Dualities and Hidden Supersymmetry in String Quantum Cosmology

19 Pages·0.2 MB·English

by  James E. Lidsey

CERN-TH/98-22
Dualities and Hidden Supersymmetry in String
Quantum Cosmology
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James E. Lidsey1a and J. Maharana2b
n ∗
a
J

7 1Astronomy Centre and Centre for Theoretical Physics,
2
University of Sussex, BRIGHTON, BN1 9QH, U. K.

1
v
0 2CERN, Geneva, SWITZERLAND
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0
8
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/
c
q
Abstract
-
r
g A supersymmetric approach to string quantum cosmology based on the non–
:
v compact, global duality symmetries of the effective action is developed. An
Xi N = 2 supersymmetric action is derived whose bosonic component is the
r Neveu–Schwarz/Neveu–Schwarz sector of the (d+1)–dimensional effective ac-
a tion compactified on a d–torus. A representation for the supercharges is found
and the form of the zero–and one–fermion quantum states is determined. The
purely bosonic component of the wavefunction is unique and manifestly invari-
ant under the symmetry of the action. The formalism applies to a wide class
of non–linear sigma–models.
PACS NUMBERS: 04.50.+h, 11.25.Mj, 11.30.Pb, 98.80.Hw
aElectronic mail: [email protected]
bElectronic mail: [email protected]
Permanent Address: Institute of Physics, Bhubaneswar - 751 005, INDIA
∗
1 Introduction
Two approaches to the subject of quantum gravity that have received considerable
attention in recent years are quantum cosmology [1] and the superstring theory [2].
In the canonical quantization of Einstein gravity, the classical Hamiltonian, H = 0,
becomes a quantum operator. The physical state, Ψ, of the universe is then identified
as the eigenstate of this operator with zero eigenvalue, HˆΨ = 0. This equation
decouples into two components, NiHˆ Ψ = 0 and NHˆ Ψ = 0, where Ni and N denote
i 0
the shift and lapse functions, respectively. The first constraint implies the invariance
of the wavefunction under spatial diffeomorphisms and the second is the Wheeler–
DeWitt equation [1]. In the minisuperspace approximation, where inhomogeneous
modes are frozen out before quantization, this is the only non–trivial constraint and
can be solved, in principle, by imposing suitable boundary conditions [3].
String theory remains the most promising theory for a unification of the fun-
damental interactions. It is now widely thought that the five separate theories are
non–perturbatively equivalent and are related by ‘duality’ symmetries, referred to as
S–, T– and U–duality, respectively [4, 5, 6]. In general, these dualities are discrete
subgroups of the non–compact, global symmetry groups of the low–energy effective
supergravity actions. T–duality is a perturbative symmetry in the string coupling,
but S–duality is non–perturbative. U–duality interchanges string and sigma–model
coupling constants and, in this sense, represents a unification of S– and T–dualities.
Quantum gravitational effects would have played a key role in the very early
universe and this represents one of the few environments where predictions of string
theory may be quantitatively tested. A central paradigm of early universe cosmology
is that of inflation, where the expansion briefly underwent a very rapid, acceleration.
The above frameworks may be employed to study the very early universe and a
question that naturally arises is whether they are compatible. At present, it is far
from clear how such a question could be fully addressed. Consequently, a more
pragmatic approach is to study how the unique features of string theory, such as its
dualitysymmetries, may beemployed togainfurther insight intoquantum cosmology,
and vice–versa.
In string quantum cosmology, one solves the Wheeler–DeWitt equation derived
from the tree–level string effective action [7, 8, 9]. The interpretive framework of
quantum cosmology may then be employed to investigate whether string theory leads
to realistic cosmologies and, in particular, whether inflation is probable. A quan-
tum cosmological approach was recently advocated for solving the problem of how
inflation ends in pre-big-bang string cosmology [10, 11, 12]. The well known factor
ordering problem is also resolved in this approach because the symmetries of the ac-
tion imply that the minisuperspace metric should be manifestly flat [10]. Moreover,
these symmetries allow the Wheeler–DeWitt equation to be solved in general for a
wide class of models [13]. For example, in the anisotropic Bianchi type IX model, the
wavefunction becomes increasingly peaked around the isotropic limit at large spatial
1
volumes [14].
When restricted to spatially flat, isotropic Friedmann–Robertson–Walker (FRW)
cosmologies, the dilaton–graviton sector of the string effective action is invariant un-
der an inversion of the scale factor and a shift in the dilaton field [15]. This ‘scale
factor duality’ is a subgroup of T–duality and leads to a supersymmetric extension
of the quantum cosmology, where the classical minisuperspace Hamiltonian may be
viewed at the quantum level as the bosonic component of an N = 2 supersymmetric
Hamiltonian [8, 16]. This is important because supersymmetric quantum cosmology
may resolve the problems that arise in the standard approach in constructing a con-
served, non–negative norm from the wavefunction. (For a recent review see, e.g.,
Refs. [17, 18]).
Thus, string quantumcosmology iswell motivated. The purposeofthepresent pa-
per is to develop a supersymmetric approach to quantum cosmology by employing the
non–compact, global symmetries of the string effective action. All ten–dimensional
string theories contain a dilaton, graviton and antisymmetric two–form potential in
the Neveu–Schwarz/Neveu–Schwarz (NS–NS) sector of the theory. Furthermore, an
interesting cosmology is the spatially flat and homogeneous, Bianchi type I universe
admitting d compact Abelian isometries. We therefore consider the NS–NS sector
of the effective action compactified on a d–torus. The reduced action is invariant
under a global O(d,d) ‘T–duality’, where the scalar fields parametrize the coset
O(d,d)/[O(d) O(d)]. This leads to an O(d,d) invariant Wheeler–DeWitt equation
×
[10].
The paper is organized as follows. After reviewing the derivation of the Wheeler–
DeWitt equation in Section 2, we proceed in Section 3 to derive an N = 2 super-
symmetric action whose bosonic component is O(d,d) invariant. The corresponding
super–constraints on the wavefunction are then derived. These constraints are solved
for the zero–fermion and one–fermion states in Section 4. We conclude in Section 5
with a discussion of how the analysis may be extended to a general class of non–linear
sigma–models.
Unless otherwise stated, units are chosen such that h¯ = c = 1.
2 O(d,d) Invariant Wheeler–DeWitt Equation
2.1 Effective Action
The NS–NS sector of the (d+1)–dimensional, tree–level string effective action is given
by [19]
1 1
S = dd+1x g e Φ R+( Φ)2 H Hαβγ +V , (2.1)
2λd 1 | | − ∇ − 12 αβγ
s− Z q (cid:20) (cid:21)
where the Yang–Mills fields are assumed to be trivial, Φ is the dilaton field, V is an
interaction potential, R is the Ricci curvature scalar of the space–time with metric
G
and signature ( ,+,+,...,+), g det , H ∂ B is the field strength of the
αβγ [α βγ]
− ≡ G ≡
2
antisymmetric two–form potential, B , and λ (α)1/2 is the fundamental string
βγ s ′
≡
length scale.
We assume a spatially closed, flat, homogeneous (Bianchi type I) space–time,
wherethedilatonandtwo–formpotentialareconstantonthesurfacesofhomogeneity,
t = constant. Without loss of generality, we may specify = 1 and = B = 0.
00 0i 0i
G − G
Integrating over the spatial variables in Eq. (2.1) then implies that
1
S = dτ Φ¯ 2 + Tr M (M 1) +Ve 2Φ¯ , (2.2)
′ ′ − ′ −
8
Z (cid:20) (cid:16) (cid:17) (cid:21)
where
1
Φ¯ Φ ln g (2.3)
≡ − 2 | |
is the shifted dilaton field,
t
τ dt eΦ¯(t1) (2.4)
1
≡
Z
is the ‘dilaton’ time parameter,
G 1 G 1B
− −
M − (2.5)
≡ BG−1 G BG−1B !
−
is a symmetric 2d 2d matrix, G is the metric on the spatial hypersurfaces, a prime
×
denotes differentiation with respect to τ and we have specified λ 2 [20]. The
s
≡
dilaton has also been shifted by the constant value Φ = ln(λ d ddx). The matrix
0 − −s
M satisfies the conditions
R
MηM = η, M = MT, (2.6)
where
0 I
η (2.7)
≡ I 0 !
and I is the d d unit matrix. It is therefore an element of the group O(d,d) and its
×
inverse is given linearly by M 1 = ηMη.
−
The kinetic sector ofaction(2.2)isinvariant under a globalO(d,d)transformation
[20]:
Φ˜¯ = Φ¯, M˜ = ΩTMΩ, ΩTηΩ = η, (2.8)
where Ω is a constant matrix. Since the shifted dilaton field transforms as a singlet
under the action of Eq. (2.8), this symmetry is respected when V is an arbitrary
function of Φ¯.
The classical Hamiltonian for this cosmological model is given by
1
H = Π2 2Tr(MΠ MΠ ) Ve 2Φ¯, (2.9)
bos 4 Φ¯ − M M − −
3
where
1
Π = 2Φ¯ , Π = M 1M M 1 (2.10)
Φ¯ ′ M −4 − ′ −
are the momenta conjugate to Φ¯ and M, respectively. The equations of motion for
the matrix M can be integrated directly to yield the first integral MηM = C [20],
′
where C is a constant, 2d 2d matrix satisfying the conditions
×
CT = C, MηC = CηM. (2.11)
− −
The first integral represents a conservation law and may also be written in terms of
the conjugate momenta (2.10):
1
MΠ = Cη. (2.12)
M
−4
2.2 Quantum Cosmology
The cosmology is quantized by identifying the momenta (2.10) with the differential
operators
δ δ
Π = i , Π = i . (2.13)
Φ¯ − δΦ¯ M − δM
Substituting Eq. (2.13) into Eq. (2.9) then leads to the Wheeler–DeWitt equation
[10]:
δ2 δ δ
+8Tr η η +4Ve 2Φ¯ Ψ(Φ¯,M) = 0. (2.14)
"δΦ¯2 δM δM! − #
However, a further constraint should also be imposed onthe wavefunction because
thematrixM belongstothe groupO(d,d). This results intheconservationlaw (2.12)
and is analogous to the ‘rigid rotator’ model for a particle moving in a spherically
symmetric potential well. In this model, angular momentum is conserved due to a
global O(3) symmetry. When states of definite angular momentum are considered,
therearisesacentrifugalbarriertermintheeffectiveactionandweencounterasimilar
situation in the model considered above. The conservation law (2.12) is responsible
for the analogue of the ‘centrifugal barrier’ term when the Wheeler-DeWitt equation
is solved subject to the requirement that the wavefunction satisfies the necessary
O(d,d) invariance properties. Thus, condition (2.12) should apply at the quantum
cosmologicallevel andthisimplies that thewavefunction should satisfythefirst–order
constraint [10]
δΨ 1
iM = CηΨ. (2.15)
δM 4
In general, the constraint (2.15) can not be solved in closed form. On the other
hand, it does imply that the wavefunction in Eq. (2.14) can be separated by spec-
ifying Ψ(M,Φ¯) = X(M)Y(Φ¯), where X(M) and Y(Φ¯) are functions of M and Φ¯,
4
respectively. The Wheeler–DeWitt equation then simplifies to an ordinary, differen-
tial equation in the shifted dilaton field:
d2
+B2 +4V(Φ¯)e 2Φ¯ Y(Φ¯) = 0, (2.16)
"dΦ¯2 − #
where
1
B2 Tr(Cη)2 (2.17)
≡ 2
is a constant and represents the ‘centrifugal barrier’ term alluded to earlier. When
V(Φ¯) is constant, Eq. (2.16) can be solved in full generality in terms of a linear
superposition of Bessel functions [10].
3 Supersymmetric String Quantum Cosmology
3.1 N=2 Supersymmetry
In this Section we derive an N = 2 supersymmetric Lagrangian whose bosonic sector
is invariant under global O(d,d) transformations. In supersymmetric point particle
mechanics, the N = 2 case allows an interaction potential to be included. Homoge-
neous N = 2 supersymmetric quantum cosmologies coupled to a single scalar field
have been studied previously in a different context [21, 22]. It has been further shown
that the scale factor duality of isotropic FRW string cosmologies is associated with
an N = 2 supersymmetry [8]. We extend previous analyses to the class of spatially
flat, anisotropic(Bianchi type I)cosmologies including a non–trivialNS–NStwo–form
potential.
In formulating an N = 2 supersymmetric action, we consider superfields of the
generic form
Xµ(τ,θ,θ¯) xµ(τ)+iψ¯µ(τ)θ +iψµ(τ)θ¯+Fµ(τ)θθ¯, (3.1)
≡
wherethebosonicfunctions, xµ(τ),Fµ(τ) ,andanticommutingcomplexspinorfunc-
tions, ψµ(τ),ψ¯µ(τ) , are arb{itrary functio}ns of the dilaton time (2.4), θ,θ¯ are con-
{ } { }
stant, anticommuting, complex spinors and µ is a parameter labelling the degrees of
freedom in minisuperspace.
The generators for the supersymmetry are defined by
∂ ∂
Qˆ iθ
1 ≡ −∂θ¯− ∂τ
∂ ∂
Qˆ +iθ¯ (3.2)
2
≡ ∂θ ∂τ
and the supersymmetry transformation rule for the superfield (3.1) is
δXµ = i ξ Qˆ +ξ Qˆ Xµ, (3.3)
1 1 2 2
−
(cid:16) (cid:17)
5
where ξ are arbitrary parameters that commute with the bosonic variables and an-
i
ticommute with all fermionic variables.
We now define the superfields
¯ ¯ ¯ ¯
m (τ,θ,θ) M (τ)+iψ (τ)θ +iψ (τ)θ +F (τ)θθ (3.4)
ij ij ij ij ij
≡
(τ,θ,θ¯) √2Φ¯(τ)+iχ¯(τ)θ+iχ(τ)θ¯+f(τ)θθ¯, (3.5)
D ≡
¯
where M (τ) is given by Eq. (2.5), ψ ,ψ ,χ,χ¯ are anticommuting, complex
ij ij ij
{ }
spinors and (i,j) = (1,2,...,2d). The spatial metric and antisymmetric, two–form
potential determine the bosonic component of the superfield (3.4) and the shifted
dilaton field (2.3) plays the equivalent role in Eq. (3.5).
We then define two further superfields in terms of Eqs. (3.4) and (3.5):
1
Σ(τ,θ,θ¯) Dˆ m ηjkDˆ m ηli (3.6)
1 ij 2 kl
≡ 8
1
Y(τ,θ,θ¯) Dˆ Dˆ W( ), (3.7)
1 2
≡ 2 D D − D
where the derivative operators are
∂ ∂
Dˆ +iθ (3.8)
1 ≡ −∂θ¯ ∂τ
∂ ∂
Dˆ iθ¯ (3.9)
2
≡ ∂θ − ∂τ
and the potential, W( ), is an arbitrary function of .
D D
The sum of Eqs. (3.6) and (3.7) is viewed as an effective Lagrangian in the action:
¯
I dτ dθdθ(Σ+Y). (3.10)
SUSY
≡
Z Z
It may be verified by substituting Eqs. (3.4) and (3.5) into Eqs. (3.6) and (3.7), ex-
panding the potential W( ) around Φ¯, and collecting coefficients in the Grassmann
¯ D
variables θ and θ that the action (3.10) is invariant under the supersymmetry trans-
¯
formations (3.3) up to a total time derivative in the θθ coefficient. This coefficient is
given by L L +L , where
g 1
≡
1
L = iψ ηjkψ¯ ηli iψ ηjkψ¯ ηli+M ηjkM ηli (3.11)
g 8 ij k′l − i′j kl i′j k′l
h i
L = Φ¯ 2 + i (χ¯χ χ¯χ)+ 1f2 1 f∂ W 1 ∂2W [χ¯,χ] (3.12)
1 ′ 2 ′ − ′ 2 − √2 Φ¯ − 4 Φ¯ −
(cid:16) (cid:17) (cid:16) (cid:17)
and W = W(Φ¯). There is an additional term in Eq. (3.11) of the form F ηjkF ηli,
ij kl
but since there is no potential contribution in the action from the matrix (2.5), its
equation of motion implies that we may specify F = 0 without loss of generality.
ij
On the other hand, the equation of motion for the auxiliary field, f, is given by
1
f = ∂ W (3.13)
√2 Φ¯
6
and substituting Eq. (3.13) into Eq. (3.12) eliminates this field from the action.
Integrating over the Grassmann variables in Eq. (3.10) therefore implies that the
N = 2 supersymmetric action is given by
1
I = dτ iψ ηjkψ¯ ηli iψ ηjkψ¯ ηli +M ηjkM ηli
SUSY 8 ij k′l − i′j kl i′j k′l
Z (cid:20) (cid:16) (cid:17)
+ Φ¯ 2 + i (χ¯χ χ¯χ) 1 (∂ W)2 1 ∂2W [χ¯,χ] . (3.14)
′ 2 ′ − ′ − 4 Φ¯ − 4 Φ¯
(cid:16) (cid:17) (cid:16) (cid:17) −(cid:21)
Theaction(3.14)reduces tothebosonicaction(2.2)inthelimitwheretheGrassmann
variables vanish if we identify the potential W:
(∂ W)2 = 4V(Φ¯)e 2Φ¯. (3.15)
Φ¯ −
−
Thus, a necessary condition for an N = 2 supersymmetric extension of the effective
action (2.2) is that the interaction potential, V, must be semi–negative definite.
The classical momenta conjugate to the bosonic and fermionic degrees of freedom
in action (3.14) are
∂L 1
Π = = ηnkM ηlm
Mmn ∂M 4 k′l
m′ n
∂L i
K = = ηnkψ¯ ηlm
mn kl
∂ψ −8
m′ n
∂L i
K¯ = = ψ ηjmηni
mn ∂ψ¯ −8 ij
m′ n
∂L
Π = = 2Φ¯
Φ¯ ∂Φ¯ ′
′
∂L i
Π = = χ¯
χ
∂χ −2
′
∂L i
Π = = χ, (3.16)
χ¯
∂χ¯ −2
′
¯
respectively, where the negative sign appears in the expressions for ψ and χ because
ij
the left derivative of the Grassmann variables is taken. The classical Hamiltonian
derived from the action (3.14) is given by
H = M ηjkΠ ηli +ψ ηjkK ηli +ψ¯ ηjkK¯ ηli +Φ¯ Π +χΠ +χ¯Π L (3.17)
i′j Mkl i′j kl i′j kl ′ Φ¯ χ χ¯ −
and substituting Eqs. (3.16) into Eq. (3.17) implies that it takes the form
1 1 1
H = 2Π ηjkΠ ηli + Π2 + (∂ W)2 + ∂2W [χ¯,χ] , (3.18)
ij kl 4 Φ¯ 4 Φ¯ 4 Φ¯
−
(cid:16) (cid:17)
where the anticommuting property of the Grassmann variables has been employed
and Π Π . The bosonic component of the Hamiltonian (3.18) corresponds to
ij ≡ Mij
Eq. (2.9). The first term in this expression describes the Hamiltonian for the matrix
M . The fermions do not appear in this component of the Hamiltonian, as is always
ij
the case in supersymmetric quantum mechanics when the fermions are free.
7
3.2 Quantum Constraints
Themodelisquantizedbyassumingthestandardoperatorrealizationsforthebosonic
variables in Eq. (2.13) and further imposing the spinor algebra
¯ ¯ ¯
[ψ ,ψ ] = ψ ,ψ = 0, ψ ,ψ = η η
ij kl + ij kl + ij kl + ik jl
h i h i
[χ,χ] = [χ¯,χ¯] = 0, [χ,χ¯] = 1
+ + +
¯ ¯
χ¯,ψ = [χ,ψ ] = χ,ψ = [χ¯,ψ ] = 0. (3.19)
ij + ij + ij + ij +
h i h i
A representation satisfying Eq. (3.19) is given in terms of the set of Grassmann
variables ζ ,β :
ij
{ }
∂
¯
ψ = η η , ψ = ζ
kl kp rl ij ij
∂ζ
pr
∂
χ = , χ¯ = β. (3.20)
∂β
¯
Notice that we have imposed [ψ ,ψ ] = η η in Eq. (3.19). The canonically
ij kl + ik jl
¯
conjugate momenta for ψ and ψ are given by Eq. (3.16) and, if we had employed
ij kl
¯
the canonical anticommutation relations between ψ and K and between ψ and
ij mn ij
K¯ , we would not have obtained the anticommutation relation presented in Eq.
mn
(3.19). We have implicitly employed [ψ ,K ] = 4η η and [ψ¯ ,K¯ ] = 4η η
ij kl + ik jl ij kl + ik jl
rather than ‘one’ on the right hand side. This is due to the fact that there is a factor
¯
of 1/8 in the kinetic energy terms of ψ and ψ rather than the conventional factor
ij kl
of 1/2 that appears in the Dirac Langrangian. (For example, the χ and χ¯ pieces in
Eq. (3.14) have the standard factor). We adopted the standard anticommutation
¯
relation between ψ and ψ because the calculations can then be performed in a
ij kl
straightforward manner without keeping track of additional numerical constants. If
theanticommutation relationinEq. (3.19)hadbeen modifiedwith anothernumerical
constant, our expressions for the supercharges Q and Q¯ given below would also have
been modified. However, anidentical expression for the Hamiltonian would have been
obtained from the anticommutator of Q and Q¯.
We now define the supercharges
1
Q 2Π ηjkψ ηli + (Π +i∂ W)χ (3.21)
≡ ij kl √2 Φ¯ Φ¯
1
Q¯ 2Π ηnrψ¯ ηpm + (Π i∂ W)χ¯, (3.22)
≡ mn rp √2 Φ¯ − Φ¯
where Q is a non–Hermitian, linear operator and Q¯ is its adjoint. Substituting Eqs.
(3.21)and(3.22)intoEq. (3.18),andemployingtheanticommutationrelations(3.19),
implies that the Hamiltonian operator may be written as
2H = Q,Q¯ , Q2 = Q¯2 = 0, (3.23)
+
h i
8
where [H,Q] = [H,Q¯] = 0. Thus, there exists an N = 2 supersymmetry in the
− −
quantum cosmology [16, 23]. This may be viewed as a direct extension of the O(d,d)
‘T–duality’ of the toroidally compactified NS–NS action (2.1).
Finally, supersymmetry implies that the wavefunction of the universe is annihi-
lated by the supercharges, QΨ = Q¯Ψ = 0. These reduce to a set of first–order
differential equations and we derive solutions to these constraints in the following
Section.
We conclude this Section by remarking that factor ordering problems in super-
symmetric quantum mechanics in curved space have been addressed previously [24].
In general, four fermion terms with the curvature tensor appear, but we have not
considered such problems here because we assumed a toroidal compactification and
this implies that the curvature is zero. Moreover, as discussed by Gasperini et al.
[10], the operator ordering issue is settled in the standard procedure by demanding
the O(d,d) invariance of the Hamiltonian.
4 Quantum States
4.1 Zero–fermion State
The quantum constraints are solved by defining the conserved ‘fermion number’:
F ψ¯ ηjkψ ηli +χ¯χ, (4.1)
ij kl
≡
where
[H,F] = 0, [Q,F] = Q, [Q¯,F] = Q¯. (4.2)
− − − −
This implies that states with a fixed fermion number may be individually considered.
The fermion vacuum, 0 , is defined:
| i
ψ 0 = χ 0 = 0 i,j. (4.3)
ij
| i | i ∀
The state with zero fermion number, ψ , is defined as ψ h(M ,Φ¯) 0 , where h
0 0 ij
| i | i ≡ | i
is an arbitrary function. This state is a function of the bosonic degrees of freedom
only and is automatically annihilated by the supercharge Q. It is annihilated by its
adjoint, Q¯, if the conditions
δh ∂ dW
= 0, + h = 0 (4.4)
δMij "∂Φ¯ dΦ¯ #
are simultaneously satisfied. Modulo a constant of proportionality, the general solu-
tion to Eq. (4.4) is
ψ = e W(Φ¯) 0 . (4.5)
0 −
| i | i
This solution is uniquely determined by the potential (3.15) and is a function of the
shifted dilaton field only. It is therefore manifestly invariant under the global O(d,d)
symmetry (2.8) of the bosonic action (2.2).
9