𝒒-invariance of the action.

9.1.5 Example: SL (2,ℝ )βˆ•SO (2) (hyperbolic space)

Let us consider the example of the coset space SL (2,β„βˆ•SO (2), which, although very simple, is nevertheless quite illustrative. Recall from Section 6.2 that the Lie algebra 𝔰𝔩(2,ℝ) is constructed from the Chevalley triple (e,h,f ),

𝔰𝔩(2,ℝ ) = ℝf ⊕ ℝh ⊕ ℝe, (9.44 )
with the following standard commutation relations
[h,e] = 2e, [h,f ] = − 2f, [e,f] = h (9.45 )
and matrix realisation
(0 1) (1 0) (0 0) e = , h = , f = . (9.46 ) 0 0 0 − 1 1 0
In the Borel gauge, V(x ) reads
[φ (x ) ] (e φ(x)βˆ•2 χ(x)eφ(x)βˆ•2) V(x ) = Exp -----h Exp [χ(x )e] = −φ(x)βˆ•2 , (9.47 ) 2 0 e
where φ(x) and χ (x) represent coordinates on the coset space, i.e., they describe the sigma model map
X ∋ x ↦−→ (φ (x ),χ(x)) ∈ SL (2,ℝ)βˆ•SO (2). (9.48 )

An arbitrary transformation on V (x) reads

V (x) ↦−→ k(x )V(x)g, k(x) ∈ SO (2), g ∈ SL (2,ℝ ), (9.49 )
which in infinitesimal form becomes
δδk(x),δgV (x) = δk(x )V (x) + V (x )δg, δk (x ) ∈ 𝔰𝔬 (2 ), δg ∈ 𝔰𝔩(2,ℝ ). (9.50 )
Let us then check how V (x ) transforms under the generators δg = e,f,h. As expected, the Borel generators h and e preserve the upper triangular structure
( φ(x)βˆ•2) δV (x) = V (x)e = 0 e , e 0 0 ( φ(x)βˆ•2 φ(x)βˆ•2) (9.51 ) δhV (x) = V (x)h = e − χ(x)e , 0 − eφ(x)βˆ•2
while the negative root generator f does not respect the form of V (x),
( χ(x )e φ(x)βˆ•2 0) δfV(x) = V (x)f = −φ(x)βˆ•2 . (9.52 ) e 0
Thus, in this case we need a compensating transformation to restore the upper triangular form. This transformation needs to cancel the factor −φ(x)βˆ•2 e in the lower left corner of the matrix δfV (x) and so it must necessarily depend on φ(x ). The transformation that does the job is
( ) 0 e−φ(x) δk (x ) = − e−φ(x) 0 ∈ 𝔰𝔬(2), (9.53 )
and we find
δδk(x),fV(x ) = δk (x)V(x ) + V (x)f ( φ(x)βˆ•2 − 3φ(x)βˆ•2 ) = χ(x )e e ∈ SL (2,ℝ )βˆ•SO (2). (9.54 ) 0 − χ (x)e−φ(x)βˆ•2

Finally, since the generalized transpose ( )𝒯 in this case reduces to the ordinary matrix transpose, the “generalized” metric becomes

( eφ(x) χ(x)eφ(x) ) M (x) = V(x)T V(x) = φ(x) 2 φ(x) − φ(x) . (9.55 ) χ(x )e χ(x) e + e
The Killing form (⋅|⋅) corresponds to taking the trace in the adjoint representation of Equation (9.46View Equation) and the action (9.35View Equation) therefore takes the form
1 ∫ √ -- [ ] SSL(2,ℝ)βˆ•SO(2) =-- dpx γγμν ∂ μφ(x)∂ νφ(x) + e2φ(x)∂μχ(x )∂νχ(x) . (9.56 ) 2 X

9.1.6 Parametrization of 𝓖 βˆ•π“š (𝓖 )

The Borel gauge choice is always accessible when the group 𝒦 (𝒒) is the maximal compact subgroup of 𝒒. 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 𝒦 (𝒒) 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 π’’βˆ•π’¦ (𝒒), with 𝒦 (𝒒) being the maximal compact subgroup. Let Π = {α ∨,⋅⋅⋅ ,α ∨} 1 n be a basis of the Cartan subalgebra π”₯ ⊂ 𝔀, and let ⋆ Δ+ ⊂ π”₯ denote the set of positive roots. The Borel subalgebra of 𝔀 can then be written as

∑n ∑ π”Ÿ = ℝ α∨ + ℝE α, (9.57 ) i=1 i α∈Δ +
where Eα is the positive root generator spanning the one-dimensional root space 𝔀α associated to the root α. The coset representative is then chosen to be
[ n ] ⌊ ⌋ ∑ ∨ ∑ V(x ) = V1 (x)V2 (x ) = Exp φi(x)αi Exp ⌈ χα(x)E α⌉ ∈ 𝒒 βˆ•π’¦ (𝒒 ). (9.58 ) i=1 α∈Δ+
Because 𝔀 is a finite Lie algebra, the sum over positive roots is finite and so this is a well-defined construction.

From Equation (9.58View Equation) we may compute the Lie algebra valued one-form ∂μV (x)V(x )− 1 explicitly. Let us do this in some detail. First, we write the general expression in terms of V1(x) and V2(x),

∂ V (x)V (x)−1 = ∂ V (x)V (x )− 1 + V (x)(∂ V (x)V (x)−1)V (x)−1. (9.59 ) μ μ 1 1 1 μ 2 2 1
To compute the individual terms in this expression we need to make use of the Baker–Hausdorff formulas:
A −A -1 -1 ∂μe e = ∂ μA + 2![A, ∂μA ] + 3![A,[A, ∂μA ]] + ⋅⋅⋅ , A −A 1 (9.60 ) e Be = B + [A,B ] + 2![A, [A,B ]] + ⋅⋅⋅ .
The first term in Equation (9.59View Equation) is easy to compute since all generators in the exponential commute. We find
n −1 ∑ ∨ ∂μV1(x )V1 (x) = ∂μφi(x)α i ∈ π”₯. (9.61 ) i=1
Secondly, we compute the corresponding expression for V2(x ). Here we must take into account all commutators between the positive root generators E α ∈ 𝔫+. Using the first of the Baker–Hausdorff formulas above, the first terms in the series become
⌊ ⌋ ⌊ ⌋ ∑ ∑ ∂μV2 (x)V2(x)− 1 = ∂ μExp ⌈ χα (x )Eα⌉ Exp ⌈− χα′(x)E α′⌉ α∈Δ α ′∈Δ + + = ∑ ∂ χ (x)E + 1- ∑ χ (x)∂ χ ′(x )[E ,E ′] μ α α 2! ′ α μ α α α α∈Δ+ α,α∈Δ+ -1 ∑ + 3! χα(x )χα′(x )∂μχα′′(x)[E α,[E α′,Eα′′]] + ⋅⋅⋅ . (9.62 ) α,α′,α′′∈Δ+
Each multi-commutator [E α,[E α′,⋅⋅⋅]⋅⋅⋅ ,E α′′′] corresponds to some new positive root generator, say E γ ∈ 𝔫+. However, since each term in the expansion (9.62View Equation) is a sum over all positive roots, the specific generator E γ 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 E γ taking the form
-1 -1- β„› γ,μ (x ) ≡ ∂μχγ(x) + 2! χβ—Ÿζ(x)∂◝μβ—œχ-ζ′(x)β—ž+ ⋅⋅⋅ + k ! χη(x)χη′(x)⋅⋅⋅χ η′′(x )∂μ&# ζ+ ζ′=γ γ η+η′+⋅⋅⋅+ η′′+ η′′′=γ
where kγ denotes the number corresponding to the last term in the Baker–Hausdorff expansion in which the generator E γ appears. The explicit form of β„›γ,μ(x) must be computed individually for each root γ ∈ Δ+.

The sum in Equation (9.62View Equation) can now be written as

∑ ∂μV2(x )V2 (x)−1 = β„›α,μ(x)E α. (9.64 ) α∈Δ+
To proceed, we must conjugate this expression with V1(x ) in order to compute the full form of Equation (9.59View Equation). This involves the use of the second Baker–Hausdorff formula in Equation (9.60View Equation) for each term in the sum, Equation (9.64View Equation). Let h denote an arbitrary element of the Cartan subalgebra,
n ∑ ∨ h = φi(x)αi ∈ π”₯. (9.65 ) i=1
Then the commutators we need are of the form
[h,E α] = α(h)E α, (9.66 )
where α(h) denotes the value of the root α ∈ π”₯⋆ acting on the Cartan element h ∈ π”₯,
∑n ∑ n ∑n α (h) = φi(x )α(α∨) = φi(x)⟨α, α∨⟩ ≡ φi(x)αi. (9.67 ) i=1 i i=1 i i=1
So, for each term in the sum in Equation (9.64View Equation) we obtain
−1 ∑ 1-∑ V1(x )E αV1 (x) = E α + φi(x)αiE α + 2 φi(x )φj (x )αiαjE α + ⋅⋅⋅ [ i ] i,j ∑ = Exp φi(x)αi E α i α(h) = e E α. (9.68 )
We can now write down the complete expression for the element ∂μV (x)V (x)−1,
∑n ∑ ∂μV (x)V (x)−1 = ∂μφi(x)α∨i + eα(h)β„› α,μ(x )E α. (9.69 ) i=1 α∈Δ+
Projection onto the coset 𝔭 gives (see Equation (9.20View Equation) and Section 6.3)
∑n ∨ 1 ∑ α(h) P μ(x) = ∂μφi(x)αi + -- e β„› α,μ(x )(E α + E −α) , (9.70 ) i=1 2 α∈Δ+
where we have used that 𝒯 E α = E −α and ∨ 𝒯 ∨ (α i ) = αi.

Next we want to compute the explicit form of the action in Equation (9.25View Equation). Choosing the following normalization for the root generators,

(Eα |E α′) = δα,−α′, (αi∨|α ∨j ) = δij, (9.71 )
which implies
𝒯 (Eα |E α′) = (E α|E− α′) = δα,α′ (9.72 )
one finds the form of the 𝒦 (𝒒)local × π’’rigid-invariant action in the parametrization of Equation (9.58View Equation),
∫ ⌊ n ⌋ p √ --μν⌈ ∑ 1-∑ 2α (h) ⌉ S π’’βˆ•π’¦(𝒒) = d x γγ ∂μφi(x)∂νφi(x) + 2 e β„›α,μ(x)β„› α,ν(x) . (9.73 ) X i=1 α∈ Δ+

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