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: Lhallabi@fsr.ac.ma

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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