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IC/2000/102
United Nations Educational Scientific and Cultural Organizationand
International Atomic Energy Agency
THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
SUPERSYMMETRY AND QUANTUM COSMOLOGY
F. Assaoui and T. Lhallabi *
Section of High Energy Physics, H. E. P. L, University Mohammed V,Scientific Faculty, Rabat, Morocco
andThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.
AbstractThe development of the N = 4 supersymmetric approach to quantum cosmology based
on the non-compact global O(d,d) symmetries of the effective action is given. The N = 4supersymmetric action whose bosonic sector is invariant under O(d,d) is determined. Arepresentation for supercharges is obtained and the form of the zero and one-fermion quantumstates leading to the Wheeler-DeWitt equation is found.
MIRAMARE - TRIESTE
August 2000
Regular Associate of the Abctus Salam ICTP. E-mail: [email protected]
1 - Introduction
The development of the quantum cosmology has been an important motivation in the
sense that the initial conditions for the emergence of the Universe as a classical result have
been explained. In principle the form of the wave function satisfying the Wheeler-DeWitt
equation must be obtained [1]. This equation describes the annihilation of the wave function
by the Hamiltonian operator and admits an infinite number of solutions. The boundary
conditions must be taken in order to specify the wave function uniquely and are viewed as an
additional physical law [2], On the other hand, in string theory there were many attempts to
develop a consistent theory of string cosmology where inflation plays an important role [3],
The solutions of the non-linear sigma model equations in a Friedman-Robertson-Walker
background for the graviton, dilaton and antisymmetric tensor have been found [4, 3].
Furthermore, for isotropic Friedman-Robertson-Walker cosmologies, which are restricted to
flat space, the dilaton-graviton sector of the string effective action is invariant under an
inversion of the scale factor and a shift in the dilaton field [5]. A supersymmetnc extension of
the quantum cosmology is obtained from the scale factor duality, which is a subgroup of T-
duality [6, 7]. Moreover, N = 2 supersymmetric approach to quantum cosmology, with
spatially flat and homogeneous Bianchi type I Universe admitting d-compact abelian
isometries, is developed by employing the non-compact global symmetries of the string
effective action [8],
The purpose of the present paper is to develop an N = 4 supersymmetric approach to
quantum cosmology whose bosonic sector is invariant under global O(d,d) transformations.
The outline of this paper is as follows: In section 2, we recall the N = 2 supersymmetric
approach to quantum cosmology given by J. E. Lidsey and I. Maharana [8] which leads to the
O(d,d) invariant Wheeler-DeWitt equation. In section 3, we present the extension of this
model to the N = 4 supersymmetric case where the bosonic sector is invariant under the global
O(d,d) transformations. The corresponding super-constraints on the wave function are then
obtained. Thereafter, we derive the classical Hamiltonian by using the classical momenta
conjugate to the bosonic and fermionic degrees of freedom. Furthermore, the N = 4
supersymmetric quantum constraints are solved for the zero fermion and one-fermion states
leading to the Wheeler-DeWitt equation. Finally, in section 4, we make concluding remarks
and discuss our results.
2 - N = 2 Supersymmetric String Quantum Cosmology
In this section we recall the N = 2 supersymmetric approach to quantum cosmology given
by J. E. Lidsey and J. Maharana in ref. [8] where the non-compact global symmetries of the
string effective action are used. However, the tree-level string effective action is given by [9].
12(2.1)
where <P is the dilaton field, H ^is the field strength of the antisymmetric torsion tensor B^v ,
R is the Ricci curvature scalar of the space-time with metric G^v, g = detG, \$ is the
fundamental string length scale and V is an interaction potential. The integration over the
spatial variables in equation (2.1), by specifying Xs= 2 [10], implies that
+ Tr(M\M-ly)+ (2.2)
where a spatially closed, flat, homogeneous ( Bianchi type I ) space-time is assumed and
where the dilaton and two-form potential are taken to be constant on the surfaces of
homogeneity
t = constant. Furthermore, the shifted dilaton field <f) and the dilaton time parameter are
respectively given by
= <!> ~-Ln\g\
T =
M is a symmetric 2d x 2d matrix taken as follows:
' r1 ~G1B
(2.3)
(2.4)
M =BG~l G-BG~]B
(2.5)
where prime denotes differentiation with respect to x. The matrix M is an element of the
group O(d,d) satisfying the conditions
Mr\ M - r\
M = MT
and its inverse is given linearly by
M~l = rj Mr)
(2.6)
(2.7)
where
7] =0
0(2.8)
and I is the d x d unit matrix. Therefore, the kinetic part of the action (2.2) is invariant under
global O(d,d) transformations [10]:
M = QTMQ, (2.9)
with Q, a constant matrix.
On the other hand, the classical Hamiltonian for this cosmological model is given by
= \n-2 - 2Tr{MnMMnM) - Ve~2* (2.10)
where
= 20'
= --M~lM'M-(2.11)
are respectively the canonical momenta of 0 and M. The cosmology is quantified by
identifying the momenta (2.11) with the following differential operators:
. S . 8(2.12)
The insertion of equations (2.12) into the expression (2.10) leads to the Wheeler-DeWitt
equation [11].
<54+87V /8M SM
v
yf((j),M) = 0 (2.13)
which is manifestly free from problems of quantum ordering [12]. In order to obtain an N = 2
supersymmetric lagrangian whose bosonic sector is invariant under the global O(d,d)
transformations J. E. Lidsey and J. Maharana [8] have defined the following superfields
,0,0) = Mtj(T) + iW
,6) = 0 (T) + ix(T)0 + iX(?)9 + f(T)60
where My(t) is given by (2.5), {\|/ij, %} are complex spinors and (ij) = (1, 2,
Furthermore, the N = 2 supersymmetric effective action [8] is written as
(2.14)
(2.15)
. . , 2d).
where
(2.16)
(2.17)
Y(r,d,6) = -b8
- W(D)
A A
with D , D2 the derivative operators namely
(2.18)
de(2.19)
(2.20)
and the potential W(D) is an arbitrary function of D. The expansion of the potential W(D)
around <j> and the use of the expressions (2.14) and (2.15) in (2.17) and (2.18) leads
respectively to
C = M L , + A ) (2-2D
where
^[w^W^W^Wu^+iM^M^} (2.22)o
(2.23)
We remark that in the expression of Li given by J. E. Lidsey and J. Maharana [8] the coupling
term
$\w)$f (2.24)
had been omitted. In this case the use of the equation of motion for the auxiliary field f gives
(2.25)
and the N = 2 supersymmetric action is as follows
UuirtJriJk}jruri" ~ ¥<#( 2 2 6 )
\{%X -XX)~
From this action, the classical Hamiltonian is written as
1 1 1 _H ~ 2n,rtJkTt,,rf' + -n\ + — (d-W)2 +-(dlW)[Y,X] (2.27)
* H 4 * 4 f 4 'where
An identical expression for the Hamiltonian (2.27) is obtained from the anticommutator of the
supercharges Q and Q , which are defined by
Q =•v Z
(2.29)
with
Q2 = 0 = Q2 , (2.30)
namely
{QiQ}=2H (2.31)
[H , Q ] = 0 = [ H , Q \ (2.32)
Therefore, there exists an N = 2 supersymmetric in quantum cosmology [13,7] which can be
considered as a direct extension of the O(d,d) T-duality of the toroidally compactified string
effective action (2.1). In the next section, we show that there exists even N = 4
supersymmetry in the quantum cosmology.
3 - N = 4 Supersymmetric String Quantum Cosmology
It is known that a hidden symmetry exists in all Bianchi class A models [14] where the
classical superspace Hamiltonian may be viewed as the bosonic part of a supersymmetric
Hamiltonian [7]. This implies that a supersymmetry can be introduced at the quantum level.
These supersymmetric extensions of the quantum theory have significant consequences for
quantum cosmology and may provide valuable insight into some of the questions relevant to a
complete theory of quantum gravity [15]. However, in order to obtain an N = 4
supersymmetric lagrangian whose bosonic sector is invariant under the global O(d,d)
transformations we consider the generators for the N = 4 supersymmetry which are defined by
d l (3.1)
where
0* = 8+e = 9aliU*,
u±auT=i, u±aut = o (3'2)a
with p and Ua ,CC =1,2 are the harmonic variables [16]. The N = 4 supersymmetric
transformation of a superfield Q, is given by
8Q = [elQ- + elQ)a (3.3)
where £7 are arbitrary superparameters. Furthermore, we define the following N = 4
superfields
26-2+2K:(T, U) +6+26-2F..(T, U)
, ) + e
+ 0+r2c+(r, u) +e-9+2c-(z, u
in terms of which the N = 4 supersymmetric effective action is expressed
d*6-dU[^+Y] (3.6)
where £ and Y are respectively given by
k ^ i ^ g l ( i ) ] ( m k l ) n u (3.7)o
Y = (D++D)(D-D) - ig20-2D)(D+2D) - W(D) (3.8)
where gt and g2are coupling constants, the potential W(D) is an arbitrary function of D and
the derivative operators are given by
D =
D+ =
d9 +
a(3.9)
9 0(3.10)
where
=U±l-
are the harmonic derivatives [16].
By using the expansions (3.4), (3.5) and by expanding the potential W(D) around^ the
(4,0) supersymmetric effective action becomes
T^.UJ _
, i *
X+ C")
(3.11)
dw -d2w- l7 Tr - .Tl
where we have used the equations of motion of auxiliary superfields (h^+\FH,Hkl) and
(d,b±j:,f) namely
(d++d- + a a+ + )M, y = o
r.=o
and where the dotted line in (3.11) indicates couplings between the superfields
(I//"*, Ky),{X±^ - + ) a n ^ higher derivative terms with respect to x and 0 have been
omitted. Let us note that we have taken into account the coupling 0 d 2-W that had been
ignored in the N = 2 supersymmetric case [8]. Furthermore, concerning the integration with
respect to the harmonic variables we consider the following harmonic expansion of each
component of the superfields (3.4) and (3.5) in terms of symmetrised products of U [16]
U ...U (3-14)
where q is the Cartan-Weyl charge. The integration over U± is defined by using the rules
(3.15)t
idu u;ai ...t/;c/- . . . t /^ = o,n+m > oOn the other hand, the action (3.11) reduces to the bosonic action (2.2) in the limit where the
Grassmann variables vanish and if we identify the potential
rdw^ (3.16)
In the remaining part of this section we take for simplicity g{ =1, g2 ~l and in order to derive
the classical Hamiltonian for the N = 4 supersymmetric case let us give the classical momenta
conjugate to the bosonic and fennionic degrees of freedom in action (3.11) namely
dLK + =
T mn
dL
dL = 1
dk+ 8mn
dL_^h
^» 8
^ " ' (3.17)dfdL _ 1
Z+' ~ 2
dL 1
07 2
X
d L 1 (
^ ac+ ' 2+ _ dL _ 1 (
^ acr' 2(
with
/^•0) = [dxdJJ LSusy J
Therefore, the N = 4 supersymmetric classical Hamiltonian is as follows
C+ 'n^ + C-'n+c_ - L
10
which can be rewritten by using (3.17) as
H = jk li
~dwdtp
1 2+
4( 7 r ^
dp _
d2W— C+'
- i 2 (3.19)
Then, the bosonic component of the N = 4 supersymmetric Hamiltonian (3.19) corresponds to
the classical Hamiltonian for the cosmological model (2.10). The fermions do not appear in
the expression of the Hamiltonian for the matrix My as in the N =2 supersymmetric case [8].
However, the N = 4 supersymmetric model is quantified by using the standard operator
realization (2.12). Furthermore, we impose the following spinor algebra
x- }=o, {r , x }=ic- }=o, {c+, c~ }=i
K hWt (3-20)
which is satisfied by introducing the set of the Grassmann variables
that
d
such
<i = n
K+ =•
X =
(3.21)
c+ = c- = t*-
An identical expression for the Hamiltonian (3.19) can be obtained from the anticommutator
of the supercharges which are defined by
11
=
71X
These supercharges satisfy the following relations
(3.22)
_2
and lead to
Q+ = 0 = Q
{Q+ , Q }=[H , Q+] = 0 = [ff , fi"]
(3.23)
(3.24)
Thus, there exists an N = 4 supers ymmetry in the quantum cosmology as in the N = 2
supersymmetric case [7,13]. Defining the conserved fermion number solves the (4,0)
supersymmetric quantum constraints
F = W (3.25)
which satisfies
[<r, F] = 2^ +
\Q-,F]=-2Q~
[H,F]= o
Furthermore, the fermion vacuum 0) is defined as follows
y: 0) = 0 = tf! 0),r 9 / y / '
(3.26)
(3.27)
and the state with zero fermion number y/0) is given by
(3.28)
12
where h is an arbitrary function of My and (f) . Since the supersymmetry implies that the wave
function of the Universe is annihilated by the supercharges Q* then the state (3.28) is
automatically annihilated by the supercharge Q+ and it is annihilated by the supercharge Q~ if
the following conditions are satisfied
dh
dM.,= 0
dh dw\~=r \hCd(j>2 )
0) = 0
The condition (3.29) shows that
and if we assume that
x~\o} * c~|o>the condition (3.29.b) leads to
which means that h = constant and
-dW0 ——
30
= 0
(3.29.a)
(3.29.b)
(3.30)
(3.31)
(3.32)
(3.33)
which implies, with the use of (3.16), that this case corresponds to a situation where the
potential V = 0. Moreover, if we take
(3.34)
then, condition (3.29.b) becomes
13
dh dT" ^ ^
80 80
where the general solution is given by
= e 0)
h = 0 (3.35)
(3.36)
with
0~d0
(3.37)
This solution is uniquely determined by the potential (3.16) as in the N = 2 supersymmetric
case [8].
On the other hand, we define the one-fermion state y/~ ) as follows
-w^ =o 0.3s)
where h^, k;j, hx, kc are arbitrary functions of the bosonic variables over the configuration
space. Such state is annihilated by the supercharge Q~ if we consider
K
K
tate V~i
V2i
f
> i .
V
rewritten
! (hi y>
)
as
(3.39)
(3.40)
and the action of the supercharge Q+ on it, with the use of equation (3.24), leads to
(3.41)
14
M.}, (j) J is a solution of the following equation
d30
h(Mir<j))=0 (3.42)
Finally, for separable solution
(3.43)
the differential equation (3.42) implies that
8 * 8 » C2
8M <5M,; 8\) id
d2 o
3 0 2V /
r(0)=o
(3.44)
(3.45)
where C is a separation constant. We note that for a constant dilaton potential equation (3.45)
can be solved. However, the duality symmetries discussed in ref. [17] can be used for the N =
4 supersymmetric quantum cosmology model. Moreover, it is known that spatially flat
isotropic cosmologies derived from the Brans-Dicke gravity action exhibit a scale factor
duality invariance [18,6]. This classical duality, associated with the hidden N = 2
supersymmetry at the quantum level [18,6], can also be studied for the N = 4 supersymmetric
case. Finally, the dualities of the Bianchi models related to N = 4 supersymmetry would be of
interest to be investigated.
15
4 - Conclusion
In this paper, we have derived an N = 4 supersymmetric quantum cosmology from the
N= 4 supersymmetric effective action. The N = 4 supersymmetric Hamiltonian operator is
obtained by using the momenta conjugate to the bosonic and fermionic degrees of freedom. In
the classical limit this operator reduces to the O(d,d) invariant Hamiltonian. The constraints
on the wave function of the Universe are imposed by supersymmetry and imply that it should
be annihilated by supercharges. On the other hand, the solutions of the N = 4 supersymmetric
constraints are found for the zero and one-fermion states. We have seen that when the
fermionic states j£~ 0^ and C~ 0) are not identical the potential V( > ) has to be cancelled.
Moreover, for the case where these two states are equivalent the Wheeler-DeWitt equation is
obtained and differs from the N = 2 supersymmetric case just by the second derivative of the
potential Wx (0 J which has been taken into account.
Acknowledgments
The authors would like to thank Professor K. S. Narain for the interesting discussions andfor reading the manuscript and Professor M. Virasoro, the International Atomic EnergyAgency and UNESCO for hospitality at the Abdus Salam International Centre for TheoreticalPhysics, Trieste. This work is supported by the program of the Associate and FederationSchemes of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.
16
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