Local Supersymmetry in One-Loop Quantum Cosmology

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0 i AA can th e condition imp osing the sup ersymmetry constrain three-sphere b oundary Massless gra p . : 3 0 S A i e

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ts conditions on nen elds. 2 provided byCERNDocumentServer brought toyouby CORE 1

Local Supersymmetry in One-Loop Quantum Cosmology

289

The PDF contribution to the full (0) value is found to b e = . Remarkably, for the

360

massless gravitino eld the PDF metho d and lo cal b oundary conditions lead to a result

for (0) which is equal to the PDF value one obtains using sp ectral b oundary conditions

on S .

PACS numb ers: 03.70.+k, 04.60.+n, 98.80.Dr 2

Local Supersymmetry in One-Loop Quantum Cosmology

1. Intro duction

The problem of one-lo op niteness of sup ergravity theories in the presence of b oundaries

110

is still receiving careful consideration in the current literature. As emphasized in Refs.

9,11-12, one can p erform one-lo op calculations paying attention to: (1) S-matrix elements;

(2) top ological invariants; (3) presence of b oundaries. For example, in the case of pure

gravity with vanishing cosmological constant:=0,itisknown that one-lo op on-shell

S-matrix elements are nite. This prop erty is shared by N = 1 sup ergravity when = 0,

and in that theory two-lo op on-shell niteness also holds. However, when 6= 0, b oth

pure gravity and N = 1 sup ergravity are no longer one-lo op nite in the sense (1) and (2),

b ecause the non-vanishing on-shell one-lo op counterterm is given by

1 2BGS

S = A : (1:1)

(1)

In equation (1.1), = n 4 is the dimensional-regularization parameter, is the Euler

87 106

9;11

;B = for pure numb er, S is the classical on-shell action, and one nds : A =

45 10

41 77

gravity, and A = ;B = for N = 1 sup ergravity.Thus, B 6= 0 is resp onsible for lack

24 12

of S-matrix one-lo op niteness, and A 6= 0 do es not yield top ological one-lo op niteness.

If any theory of quantum gravity can b e studied from a p erturbative p oint of view,

b oundary e ects playa key role in understanding whether it has interesting and useful

niteness prop erties. It is therefore necessary to analyze in detail the structure of the one-

lo op b oundary counterterms for elds of various spins. This problem has b een recently

studied within the framework of one-lo op quantum cosmology, where the b oundary is 3

Local Supersymmetry in One-Loop Quantum Cosmology

usually taken to b e a three-sphere, and the background is at Euclidean space or a de

210

Sitter four-sphere or a more general curved four-geometry.

Our pap er describ es one-lo op prop erties of spin- elds to present a calculation which

5;9

was previously studied in research b o oks but not in physics journals (see, however, re-

marks at the end of Ref. 10). In the Euclidean-time regime, the spin- eld is represented,

using two-comp onent spinor notation, by a pair of indep endent spinor-valued one-forms

0 0

4;9 A A A A

e e

. After imp osing the gauge conditions with spatial comp onents ; ;

i i

(hereafter e is the tetrad)

i A i A

e =0 ; e =0 : (1:2)

0 0

AA i AA i

A A

and the linearized sup ersymmetry constraints, the expansion of ; on a family of

i i

4;9

three-spheres centred on the origin takes the form

(n+1)(n+4)

X X

0 0

A nq AB B pq nq AB B

= + re ( ) e ; (1:3) m ( )

np BB i np

i n

n=0 p;q =1

(n+1)(n+4)

X 0 0 X

0 0 0

nq B A B

A nq B A B pq

= me ( ) + r ( ) e : (1:4)

np np BB i

i n

n=0 p;q =1

With our notation, is the radial distance from the origin in at Euclidean four-space,

1 1

the matrix is blo ck-diagonal in the indices pq , with blo cks . Note also that

1 1

the mo des me ( ); re ( ) are not the complex conjugates of m ( );r ( ) resp ectively.

np np np np

0 0 0 0 0 0 0

4;9 nq AB B nq (AB C ) B nq B A B nq (A B C ) B

n n Moreover, one has = ; = where the

C C

0 0 0

nq (AB C ) nq (A B C )

harmonics and are symmetric in their three spinor indices and have 4

Local Supersymmetry in One-Loop Quantum Cosmology

1 5

3 4

p ositive eigenvalues n + of the intrinsic three-dimensional Dirac op erator on S ,

2 2

3 3;9 CB

is the Lorentzian normal to S . and n

Sec. 2 studies lo cally sup ersymmetric b oundary conditions on S for the spin- po-

tential, and the equation ob eyed by the eigenvalues by virtue of these b oundary conditions

is derived. Sec. 3 uses zeta-function regularization and obtains the contribution of phys-

ical degrees of freedom (hereafter referred to as PDF) to the full (0) value. Concluding

remarks and op en research problems are presented in Sec. 4.

2. Lo cal Boundary Conditions for the Spin- Potential

In Euclidean sup ergravity, the mathematical description of the gravitino leads to the intro-

A A

duction of the indep endent spinor-valued one-forms ; with spatial comp onents

A A

; .We are here interested in a generalization to simple sup ergravity of the cal-

i i

culations in Ref. 3 for the spin- eld. Thus, we consider a at Euclidean background,

requiring on the b ounding S that

0 0

A A A

2 n = ; (2:1)

A i i

where = 1. The consideration of (2.1) is suggested by the work in Ref. 1, where it is

0 0 0

A A AA A

2 n transform shown that the spatial tetrad e and the pro jection

A i i i

into each other under half of the lo cal sup ersymmetry transformations at the b oundary, 5

Local Supersymmetry in One-Loop Quantum Cosmology

and that after adding a suitable b oundary term, the sup ergravity action is invariant under

3;9

these lo cal sup ersymmetry transformations.

Indeed, from Sec. 1 we already know that, imp osing the sup ersymmetry constraints

and cho osing the gauge condition (1.2), the spin- p otential nally assumes the form (1.3)-

(1.4). It is therefore useful to derive identities relating barred to unbarred harmonics,

generalizing the technique in Ref. 13. This is achieved by using the relations

0 0 0

np mq

CC BB AA nm pq

n n d n = H ; (2:2)

0 0 0

AB C A B C

np mq

AD BE CF nm pq

d = A ; (2:3)

AB C DE F

and the expansion of the totally symmetric eld strength

(n+1)(n+4)

X X

np np

(x) : (2:4) (x)= ba (x)+b

AB C np np

AB C AB C

n=0 p=1

Thus, we can express the ba co ecients in two equivalentways using (2.4), and (2.2) or

(2.3). The equality of the two resulting formulae leads to

(n+1)(n+4) (n+1)(n+4)

X X

0 0 0

qp qp

nq nq

1 AD BE CF AA BB CC 1

H = n n n A ; (2:5)

0 0 0

n n

A B C DE F

q =1 q =1

which is nally cast in the form

(n+1)(n+4)

nq np

1 F E D

A H n : (2:6) n = 8n

0 0 0

n F E D

DE F D E F

q =1 6

Local Supersymmetry in One-Loop Quantum Cosmology

In a similar way,we obtain

(n+1)(n+4)

0 0 0

np nq

D E F 1

= 8n n n A H : (2:7)

0 0 0

D E F n

DE F D E F

q =1

pq pq

The form of the matrices A and H is obtained taking the complex conjugate of (2.6),

n n

and then inserting the form of so obtained into the right-hand side of (2.6). This

DE F

yields the consistency condition

1 1

A H A H = 1 ; (2:8)

n n n

n n

0 1 1 0 0 1

which is solved by A = , H = , so that A =2 2 .

n n

2 2

1 0 0 1 1 0

We can now remark that (1.3)-(1.4) and (2.1) imply

(n+1)(n+4) (n+1)(n+4)

X X

0 0 0 0 0

pq ( ) B B pq ( ) nq AB D A nq A B D

= me (a) n n ; i 2 m (a) n

D n np A D n np

p;q =1 p;q =1

(2:9)

(n+1)(n+4) (n+1)(n+4)

X X

0 0 0 0 0

B pq () nq AB D A B pq () nq A B D

: i n 2 re (a) n n = r (a)

D n np A D n np

p;q =1 p;q =1

(2:10)

This is why Eqs. (2.6)-(2.7), (2.9)-(2.10) and the formulae for A H lead to the b oundary

conditions

pq pq

X X

1 1 1 1

( ) nq AB C ( )

i m (a) = me (a)

np np

1 1 1 1

pq pq

0 1

ndAB C

; (2:11)

1 0

d 7

Local Supersymmetry in One-Loop Quantum Cosmology

pq pq

X X

0 0 0

1 1 1 1

() nq A B C ()

r (a) = i er (a)

np np

1 1 1 1

pq pq

0 0 0

0 1

ndA B C

: (2:12)

1 0

Since the - and -harmonics on the b ounding three-sphere of radius a are linearly inde-

3;9

p endent, the typical case of the indices p; q =1;2 yields

( ) ( )

im (a)=me (a) ; (2:13)

n1 n2

( ) ( )

im (a)=me (a) ; (2:14)

n2 n1

() ()

ire (a)=r (a) ; (2:15)

n1 n2

() ()

ire (a)=r (a) : (2:16)

n2 n1

If wenow set n + and de ne, 8n 0, the op erators

L ; (2:17)

M + ; (2:18)

the coupled eigenvalue equations take, in light of the mo de-by-mo de expansion of the action

4;9

integral, the form

L x = E xe ; M xe = Ex ; (2:19)

n n

L y = E ye ; M ye = Ey ; (2:20)

n n

e e

X ; M X = EX ; (2:21) L X = E

n n 8

Local Supersymmetry in One-Loop Quantum Cosmology

e e

L Y = E Y ; M Y = EY ; (2:22)

n n

where

( ) ( )

x m ; X m ; (2:23)

n1 n2

( ) ( )

xe me ; X me ; (2:24)

n1 n2

() ()

y r ; Y r ; (2:25)

n1 n2

() ()

ye re ; Y re : (2:26)

n1 n2

Wenow de ne 8n 0 the di erential op erators

2 3

2 (n +3)

d 4

4 5

P + E ; (2:27)

2 2

2 3

2 (n +2)

4 5

Q + E : (2:28)

2 2

Eqs. (2.19)-(2.22) lead to the following second-order equations, 8n 0:

e e

P xe = P X = P ye = P Y =0 ; (2:29)

n n n n

Q y = Q Y = Q x = Q X =0 : (2:30)

n n n n

The solutions of (2.29)-(2.30) regular at the origin are

p p

xe = C J (E ) ; X = C J (E ) ; (2:31)

1 n+3 2 n+3

p p

x = C J (E ) ; X = C J (E ) ; (2:32)

3 n+2 4 n+2 9

Local Supersymmetry in One-Loop Quantum Cosmology

p p

ye = C J (E ) ; Y = C J (E ) ; (2:33)

5 n+3 6 n+3

p p

y = C J (E ) ; Y = C J (E ) : (2:34)

7 n+2 8 n+2

To nd the condition ob eyed by the eigenvalues E ,wenow insert (2.31)-(2.34) into the

b oundary conditions (2.13)-(2.16), taking into account also the rst-order system given by

(2.19)-(2.22). This gives the eight equations

iC J (Ea)=C J (Ea) ; (2:35)

3 n+2 2 n+3

iC J (Ea)=C J (Ea) ; (2:36)

4 n+2 1 n+3

iC J (Ea)=C J (Ea) ; (2:37)

5 n+3 8 n+2

iC J (Ea)=C J (Ea) ; (2:38)

6 n+3 7 n+2

EC J (Ea)

3 n+2

C = ; (2:39)

J (Ea)

n+3

EJ (Ea)+(n+3)

n+3

EC J (Ea)

4 n+2

C = ; (2:40)

J (Ea)

n+3

EJ (Ea)+(n+3)

n+3

EC J (Ea)

5 n+3

C = ; (2:41)

J (Ea)

n+2

EJ (Ea) (n +2)

n+2

EC J (Ea)

6 n+3

C = : (2:42)

J (Ea)

n+2

EJ (Ea) (n +2)

n+2

a 10

Local Supersymmetry in One-Loop Quantum Cosmology

Interestingly, these give separate relations among the constants C ;C ;C ;C and among

1 2 3 4

C ;C ;C ;C [3,9]. For example, eliminating C ;C ;C ;C , using (2.35)-(2.36), (2.39)-

5 6 7 8 1 2 3 4

(2.40) and the useful identities

EaJ (Ea) (n +2)J (Ea)=EaJ (Ea) ; (2:43)

n+2 n+2 n+3

EaJ (Ea)+(n+3)J (Ea)=EaJ (Ea) ; (2:44)

n+3 n+3 n+2

one nds

J (Ea) C C J (Ea)

n+2 2 4 n+3

2 2 3

i = = = i ; (2:45)

J (Ea) C C J (Ea)

n+3 3 1 n+2

which implies (since = 1)

h i h i

2 2

J (E ) J (E ) =0 ; 8n0 ; (2:46)

n+2 n+3

where we set a = 1 for simplicity.

3. Physical-Degrees-of-Freedom Contribution to (0)

, The eigenvalue condition (2.46) is very similar to the formula found in Refs. 3,9 for spin

h i h i

2 2

i.e. J (E ) J (E ) =0;8n0. Thus, the same technique can b e now applied

n+1 n+2

to derive the PDF contribution to (0) in the case of gravitinos. As we know from Refs.

4,9, the completely symmetric harmonics have degeneracy d(n)=(n+ 4)(n + 1), 8n 0.

This is the full degeneracy in the case of lo cal b oundary conditions (2.1), since we need

twice as many mo des to get the same numb er of eigenvalue conditions as in the sp ectral 11

Local Supersymmetry in One-Loop Quantum Cosmology

34;9

case. The (0) calculation is now p erformed using ideas rst describ ed in Ref. 15, and

then used in Refs. 3,9. Given the zeta-function at large x

2 2

(s; x ) + x ; (3:1)

j =1

2 3;9

where = E are the squared eigenvalues of the Dirac op erator in our case, one has in

four dimensions

2 2 x T l2

(3) (3;x )= T e G(T) dT C 1+ x ; (3:2)

l=0

9 +

where wehave used the asymptotic expansion of the heat kernel for T ! 0

G(T ) : (3:3) C T

l=0

3;9

On the other hand, de ning m n +3,we nd

i h

1 d

2 2 2(m1) 2

(ix) J J log (ix) (3) (3;x )= (m + 1)(m 2)

m1 m

2x dx

m=3

" #

1 5

X X

1 d

m m S (m; (x))

i m

2x dx

m=0

i=1

+ Z + Z + q x ; (3:4)

1 2 n

n=5

3;9

where

2 2

(x) m + x ; (3:5)

S (m; (x)) log( )+2 ; (3:6)

1 m m 12

Local Supersymmetry in One-Loop Quantum Cosmology

S (m; (x)) (2m 1) log (m + ) ; (3:7)

2 m m

r r 1

S (m; (x)) k m ; (3:8)

3 m 1r

r =0

r r 2

S (m; (x)) k m ; (3:9)

4 m 2r

r =0

r r 3

S (m; (x)) k m ; (3:10)

5 m 3r

r =0

" #

5 5 1

X X X

d 1

(i)

S (m; (x)) = X ; (3:11) Z 2

i m 1

2x dx

m=0

i=1 i=1

" #

1 5 5

X X X

1 d

(i)

Z 2 S (m; (x)) = Y : (3:12)

2 i m

2x dx

m=0

i=1 i=1

One can thus obtain (0) = C as half the co ecientof x in the asymptotic expansion of

3;9

the right-hand side of (3.4), by comparison of (3.2) and (3.4), and b earing in mind that

1 1

k = ; k =0 ; k = ; (3:13)

10 11 12

4 12

1 1 1

; k = k = ; k = ; (3:14) k =0 ; k =

22 23 24 20 21

8 8 8

5 1 9 1

k = ; k = ; k = ; k = ; (3:15a)

30 31 32 33

192 8 320 2

23 3 179

k = ; k = ; k = : (3:15b)

34 35 36

64 8 576 13

Local Supersymmetry in One-Loop Quantum Cosmology

3 1

The PDF (0) value for spin is thus given by the spin- value rst found in Ref. 3 plus

2 2

3;9;15

the contributions of Z and Z .For this purp ose, we also use the identities

1 2

1 d 1 9 3

3 3 4 2 5

log =(m+ ) m m ; (3:16)

m m m

2x dx m + 8 8

( m)

(m + ) = : (3:17)

3;9;15

The insertion of (3.17) into (3.16) yields

h i

1 m 3 d

6 2 6 1 4 3 2 2 5

m log (m + ) = mx + m x + x + m x : (3:18)

m m m

2x dx 2 8

This further identity leads to divergences in the calculation, but these are only ctitious in

light of (3.16). Such ctitious divergences are regularized dividing by , summing using

3;9;15

the contour formulae

1 1

k +

2 2 2

2k 2k q 1q

m = x ; 8k =1;2;3; ::: (3:19)

2 k +

m=0

1 1 r r

1 1

r + + 1q

X X

2 2 2 2 2

2 x r

1q r

m B x cos ; (3:20)

r ! 2

2 +

m=0 r =0

2 2

3;9;15

where B are Bernoulli numb ers, and then taking the limit s ! 0.

Indeed, from (3.11) we nd

(1) 5

X = ; (3:21)

1 m

m=0 14

Local Supersymmetry in One-Loop Quantum Cosmology

which do es not contain x and hence do es not contribute to (0). However, (3.18) and

(3.7) imply

3 3

(2) 6 6 4 2 6 6 4 2

X =4x 4x 2x x 2x +2x +x + x ; (3:22)

1 2 3 4 5 6 7 8

2 4

where

m ; (3:23)

m=0

2 1

m ; (3:24)

m=0

2 3

m ; (3:25)

m=0

2 5

m ; (3:26)

m=0

lim ; (3:27)

s!0

m=0

12s

lim m ; (3:28)

s!0

m=0

m ; (3:29)

m=0

: (3:30) m

m=0

Note that only and contribute to (0). This is proved using (3.19)-(3.20) and the

1 5

Euler-Maclaurin formula. According to this algorithm, if f is a real- or complex-valued 15

Local Supersymmetry in One-Loop Quantum Cosmology

(2m)

function de ned on 0 t 1, and if f (t) is absolutely integrable on (0; 1) then, for

3;9;16

u =1;2; :::

u m1

u h i

X X

1 B

(2s1) (2s1)

f (u) f (0) + R (u) ; f (i) f (x) dx = f (0) + f (u) +

2 (2s)!

s=1

i=0

(3:31)

where the remainder R satis es the inequality

j B j

12m (2m)

j R (u) j 2 2 j f (x) j dx : (3:32)

(2m)!

The asymptotic expansion (3.20) implies that gives the contribution

(a)

= 2 cos( ) B = ; (3:33)

and the Euler-Maclaurin formula shows that contributes

(b)

= : (3:34)

By virtue of (3.13), (3.8) and (3.11), we also nd that

1 1

X X

15 105

(3) 7 2 9

= X + : (3:35) k k m

10 12

1 m m

4 4

m=0 m=0

Thus, using (3.19) and (3.13), we derive the following contribution to (0):

(c)

=(k + k )= : (3:36)

10 12

6 16

Local Supersymmetry in One-Loop Quantum Cosmology

Finally, using (3.9)-(3.11) we obtain

" #

4 1

X X

(4) r r 8

X = k (r + 2)(r + 4)(r +6) m ; (3:37)

1 m

r =0 m=0

" #

6 1

X X

(5) r r 9

X = k (r + 3)(r + 5)(r +7) m ; (3:38)

1 m

r =0 m=0

(4)

and in light of (3.19)-(3.20) we derive that the asymptotic b ehaviour of X is O(x ),

(5)

and the asymptotic form of X is O(x ). Thus, they do not a ect the (0) value.

Moreover, the whole of Z (cf (3.12)) do es not a ect (0). In fact one nds

5 2 (1)

x 1+ 1+ x ; (3:39) Y =

5 1 3

2 2 2

2 2 2 7 (2) 7 7

1+x 1+x 1+x x Y =2x +x + ; (3:40)

7 9

15 2 105 2

7 2 9 2 (3)

k x 1+ 1+x k x 1+x Y = ; (3:41)

10 12

4 4

r 8 2 (4)

x k (r + 2)(r + 4)(r +6) 1+x ; (3:42) Y =

r =1

r 9

105 1 2 2

(5) 9 r 9 2

= Y k x x k (r + 3)(r + 5)(r +7) 1+x ; (3:43)

30 3r

4 4

r =0

and the reader can now easily see that the formulae (3.39)-(3.43) do not contain terms

prop ortional to x . 17

Local Supersymmetry in One-Loop Quantum Cosmology

At the end, wehave to consider more carefully the e ect of higher-order terms in the

2(m1) 2 2

asymptotic expansion of log (ix) J J (ix) . In light of Refs. 3,9 and of

m1 m

Eqs. (3.4)-(3.12) we study, 8n> 3

l 1

X X

n;A pn6 p pn5 p1

H h a (m + ) + b (m + )

np np m np m

1 m m

p=1 m=0

pn4 p2 pn3 p3

+ c (m + ) + d (m + ) ; (3:44)

np m np m

m m

2n 1

X X

r r n6 n;B

k (r + n)(r + n + 2)(r + n +4) m ; (3:45) H

m 1

r =0 m=0

l 1

X X

n;C pn6 p pn5 p1

H h a (m + ) + b (m + )

np np m np m

1 m m

p=1 m=0

pn4 p2 pn3 p3

; (3:46) + c (m + ) + d (m + )

np m np m

m m

2n 1

X X

n;D r r n6

H k (r + n)(r + n + 2)(r + n +4) m ; (3:47)

1 m

r=0 m=0

where a ;b ;c ;d ;h are constant co ecients. In (3.44)-(3.47), n should not b e

np np np np np

confused with the integer app earing in (2.46) and in the de nition of m. Again, the Euler-

n;A

Maclaurin formula is very useful in studying H . The equivalentoff(0) in (3.31) gives

a contribution prop ortional to x . Bernoulli numb ers and derivatives of o dd order give

a contribution prop ortional to x plus higher-order terms. The conversion of (3.44)

, as it is evident studying the integrals into an integral yields a term prop ortional to x

2 2

(np)

2 2

x + y x + y dy ; (3:48) y + I

0 18

Local Supersymmetry in One-Loop Quantum Cosmology

n 5

2 2 2

(np)

2 2

y + dy ; (3:49) I x + y x + y

2 2

(np)

2 2

y + I x + y x + y dy ; (3:50)

n 3

2 2 2

(np)

2 2

y + I x + y x + y dy : (3:51)

n;B

The e ect of H is derived by using (3.19)-(3.20). When r =0wehave to consider

n6 6

, which do es not contain x . When r =2k>0, (3.19) leads to a contribution

m=0

prop ortional to x , and when r =2k+ 1, (3.20) leads to a contribution prop ortional

to x plus higher-order terms. One also nds that

n;C

H = h (a + b + c + d )

np np np np np

p=1

p 3

2 2

2 1

+ a 1+x 1+x x +

n 5

2 2 2

2 1

+b 1+x x + 1+x

2 p2

2 2

2 1

+c 1+x x + 1+x

n 3

2 2 2

2 1

+d 1+x x + 1+x ; (3:52)

1 2

n;D n6 2

= H k n(n + 2)(n +4)x 1+ 1+x

r n

2 2

rn6 2

k (r + n)(r + n + 2)(r + n +4)x 1+x : (3:53)

r=1 19

Local Supersymmetry in One-Loop Quantum Cosmology

n;A n;B n;C n;D 6

e e e e

This is why H , H , H and H do not contain terms prop ortional to x , and

1 1 1 1

hence do not contribute to (0).

To sum up, in light of (3.4), (3.33)-(3.34), (3.36), (3.44)-(3.47), and using the (0)

value obtained in Ref. 3, we nd

11 5 289

(0) = = ; (3:54)

360 6 360

all untwiddled which is equal to the PDF value found in Ref. 4 when one sets to zero on S

A A

co ecients of and . However, as shown in Ref. 10, (0) values dep end on the

i i

b oundary conditions if Ma jorana fermions and gravitinos are massive.

4. Concluding Remarks

The calculation app earing in our pap er was not p erformed explicitly in Refs. 5,10, and

was only available in Ref. 9. Wehave therefore tried to present it in a self-contained

way in this journal, to make it accessible to a wider audience. Interestingly, if the gauge

constraints (1.2) and sup ersymmetry constraints are imp osed before quantization, the PDF

289

(PDF)

value is found to b e (0) = . However, Becchi-Rouet-Stora-Tyutin-invariant

360

quantization techniques might lead to di erent (0) values. This is indeed what happ ens

in Ref. 2, where, studying the e ect of ghost elds and gauge degrees of freedom, the

197

(0) = . In this case the di erence with resp ect to the PDF value (3.54) author nds

180

is substantial, at least b ecause the signs are opp osite. However, one should b ear in mind

1 3

that the discrepancy found in Ref. 3 for the spin- result also a ects the spin- calculation.

2 2 20

Local Supersymmetry in One-Loop Quantum Cosmology

Moreover, it is also worth remarking that in Ref. 2 the gravitino contribution to (0) in

simple sup ergravity makes the one-lo op amplitude even more divergent, when p erturbative

mo des for the three-metric are set to zero on S . By contrast, within the PDF approach,

the gravitino contribution to (0) in N = 1 sup ergravity partially cancels the contribution

of the gravitational eld in such a case.

Our result (3.54) may not only add evidence in favour of di erent quantization tech-

niques for gauge elds b eing inequivalent, but remains of some value if a mo de-by-mo de

gauge-invariant (0) calculation is p erformed. In that case, the physical degrees of free-

dom decouple from gauge and ghost mo des, so that their contribution to (0) is again

given by equation (3.54) if the b oundary conditions (2.1) are required. Unfortunately,

already in the simpler case of Euclidean Maxwell theory in four dimensions, gauge mo des

are then found to ob ey a very complicated set of coupled eigenvalue equations, and it is

not yet clear howtoevaluate their contribution to the full (0) value in a mo de-by-mo de

analysis. If this last technical problem could b e solved, one would then obtain a very rele-

vantcheckof(0) values for gauge elds in the presence of b oundaries previously found in

the literature. Of course, sup ergravitymultiplets cannot b e studied at one-lo op ab out at

Euclidean four-space, since the existence of a cosmological constant is incompatible with a

at background geometry. However, we hop e that the calculations in our pap er (see also

Ref. 10) can b e used as a rst step towards a mo de-by-mo de p erturbative analysis in the

9;17

presence of curved backgrounds, at least in the limit of small b oundary three-geometry.

A further interesting question, arising from the work in Refs. 9,18-19, is whether lo cal

b oundary conditions involving eld strengths rather than p otentials can b e used for spin 21

Local Supersymmetry in One-Loop Quantum Cosmology

. It is not yet clear whether, and eventually how, the corresp onding one-lo op calculation

might b e p erformed.

Acknowledgments

Iammuch indebted to Peter D'Eath for enlightening conversations and to Alexander

Kamenshchik for corresp ondence.

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Local Supersymmetry in One-Loop Quantum Cosmology

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