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2.7 The torus universe

The simplest nontrivial vacuum cosmology occurs for a spacetime with the topology R × T 2, where 2 T is the two-dimensional torus. This case is in some ways exceptional - for example, the standard metric gij of Equation (31View Equation) is flat rather than hyperbolic - but it is also simple enough that a great deal can be done explicitly. Later in this review, the torus universe will be a canonical test of quantization; here we review classical aspects. The problem of finding the classical solutions, as well as an approach to the quantization, was, I believe, first discussed by Martinec  [193Jump To The Next Citation Point] . I refer the reader to  [7681Jump To The Next Citation Point87Jump To The Next Citation Point86Jump To The Next Citation Point] for further details. A similarly detailed analysis may be possible when the spatial topology is that of a Klein bottle (see, for instance, [181Jump To The Next Citation Point]) but so far, this and other nonorientable examples have been studied in much less detail.

For simplicity, let us initially restrict our attention to the case /\ = -1/l2 < 0 . The group G of Section  2.2, or, equivalently, the gauge group in the Chern-Simons formalism of Section  2.3, is then SO(2, 2) . The fundamental group 2 p1(R × T ) has two generators, [g1] and [g2], satisfying a single relation similar to Equation (12View Equation):

[g1] .[g2] = [g2] .[g1]. (37)
The holonomy group (11View Equation) is therefore generated by two commuting SO(2, 2) matrices, unique up to overall conjugation.

It is a bit more convenient to describe the holonomies as elements of the covering group SL(2, R) × SL(2, R)   [211Jump To The Next Citation Point] . Let r±[ga] denote the two SL(2,R) holonomies corresponding to the curve ga . An SL(2,R) matrix S is called hyperbolic, elliptic, or parabolic according to whether |trS | is greater than, equal to, or less than 2, and the space of holonomies correspondingly splits into nine sectors. It may be shown that only the hyperbolic-hyperbolic sector corresponds to a spacetime in which the T2 slices are spacelike  [117Jump To The Next Citation Point119182Jump To The Next Citation Point209] . By suitable overall conjugation, the two generators of the holonomy group in this sector can then be taken to be

( ± ) ( ± ) ± er1 /2 0 ± er2 /2 0 r [g1] = 0 e- r±1 /2 , r [g2] = 0 e-r±2 /2 , (38)
where the ra± are four arbitrary parameters. Note that this gives the right counting: The Riemann moduli space of the torus is two dimensional, so from Section  2.4 we expect a four-dimensional space of solutions.

To obtain the corresponding geometry, we can use the quotient space construction of Section  2.2 . Note first that three-dimensional anti-de Sitter space can be represented as the submanifold of flat R2,2 (with coordinates (X1, X2, T1,T2) and metric dS2 = dX21 + dX22 - dT 21 - dT22) defined by the condition that

( ) det |X |= 1, X = 1- X1 + T1 X2 + T2 . (39) l - X2 + T2 X1 - T1
This gives an isometry between AdS3 and the group manifold of SL(2,R) . The quotient of AdS3 by the holonomy group (38View Equation) may now be obtained by allowing the + r [ga] to act on X by left multiplication and the r-[ga] to act by right multiplication.

It is straightforward to show that the resulting induced metric is

[ ] 2 2 l2 + 2 - 2 + - 2t 2 ds = dt - -- (r1 ) + (r1 ) + 2r1 r1 cos-- dx 4 [ l ] l2 + + - - + - - + 2t - 2 r1 r2 + r1 r 2 + (r1 r2 + r1 r2 )cos l dxdy 2[ ] - l- (r+ )2 + (r- )2 + 2r+ r-cos 2t dy2, (40) 4 2 2 2 2 l
where x and y are coordinates with period 1 . An easy calculation confirms that this is a space of constant negative curvature. The triad may be read off directly from Equation (40View Equation), and it is easy to solve Equation (15View Equation) for the spin connection w . The resulting Chern-Simons connections A( ±) of Equation (23View Equation) are flat, and their holonomies reproduce the holonomies (38View Equation) of the geometric structure we began with.

To relate these expressions to the ADM formalism of Section  2.4, we must first find the slices of constant extrinsic curvature T . For the metric (40View Equation), the extrinsic curvature of a slice of constant t is T = - 2l cot 2lt, which is independent of x and y . A constant t slice is thus also a slice of constant York time. The standard flat metric on T 2, the genus one version of the standard metric (31View Equation), is

2 -1 2 ds = t2 |dx + t dy| , (41)
where t = t + it 1 2 is the modulus. Comparing (40View Equation), we see that a slice of constant t has a modulus
t = (r- eit/l + r+ e-it/l)(r- eit/l + r+e -it/l).-1 (42) 1 1 2 2
The conjugate momentum p = p1 + ip2 can be similarly computed from Equation (33View Equation),
il ( )2 p = - -----2t- r+2 eit/l + r-2 e- it/l , (43) 2sin l
while the ADM Hamiltonian H of Equation (35View Equation) becomes
( )- 1/2 l2 2t - + + - 2 4- [ 2 ]1/2 H = 4 sin l (r1 r2- r1 r2 ) = T + l2 t2pp . (44)
In the limit of vanishing /\, these relations go over to those of  [66Jump To The Next Citation Point] .

To quantize this system, we will need the classical Poisson brackets, which can be obtained from Equation (26View Equation):

± ± 1- + - {r1 ,r2 }= ± l, {ra ,rb }= 0. (45)
These, in turn, determine the brackets among the moduli and momenta t and p,
{t, p}= {t,p}= 2, {t, p}= {t,p}= 0, (46)
a result consistent with Equation (36View Equation). It may be shown that the version of Hamilton’s equations of motion coming from these brackets reproduces the time dependence (42View Equation) of the moduli; see  [87Jump To The Next Citation Point135Jump To The Next Citation Point] for details. The Poisson brackets among the traces of the holonomies (38View Equation) are also easy to compute. If we let
± 1 ± r±1 R 1 = -trr [g1] = cosh---, 2 2 ± 1- ± r±2- R 2 = 2 trr [g2] = cosh 2 , (47) ± ± ± 1- ± (r1-+-r2 ) R 12 = 2 trr [g1 .g2] = cosh 2 ,
it is not hard to check that
1 {R1±,R2± }= ± --(R1±2 - R1±R2±) and cyclical permutations, (48) 4l
reproducing the Poisson algebra of Nelson, Regge, and Zertuche  [211Jump To The Next Citation Point] .

Finally, let us consider the action of the torus mapping class group. This group is generated by two Dehn twists, which act on p (T2) 1 by

- 1 S : g1 --> g2 , g2 --> g1, (49) T : g1 --> g1 .g2, g2 --> g2.
These transformations act on the parameters r±a and the ADM moduli and momenta as
S : r± --> r± r± --> - r± t --> - 1- p --> t2p 1 2 2 1 t (50) ± ± ± ± ± T : r1 --> r1 + r2 r2 --> r2 t --> t + 1 p --> p.
These transformations are consistent with the relationships between the ADM and holonomy variables, and that they preserve all Poisson brackets.

For a torus universe with zero or positive cosmological constant, similar constructions are possible. I refer the reader to  [81Jump To The Next Citation Point] for details.


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