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B.1 Homogeneous models

In Bianchi models the transitive symmetry group acts freely on S, which implies that S can locally be identified with the group manifold S. The three generators of LS will be denoted as TI, 1 < I < 3, with relations K [TI,TJ ] = CIJTK, where K C IJ are the structure constants of LS fulfilling CJIJ = 0 for class A models by definition. The Maurer-Cartan form on S is given by hMC = wITI with left invariant one-forms wI on S, which fulfill the Maurer-Cartan equations
dwI = - 1CI wJ /\ wK . (71) 2 JK
Due to F = {1}, all homomorphisms c : F --> G are given by 1 '--> 1, and we can use the embedding i = id: S/F --> S. An invariant connection then takes the form A = f~o hMC = ~fiItiwI = Aiatidxa with matrices ti generating LSU(2). The scalar field is given by f~: LS --> LG, TI '--> ~f(TI ) =: ~fiti I already in its final form, because condition (69View Equation) is empty for a trivial isotropy group.

Using left invariant vector fields XI obeying wI (XJ ) = dI J and with Lie brackets [XI ,XJ ] = CK XK IJ the momenta canonically conjugate to Ai = ~fiwI a I a can be written as Ea = V~ g-~pIXa i 0 i I with ~pI i being canonically conjugate to i ~fI. Here, I 2 g0 = det(w a) is the determinant of the left invariant metric sum (g0)ab := I wIawIb on S, which is used to provide the density weight of Eai. The symplectic structure can be derived from

integral integral --1--- 3 i a --1--- 3 V~ --~i J I --V0--~i I 8pgG S d xA aE i = 8pgG S d x g0fIp~i w (XJ ) = 8pgG fI~pi ,

to obtain

{~fi, ~pJ}= 8pgGV0di dJ (72) I j j I
with the volume integral V ~ -- V0 := Sd3x g0 of S measured with the invariant metric g0.

It is convenient to absorb the coordinate volume V 0 into the fields by redefining fi := V 1/3f~i I 0 I and I 2/3 I pi := V0 ~pi. This makes the symplectic structure independent of V0 in accordance with background independence. These redefined variables automatically appear in holonomies and fluxes through coordinate integrations.

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