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B.2 Isotropic models

On Bianchi models, additional symmetries can be imposed, which corresponds to a further symmetry reduction and introduces non-trivial isotropy subgroups. These models with enhanced symmetry can be treated on an equal footing by writing the symmetry group as a semidirect product S = N ><| F r, with the isotropy subgroup F and the translational subgroup N, which is one of the Bianchi groups. Composition in this group is defined as (n1, f1)(n2, f2) := (n1r(f1)(n2),f1f2), which depends on the group homomorphism r : F --> AutN into the automorphism group of N (which will be denoted by the same letter as the representation on AutLN used below). Inverse elements are given by (n,f)-1 = (r(f -1)(n -1),f-1). To determine the form of invariant connections we have to compute the Maurer-Cartan form on S (using the usual notation):
(S) hMC(n, f) = (n,f )-1d(n,f) = (r(f -1)(n-1),f- 1)(dn, df) = (r(f- 1)(n -1)r(f-1)(dn),f -1df) = (r(f -1)(n -1dn),f -1df ) ( ) = r(f -1)(h(NM)C (n)),h(FMC)(f ) . (73)
Here the Maurer-Cartan forms h(MNC) on N and h(FMC) on F appear. We then choose an embedding i: S/F = N --> S, which can most easily be done as i: n '--> (n,1). Thus, * (S) (N) i hMC = hMC, and a reconstructed connection takes the form (S) ~f o i*hMC = ~fiIwIti, which is the same as for anisotropic models before (where now wI are left invariant one-forms on the translation group N). However, here f~ is constrained by equation (69View Equation) and we get only a subset as isotropic connections.

To solve Equation (69View Equation) we have to treat LRS (locally rotationally symmetric) models with a single rotational symmetry and isotropic models separately. In the first case we choose LF = <t3>, whereas in the second case we have LF = <t1, t2,t3> (<.> denotes the linear span). Equation (69View Equation) can be written infinitesimally as

f~(adti(TI)) = addc(ti)~f(TI ) = [dc(ti), ~f(TI)]

(i = 3 for LRS, 1 < i < 3 for isotropy). The TI are generators of LN = LF_ L, on which the isotropy subgroup F acts by rotation, adt (TI) = eiIK TK i. This is the derivative of the representation r defining the semidirect product S: Conjugation on the left hand side of (69View Equation) is -1 Ad(1,f)(n,1) = (1,f )(n, 1)(1,f ) = (r(f)(n),1), which follows from the composition in S.

Next, we have to determine the possible conjugacy classes of homomorphisms c : F --> G. For LRS models their representatives are given by

c : U(1) --> SU(2), exp tt '--> exp ktt k 3 3

for k (- N = {0,1,...} 0 (as will be shown in detail below for spherically symmetric connections). For the components ~i fI of ~ f defined by ~ ~i f(TI ) = fIti, Equation (69View Equation) takes the form ~j ~l e3IK fK = ke3ljfI. This has a non-trivial solution only for k = 1, in which case ~f can be written as

f~1 = ~at1 + ~bt2, ~f2 = - ~bt1 + ~at2, ~f3 = ~ct3

with arbitrary numbers ~a, ~b, ~c (the factors of 1 2- 2 are introduced for the sake of normalization). Their conjugate momenta take the form

~p1 = 1(~pat1 + ~pbt2), ~p2 = 1(- ~pbt1 + ~pat2), ~p3 = ~pct3, 2 2

and the symplectic structure is given by

{~a, ~p }= {~b,p~}= {~c, ~p }= 8pgGV a b c 0

and vanishes in all other cases. There is remaining gauge freedom from the reduced structure group Z ~= U(1) c which rotates the pairs (~a,~b) and (p~ , ~p) a b. Gauge invariant are then only V~ ------- ~a2 + ~b2 and its momentum V~ ------- (~a~pa + ~b~pb)/ ~a2 + ~b2.

In the case of isotropic models we have only two homomorphisms c0 : SU(2) --> SU(2), f '--> 1 and c1 = id up to conjugation (to simplify notation we use the same letters for the homomorphisms as in the LRS case, which is justified by the fact that the LRS homomorphisms are restrictions of those appearing here). Equation (69View Equation) takes the form ~ j eiIKf K = 0 for c0 without non-trivial solutions, and j eiIKf~K = eilj~flI for c1. Each of the last equations has the same form as for LRS models with k = 1, and their solution is ~fiI = ~cdiI with an arbitrary ~c. In this case the conjugate momenta can be written as ~pI = ~pdI i i, and we have the symplectic structure {~c,p~}= 8pGgV 3 0.

Thus, in both cases there is a unique non-trivial sector, and no topological charge appears. The symplectic structure can again be made independent of V0 by redefining 1/3 a := V 0 ~a, 1/3 b := V0 ~b, c := V01/3~c and pa := V02/3p~a, pb := V20/3~pb, pc := V02/3~pc, p := V20/3~p. If one computes the isotropic reduction of a Bianchi IX metric following from the left-invariant 1-forms of SU(2), one obtains a closed Friedmann-Robertson-Walker metric with scale factor V~ --- a = 2~a = 2 |~p| (see, e.g., [36] for the calculation). Thus, we obtain the identification (18View Equation) used in isotropic loop cosmology. (Such a normalization can only be obtained in curved models.)

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