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The commutation relations of the so(n) generators are very simple and were already considered several times in Volume 1. We have:
[Jij , Jk ] = −δik Jj + δj k Ji − δj Jik + δi Jj k
The coset generators can instead be chosen as the following matrices:
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− |
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and satisfy the following commutation relations:
[Jij , Pk ] = −δik Pj + δj k Pi
[Pi , Pj ] = −κJij
(5.2.49)
(5.2.50)
(5.2.51)
(5.2.52)
Equation (5.2.51) states that the generators Pi transform as an n-vector under so(n) rotations (reductivity) while (5.2.52) shows that for both signs κ = ±1 the considered coset manifold is a symmetric space. Correspondingly we name kij = kij (y) ∂y∂ the Killing vector fields associated with the action of the generators Jij :
Jij Lκ (y) = kij Lκ (y) + Lκ (y)Jpq Wijpq (y) |
(5.2.53) |
while we name ki = ki (y) ∂y∂ the Killing vector fields associated with the action of the generators Pi :
Pi Lκ (y) = ki Lκ (y) + Lκ (y)Jpq Wipq (y) |
(5.2.54) |
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Resolving conditions (5.2.53) and (5.2.54) we obtain: |
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kij = yi ∂j |
− yj ∂i |
(5.2.55) |
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− κy2 ∂i + κyi y · ∂ |
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ki = |
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(5.2.56) |
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The H-compensators Wipq and Wijpq can also be extracted from the same calculation but since their explicit form is not essential for our future discussion we skip them.
5.2.3.2 Vielbeins, Connections and Metrics on G/H
Consider next the following 1-form defined over the reductive coset manifold G/H:
Σ(y) = L−1(y) dL(y) |
(5.2.57) |
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which generalizes the Maurer Cartan form defined over the group manifold G, discussed in Sect. 3.3.1.2 of Volume 1. As a consequence of its own definition the 1-form Σ satisfies the equation:
0 = dΣ + Σ Σ |
(5.2.58) |
which provides the clue to the entire (pseudo-)Riemannian geometry of the coset manifold. To work out this latter we start by decomposing Σ along a complete set of generators of the Lie algebra G. According with the notations introduced in the previous subsection we put:
Σ = V a Ta + ωi Ti |
(5.2.59) |
The set of (n − m) 1-forms V a = Vαa (y) dyα provides a covariant frame for the cotangent bundle CT(G/H), namely a complete basis of sections of this vector bundle that transform in a proper way under the action of the group G. On the other hand ω = ωi Ti = ωαi (y) dyα Ti is called the H-connection. Indeed, according to the theory exposed in Chap. 3 of Volume 1, ω turns out to be the 1-form of a bona-fide principal connection on the principal fibre bundle:
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(5.2.60) |
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which has the Lie group G as total space, the coset manifold G/H as base space and the closed Lie subgroup H G as structural group. The bundle P( GH , H) is uniquely defined by the projection that associates to each group element g G the equivalence class gH it belongs to.
Introducing the decomposition (5.2.59) into the Maurer Cartan equation (5.2.58), this latter can be rewritten as the following pair of equations:
dV a + Caib ωi V b = − |
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Cabc V b V c |
(5.2.61) |
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dωi + |
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Cij k ωj ωk = − |
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Cibc V b V c |
(5.2.62) |
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where we have used the Lie algebra structure constants organized as in (5.2.30)– (5.2.32).
Let us now consider the transformations of the 1-forms we have introduced. Under left multiplication by a constant group element g G the 1-form Σ(y)
transforms as follows:
Σ y = h(y, g)L−1(y)g−1 d gL(y)h−1 |
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= h(y, g)−1Σ(y)h(y, g) + h(y, g)−1 dh(y, g) |
(5.2.63) |
where y = g.y is the new point in the manifold G/H whereto y is moved by the action of g. Projecting the above equation on the coset generators Ta we obtain:
V a y = V b(y)Db a h(y, g) |
(5.2.64) |
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where D = exp[DH], having denoted by DH the (n− m) dimensional representation of the subalgebra H which occurs in the decomposition of the adjoint representation of G:
adj G = adj H DH |
(5.2.65) |
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Projecting on the other hand on the H-subalgebra generators Ti we get: |
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ω y = A h(y, g) ω(y)A −1 h(y, g) + A h(y, g) dA −1 h(y, g) |
(5.2.66) |
where we have set: |
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A = exp[AH] |
(5.2.67) |
Considering a complete basis TA of generators for the full Lie algebra G, the adjoint representation is defined as follows:
g G : g−1TAg ≡ adj(g)AB TB |
(5.2.68) |
In the explicit basis of TA generators the decomposition (5.2.65) means that, once restricted to the elements of the subgroup H G, the adjoint representation becomes block-diagonal:
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adj(h) = D0 |
A (h) |
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(5.2.69) |
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Note that for such decomposition to hold true the coset manifold has to be reductive according to definition (5.2.27).
The infinitesimal form of (5.2.64) is the following one:
V a (y + δy) − V a (y) = −εAWAi (y)Caib V b(y) |
(5.2.70) |
δyα = εAkAα (y) |
(5.2.71) |
for a group element g G very close to the identity as in (5.2.33). |
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Similarly the infinitesimal form of (5.2.66) is: |
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ωi (y + δy) − ωi (y) = −εA Cikj WAk ωj + dWAi |
(5.2.72) |
5.2.3.3 Lie Derivatives
The Lie derivative of a tensor Tα1...αp along a vector field vμ provides the change in shape of that tensor under an infinitesimal diffeomorphism:
yμ → yμ + vμ(y) |
(5.2.73) |
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Explicitly one sets: |
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vTα1...αp (y) = vμ∂μTα1...αp + ∂α1 vγ Tγ α2...αp + · · · + ∂αp vγ Tα1α2...γ |
(5.2.74) |
In the case of p-forms, namely of antisymmetric tensors the definition (5.2.74) of Lie derivative can be recast into a more intrinsic form using both the exterior differential d and the contraction operator.
Definition 5.2.5 Let M be a differentiable manifold and let Λk (M ) be the vector bundles of differential k-forms on M , let v Γ (T M , M ) be a vector field. The contraction ik is a linear map:
iv : Λk (M ) → Λk−1(M ) |
(5.2.75) |
such that for any ω(k) Λk (M ) and for any set of k − 1 vector fields w1, . . . , wk−1, we have:
ivω(k)(w1, . . . , wk−1) ≡ kω(k)(v, w1, . . . , wk−1) |
(5.2.76) |
Then by going to components we can verify that the tensor definition (5.2.74) is equivalent to the following one:
Definition 5.2.6 Let M be a differentiable manifold and let Λk (M ) be the vector bundles of differential k-forms on M , let v Γ (T M , M ) be a vector field. The Lie derivative v is a linear map:
v : Λk (M ) → Λk (M ) |
(5.2.77) |
such that for any ω(k) Λk (M ) we have:
vω(k) ≡ iv dω(k) + divω(k) |
(5.2.78) |
On the other hand for vector fields the tensor definition (5.2.74) is equivalent to the following one.
Definition 5.2.7 Let M be a differentiable manifold and let T M → M be the tangent bundle, whose sections are the vector fields. Let v Γ (T M , M ) be a vector field. The Lie derivative v is a linear map:
v : Γ (T M , M ) → Γ (T M , M ) |
(5.2.79) |
such that for any w Γ (T M , M ) we have:
vw ≡ [v, w] |
(5.2.80) |