SELF-DUAL SOLUTIONS TO EUCLIDEAN GRAVITY ... - CiteSeerX

SLAC-PUB-2213 LBL-8265 October 1978 CT)

SELF-DUAL SOLUTIONS TO EUCLIDEAN GRAVITY *+

Tohru Eguchi Stanford

Linear

Accelerator Center,, Stanford 94305 Stanford, California

University

and ** Andrew J. Hanson Lawrence Berkeley Laboratory, University of California Berkeley, California 94720

ABSTRACT Recent work on Euclidean reviewed. treat

We discuss

asymptotically

These latter

solutions

topological resembling

invariants, that

various locally

self-dual

solutions Euclidean

have vanishing

gravitational to the Einstein

self-dual

of the Yang-Mills

(Sub.

action

and

in detail. and nontrivial

in quantum gravity

instantons.

to Annals

of Physics)

Jc

Research contract

**

Research supported in part by the High Energy Physics of the United States Department of Energy.

t

Present address: Chicago, Illinois

supported in part by the Department number EY-76-C-03-0515.

Enrico 60637.

is

equations

metrics

classical

and so may play a role

fields

Fermi

Institute,

of Energy

University

under Division

of Chicago,

-2-

INTRODUCTION

1. The discovery Yang-Mills

of self-dual

theory

interest

[l]

has recently

in self-dual

solutions

One would expect that

and free

fields

a flat

"asymptotically

differs

metric

flat

from that

instanton

local

of Einstein

metric

II

resembles

solutions

[2]

Bianchi

V-cry general

dual asymptotically identified

could describe

classes

locally

by Hitchin

Asymptotically number of special

space.

that

are called

topology

the Yang-Mills locally

then studied

their

at infinity

this

class

case.

Euclidean in ref.

metrics

121.

the general

and showed that

a nonsingular

class only Gibbons

manifold.

an entire

series

manifolds

which could admit self-

Euclidean

metrics

of such metrics.

have recently

been

[ 51 . locally

properties.

and so must be quite

of

the metric

Since the Yang-Mills

type IX metrics

and Hawking [ 41 have now exhibited In fact,

global

have been

in spiteof

given by the authors

Page and Pope [3]

Euclidean of ref.

Euclidean

closely

Gibbons,

of self-dual

because,

their

character,

would be in Euclidean

solutions

These solutions

examples of asymptotically

were the self-dual Belinskii,

metrics

approaches a pure gauge at infinity,

solutions

The first

of gravitation.

property

metrics

deal of

localized

In fact, interesting

of ordinary

potential

theory

are self-dual,

Euclidean"

in Euclidean

a great

instanton-like

at infinity.

locally

asymptotically

stimulated

of singularities.

found which have the additional approaches

solutions

to Einstein's

the relevant

those whose gravitational spacetime

instanton

important

Euclidean

self-dual

For one thing,

metrics

have a

they have zero action

in the path integral.

Secondly,

since

-3-

the metrics switched

become flat

off

applied

and the gravitational

at infinity,

standard

for gravitational

let

locally

identification

of the Euler

of a manifold

manifolds

invariants

instantons.

Einstein's

metric

has so far The first

solutions arise

metrics

Unfortunately,

of black

time variable,

spacetime,

equations

which is associated

In this

zero cosmological

[ 81;

Gth

the explicit

without this

vanishing

form of the

While all

black

K3

metrics with

spatial Euclidean

case Einstein's

constant,

hole solutions

$hey can be continued

These continued

is the self-dual

Hawking [lo].

K3

which come to mind are the standard

hole physics.

and decay only in the three a metric

the

manifold

curvature

regime to produce positive-definite 191 [ 101.

for

eluded discovery. known metrics

in Minkowski

Euclidean

satisfy

of these,

four-dimensional self-dual

Riemannian

candidates

The most remarkable

metric

constant.

signature

analogs of the Yang-

as logical

with

of

step was the

[ 61 171. A number of standard

boundary which admits a metric

cosmological

the discovery

and Hirzebruch

gravitational

is the only compact regular

would therefore

stages of the search

The first

characteristic

were of course considered

gravitational surface,

metrics.

as the appropriate

topological

of such metrics.

which took place before

Euclidean

are

methods can be

us summarize various

instantons

asymptotically

iWills

asymptotic-state

to analyze the quantum effects For completeness,

interactions

also to the

singularity-free are periodic

in the new

the thermodynamic

directions.

One example of such

Taub-NUT solution equations

and the manifold

temperature,

is

examined by

are satisfied 0X4 with

with

a boundary

-4-

twisted

which is a 4

metric.

The metric

fall

in all

off

Another

rather

a Maxwell

type of acceptable

of the metrics

are regular

The gravitational and make it necessary

difficult for

ordinary

very interesting,

described

Euclidean

scattering

gravitational

contains

asymptotically

P,(G)

by adding

throughout

coincide

spacetime

plane-wave

Although with

excitations

flat.

states

these metrics

are

our intuitive

in Euclidean

In contrast,

seem to be very naturally

spacetime

the asymptotically identifiable

as

instantons.

The remainder II

persist

theory.

as localized

metrics

one can constructa on

the asymptotic

which approach the vacuum at infinity. locally

does not

but are not asymptotically

they do not quite

of instantons

P,(a)

are in some sense self-dual,

such metrics

to define

equations

.

action,

fieldsof

Weyl tensor

Einstein's

spin structure

just

and have finite

has self-dual

Nevertheless,

[ll]

on P2(@),

by Eguchi and

One drawback is that

spinors.

to the theory

metric

boundary and has

and so solves

term.

Dirac

field

All

The metric

does not

directions.

space, studied

curvature,

cosmological

more general

because it

is compact without

curvature.

admit well-defined

a distorted

case is the Fubini-Study

manifold

than self-dual

flat

spacetime

complex projective

scalar

with nonzero

picture

asymptotic

interesting

Freund[71[2C,!..This constant

is not asymptotically

four

two-dimensional

S5 possessing

three-sphere

of the paper is organized

a complete explanation flat

self-dual

as follows:

of the derivation

solution

presented

Section

of the regular

in ref.

[2].

-5-

In Section

III,

we examine the properties

which have instanton-like self-dual remarks.

multicenter

properties. metrics

of various Section

and Section

other metrics

IV is devoted to

V contains

concluding

-6-

11.

AN ASYMPTOTICALLYFLAT SELF-DUAL SOLUTION OF EUCLIDEAN GRAVITY

We now derive dual solution ref.

[2].

the simplest

of Euclidean

gravity,

equations

and noting

possible

instanton,

we do the following:

a procedure

to obtain

gravitational

Observe that

the Yang-Mills

(2)

self-

as metric

II in

by which one can solution

To obtain

[l]

the

equations = 0,

F =aA - a,& P I-iv

due to the Bianchi

flat

the instanton

parallels.

3F + lA,&Vl 1-1w where

asymptotically

which was labeled

Let us begin by reviewing

solve the Yang-Mills

(1)

regular

+ ~Av,Avl , are solved at once

identities

if

.

Choose the Ansatz A 1-I for the g = (t

(3)

SU(2) - i':

l

gauge potential, ;)/r,

Solve the first-order

and

by setting P = r2/(r2

equation

= 0 -3 F l.lv j-2 + a2)

r2 = t2 + z2,

are the Pauli

differential

P’(r) +$0-l) obtained

{T]

where

.

we find

matrices.

In this

way, we find

SU(2) Yang-Mills

solution

with

localized at r = 0 and falling like F PJ a pure gauge at infinity. at infinity, and A asymptotically ?J We wish to find a Euclidean gravity solution with finite

finite

action,

l/r4

self-dual

action,

self-dual

rapidly

at infinity,

Euclidean by

a Euclidean

curvature and with

inside

the metric

the following

the manifold

and falling

asymptotically

We might therefore

at infinity.

undertaking

localized

search

locally

for such a solution

gravitationalanalogsof

the Yang-Mills

procedure: (1)

Observe that self-dual

(i.

satisfied

metric

obtained

the essential let flat

we establish concepts

the four Euclidean metric

is

the curvature so Einstein's

guv(x)

equations

differential

by requiring

from a flat

r2 = t2 + "2

of

equations

alone.

in the

wab to be self-dual.

Preliminaries. some useful

appearing coordinates

notation

and explain

in the procedure be

x1-l

=

just

(t,x,y,z)

more fully

outlined. so that

is given by ds2

=

are

identities.

which'differs

by functions

Solve the first-order

A. First

tiab

e.,

Choose an Ansatz for

metric

l-form

at once due to the cyclic

Euclidean (3)

the spin connection

Rab is self-dual,

2-form

(2)

if

dt2 + dx2 + dy2 +dz

2

e

(2.1)

We the

-8-

We next change to four-dimensional 4

t2

+ x2 + y2 + z2

0

X

=

$

(xdt

- tdx + ydz - zdy)

= $ (ydt -tdy+

Cl Z

=

1 (zdt r2

coordinates

with

r2 =

and define

cl

Y

polar

zdx-

= $- (sin

xdz)

=

- tdz + xdy - ydx)

$de - sin 0 cos $d@)

-cos @de - sin 0 sin $d!$)

= $- (d$ + cos ed@). (2.2)

The variables with

8, $I, $

are Euler

angles on the three-sphere

S5

ranges

O Suggestively, the l/r4 as that

with

asymptotic

behavior

of the Maxwell field

of the Yang-Mills

cJ.

Properties

in the new metric

complete manifold alternative

(2.26)

[3].

forms.

is -the same

instanton.

We now need to determine ities

(2.30)

of the Manifold -whether there

and whether it

We begin by writing

First,

are any true describes the metric

singular-

a geodesically in several

let

p4 = r4 _ a4-,

(2.31)

then ds2

=

[l+(a/p)4]w4

{

+ El + (a/p)41'{p20x2 These coordinates

are well-adapted

dp2 +Pa2 Z2 +Po2 ,"}

to converting

.

the metric

complex form using z1

L

=

x + iy

=

zlzl

A

p

+ z2i2 .

"2

=

z + it

(2.32)

into

I -17-

One then finds a Kahler

form

1)

=

two equivalent 151 [131

4\ K

[ P4

ways of writing on

the metric

in terms of

CC2- (01:

P2 p ( dzp$ + dzp*)

+ I p4

+ a4l

a4

1 a;ib(p”) + a4]’ (2.33a)

2)

K

+ a4]’ $I = P2 exp [ P4 1 a2 + IP4 + a4]H

= aa Rn c$,

The

aa Rn p2

at

r = a).

one identifies

term in these forms causes problems However, this opposite

apparent

points

(zp,)’

(2.33b)

*

singularity

at

P = 0 (i.

e.

can be removed if

of the manifold,

‘L (-zp,)

(2.34)

or 'xv'

'L (-x,).

We next give a more elementary First,

let

us change radial

u2 Then the metric dS2

explanation variables

of this

fact.

once again by defining

= r211 - (a/r)4].

(2.35)

can be written = du2/[l

+ (a/r)4]2

+ u20z2 +r2(o

2 +cs 2 . Y)

X

(2.36)

-1s

Very near to the apparent .-

singularity

at

r = a, or, equivalently,

u = 0, we have ds2

=

; du2 + T' u2(dJ, + cos ed$)2 + t2(de2

+ sin28d$2)m (2.37)

For fixed

8 and

ds2 =

u=o

$,

$du2

we obtain

+ u2d$2).

A short

exercise

is real

or is a removable polar

simply note that ds2

=

tells

(2.3g) us whether

the apparent dx2 + dy2

-is removable provided

r = 0

or not the singularity coordinate

singularity

singularity. in the

Eq. (2.5)

= dr2 + r2dQ2

(2.39)

that (2.40)

conclude that

if

the range

0 < $ < 4~r

given by

is changed to

0 < * < 27r,

(2.41)

we can remove the apparent desically

singularity

at

r=a

and obtain

a geo-

complete manifold. The global

Near

We

B' metric

0 < Q < 27r. We therefore

at

topology

r = a, the manifold

of our manifold

is now the following:

has the topology

p2 x S2 indicated

by

-19Equation

(2.3'7).

parametrized

To be precise,

by ( e,$),

to a point

as

there

r +a (u -t 0).

and has the same Euler For

large

r.,

points

identified, of

covering.

the metric

SO( 3) = P,(a),

Euclidean It

(i.

e. not

= s2,

that

is in fact

an JR2 which shrinks is thus homotopic

9,

to

metric.

the constant-r with

However, hyper-

opposite

r -f 00 is thus the familiar S3 = SU(2)

example of a metric (P$R)

whose topology

= S3/Z2),

but not

S3). the entire

the cotangent

manifold

bundle

M we have

of the complex plane,

M = T*(Pl(C)) (2.42)

In Figure manifold regularity

1, we present

. a description

of the topology

which we have deduced from the metric requirements.

group

is the double

and so we may write

aiu = Pi

S2

x = 2.

S2,

but three-spheres

Euclidean

can be shown [5]

described

of

for which

locally

of the two-sphere

approaches a flat

range (2.41)

This is an explicit

globally

as

The boundary as

is asymptotically

P,(C)

The manifold

are not three-spheres,

manifold

just

is attached

characteristic

because of the altered surfaces

at each point

(2.26)

of the and the

-2o-

Action

D. .-

and Topological

Using the connections

4

the metric various

(2.26)

integrals

with

and curvatures

the

$-range

characterizing

(2.27)

(2.41),

and (2.28)

for

we now calculate

the

Since our manifold

the solution. we will

a non-empty boundary surface, fundamental

Invariants

repeatedly

need the second

form,

cab = wab- (“o)a,, to compute boundary

everywhere

the connection

of the product

ds2

(2.43)

If we choose the radial

corrections.

as the direction

normal to the boundary, metric

for fixed

direction

(w~)"~

is

ro,

= [1 - (a/r 0 141m1dr2 + r ox'(0 2 + oy 2 + [l _ (a/r,)4]o (2.44)

Since our scalar

curvature

is identically

action comes from the surface term [9]. eo *= -KljeJ, we calculate the surface i 1 -5

has

J

Kii

dx

= i [3r2

- ?4

aM(d

zero, . Defining Klj

action

at large

- 3r2(l

the entire by

[ 131

r

to be

- (a/r)4)']

4 (2.45)

-Q* Since the surface

term falls

vanishing

for

action

like

the metric

l/r2

as

r -f ~0, we find

(2.26),

S[ .!?I = 0.

(2.46)

3.

Z

-21We have already manifold

stated

the same Euler

We confirm

this

fact

the topological x=2,

characteristic,

using Chern's

arguments giving

formula

a "Red E abcdR b

/

as

our

S2.

[141[151 Eabcd(2eab A Red

aM(c0)

ZZ

3

- (-$)

= 2.

(2.47)

z

Had we allowed

$

to range over all

we would have found disagreement

twice

singularities

necessity

corrections

of cone-tip

in the curvature istic

to its

correct

very satisfactory $

at

must indeed be

r = a)

physical

0 < $ < 4T, this

at r = a;

in order to adjust

situation,

theorem

the manifold implies

the _

delta-functions

the Euler

character-

This does not seem to be a

value.

so that

the proper

range of

0 < $J < 27~.

To compute the signature pute the integral

for

(such as effective

topological

P3(lR),

and the Chern-Gauss-Bonnet

11 range shows that

would have "cone-tip"

of

The apparent

answer, X = 4.

this

between the topology

for the wrong

S3 instead

of

of the first

P,[M]

=

of our manifold

Pontrjagin

J

p1

M =

T

-3.

M,

we first

class,

= -+ "

J M

Tr(R A R) (2.48)

com-

-22The Chern-Simons boundary

- Q,[aM]

correction

Tr(0' A R)

= 1

[16]

vanishes,

(2.49)

= 0.

8n2

aiv The signature

q-invariant

q,

has been computed by Atiyah, qs(P3(/R))

Thus the signature

of

-c[M]

Patodi

boundary

Atiyah

I,

2 of the metric

2

[18]

case, with

=

P3(/R)

1171 to vanish

also:

(2.50)

[17]

one anti-self-dual

of the Hirzebruch harmonic

signature

2-form with

conditions.

of the spin

-$-(P1[ti

(2.51)

= -1.

extension

the appropriate

I,

on

= 0.

= +(P,-Q&n,

is exactly

presence

metric

M is

there

The index

and Singer

= $ cot2 ( 3)

Ey the Atiyah-Patodi-Singer theorem,

for the canonical

(2.26)

f

Dirac

is given by - Q,[aid)

operator [17]

in the

[15]. , (2.52)

- +(v, 2 + h$.

has extended the computation

of ref.

]17]

to the Dirac

the result

h,z = 0.

(2.53)

-23Thus I

and there

ZZ _ $

3

(-3-o),

-i(i)

= 0 '

is no asymmetry between the right

zero-frequency

modes of the Dirac

For the spin j/2

(2.54)

and left

chirality

operator.

Rarita-Schwinger

operator,

the index is

given by

‘3/z

=

$$

-Q,[aMI > - $(n3/2

(Pl[MI

+

h3/2) > (2.55)

where we have corrected

the result

of ref.

1191

to include

boundary terms in the obvious way.

Hanson and Rgmer [ 201

calculated

the Atiyah-Patodi-Singer

n-invariant

the expression with

‘3/2

involving

the result

+ h3,2

=

- $ .

There are thus two excess negative obeying the Atiyah-Patodi-Singer

*3/z

1211,

covariant

constant

chirality boundary

= s (-3-o) +2,

This is in agreement with and Pope

have

who build spinors

the explicit two spin

spin 3/2 conditions,

=

4.

construction 3/2

and the single

fields

(2.57) of Hawking

wave functions

out of two

anti-self-dual

Maxwell

-24field

whose existence

The indicated

is required

spin j/2

an appropriate

solutions

supersymmetry

While the spin

2

by Eq. (2.51) may in fact

index characterizing

perturbations

about the metric

calculated

at this

there

corresponding

follow

toadilatation,which

directly

from

transformation.

self-dual

time,

for the signature.

is at least

the number of anti-

(2.26)

has not been

one zero-frequency

is not a gauge transformation

mode, [22].

-25-

PROPERTIESOF MOREGENERALMETRICS

111.

General Bianchi

A. One might naturally replaced

ask what happens if

by the most general

ds2

=

f2(r)dr2

IX Ansatz

Ansatz giving

+ a2(r)ox2

the Ansatz (2.21)

a Bianchi

+ b2(r)oy2

IX metric

is I 31 ,

+ c2(r)oz2*

(3.1) For the case a(r)

we find a flat

that

self-duality

metric.

implies

(3.2)

c(r),

a vanishing

curvature

and hence

II

= b(r)

of ref.

[21,

c(r)

= 1,

= g(r)

which we studied

(3.3)

in the previous

the choice

section;

4d

= b(r)

E g(r)

I

of ref.

[21.

was case describe of

=

The case

a(r) was our choice

= b(r)

the regular

(3.4)

are therefore

in fact

manifold

,

While self-dual of Fig.

have a singularity

unacceptable.

4 d

=l

(3.4)

solutions

1, the self-dual at finite

proper

of

(3.3) solutions

distance

and

-26A general Ansatz (3.1) -

131 *

solution

of the self-duality

equations

has been given by Belinskii,

Gibbons,

for the

Page and Pope

They find f2W

= F-'(r)

= r6a-"(r)be2(r)cB2(r)

a2(d

= r2Fg(r)/(l

- (al/r)4) (3.5)

b2(d

= r2Ff(r)/(l

- (a2/r)4)

c2W

= r2FS.(r)/(l

- (a,/r)')

where F(r)

= (1 - (al/r)4)(l

are constants.

a.&. y-2Ja3 for general finite

parameters

proper

manifolds,

II

inside

to it.

distance

ai,

They find

allows the

S2

and so describe

r

=

a

(3.6)

(see also ref. all'have

physically

degenerate

a mechanism for at

- (a3/r-j4)

these metrics

Only the particular

in Section larity

- (a,/r)4)(1

so that

that

singularities unacceptable

case (3.3)

"shielding"

f 131 )

described

the naked singu-

geodesics

cannot get

(Th'1s is of course analogous to what happens in the

Euclidean

continuation

of the Schwarzschild B.

Given a metric, know is whether

and Taub-NUT metrics.)

Nuts and Bolts

one of the most important

or not its

apparent

singularities

The two known types of removable singularities

things

one must

are removable. have been christened

at

.

-27-

"nuts "

and "bolts"

by Gibbons and Hawking [231 .

four-dimensional is flat

)R4 polar

at the origin,

A "bolt"

like

looking

like

one unit

of

singularity

the self-dual

is a two-dimensional

a metric carry

coordinate

Euclidean

JR2 polar

characteristic,

is a

in a metric

while

which

Taub-NW metric.

coordinate

jR2 X S2 near the origin, Euler

A "nut"

singularity

like

(2.26).

bolts

carry

in Nuts two

units. A precise follows

formulation

[131 [ 231 : Consider

ds2

=

coordinate In general,

r

into

'I

A metric

2 + c2(~)oz2

distance a,b,c

the manifold

,

(3.5)

to convert

the

(or proper

time)

be finite

manifold.

'1.

and nonsingular

(For infinite has a suitable

can be regular

consider

has a removable nut

near

T = 0,

In this

case at

T = 0

the flat

polar

coordinate

singularity

is as

rc,

this

boundary at

even in the presence

singularities.

Let us for simplicity T = 0.

if

However, the manifold

of apparent

that

to get a regular

can be relaxed

'I = a.)

Y

the proper

one would require

finite

restriction

+ b2(-r)o

change has been made on (3.1)

radius

and bolts

the metric

dT2 + a2(T)ox2

where a variable

for

of the concepts of nuts

singularities singularity

occurring provided

2 a2 = b2 = c2 =+r.

we have simply

a coordinate

system on an iR4 centered

is removed by changing to a local

at

that (3.6)

singularity at

Cartesian

in

T = 0. coordinate

The

-28-

system near Near

'r = 0

~=0,

and adding the point

the manifold

A metric

is topologically

has a removable bolt

a2 = b2

while

at constant

TR4. if

= finite

= 0,

(3.7) c2

Here

to the manifold.

singularity

= b2

a2 near T

T = 0

multiplies

the canonical

(f3,@), the

n = integer.

>

S2 metric

(d-r2 + c2(1)o z2)

$ (de2 + sin2Bd$2),

Piece of (3.5)

looks

like 2 dT + n2-c2$

Provided

the range of

singularity the flat

at polar

n Q/2

T = 0

is adjusted

is nothing

coordinate

d$2 ,

(3.8)

to

0 -+ 2~,

but a coordinate

system on B2.

Again,

the apparent

singularity this

in

singularity

be removed by using fold

is locally

as

-c + 0. C.

Cartesian coordinates. The topology of the manic E2 x SL with the BL shrinking to a point on S2

The Fundamental Triplet

The prototype (2.26)

introduced

of a metric

in ref.

[2].

in detail

with

nut is the self-dual

ds2

lr+m = ~i;r-m

of Self-Dual

with

a single

Metrics

bolt

The removal of this

was discussed a single

can

in Section

II.

is the metric singularity

The prototype

Euclidean

of a metric

Taub-NUT metric

[XC]

dr2 + $ (r2 - m2)(dB2 + sin20d$2) (3.9) + m2 $+

(d+ + cos ed@f

-29-

To remove the apparent proper

distance

singularity

at

r = m,

we first

change to the

coordinate

(.3.N

and consider

only the region

r = m + a, E

where dr,

2(r2 - rn2)$ ox, 2(r2 - m2)" 0 , Y

-34-

The Maxwell field

our metric

has the characteristic while

(2.30)

The P,(G)

l/r2

field

(3.25),

was presented

triplet

its

Euler

is

-r = 1.

of the Yang-Mills

instanton.

hand, is constant

of metrics. without

p2w

boundary.

everywhere.

is

x

invariants

is the easiest, Because

P,(g)

= 2 + 1 = 3,

were a spin manifold,

p2w

in

the Taub-NUT Maxwell field

summary of the topological

characteristic If

earlier

of a magnetic monopole,

on the &her

is a compact manifold

signature

behavior

we give a brief

of the fundamental

and a nut,

behavior

has the l/r4

Finally,

it

(2.26)

We note for comparison that

Eq. (2,30). *

for

since

has a bolt while

the

the spin B

index would be (3.27)

must be an integer for a manifold admitting well-defined 3 spinor structure, we confirm the fact that P ((c) has no spin 2 structure. For the Taub-NUT metric, x = 1 and the spin f index

Since

I

with boundary

corrections

the topological A tabulation

invariants

Introduction.

[151-

were previously

of the properties

in Table 2 alongside the

vanishes

For the metric

given in Eqs. (2.47-54).

of the fundamental

the properties

(2.26),

triplet

of the K3 manifold

is presented mentioned

in

IV. -

Hawking's

A" Just as there Yang-Mills multiple

problem removable ds2

MULTICENTERMETRICS E = 1 Multicenter

are multiple [25],

instanton

there

for the metric

SU(2)

with

Hawking examined the Ansatz [lo]

(d+ + ;

and found (anti)self-dual

solutions

is a gravitational

singularities. = V-l(;)

Nitric

l

connections

d;,2

+ V&d;

l

d;:

(4.1)

(and hence an Einstein

solution)

provided -P

w

+ self-dual

= k$x;

- anti-self-dual.

(4.2)

Clearly,

$“v = 0, so that,

modulo delta

(4.3)

functions,

v(Z)

=

a solution n

& +

is

2mi]Z - si]-'

c

,

(4.4)

i=l

where ities all

E is an integration at

the

Z = Z mi

i

into

constant.

removable

to be equal,

In order to make the singular-

nut singularities,

mi = M,

and make $

one must take periodic

with

range 0 < $ 6 &rM/n

(4.5)

The case E

= 1,

(4.6)

the

-36-

which reproduces

the self-dual

examined in ref.

S E=l

Since all unit

action

[nl

=

B. in (4.4)

Eql.

term

are nuts and each nut contributes we find

characteristic,

s

[26] (4*7)

comes from the surface

x

Since

has been computed to be

47r n 2,

contribution

the singularities

of Euler

for n = 1,was

[lo].

The gravitational

where the entire

Taub-NUT metric

the Euler

one

characteristic

= n.

(4.8)

The E = 0 Metric is

an arbitrary

constant,

one might ask

what happens when we set E

This case has recently Clearly, altered

0.

(4.9)

been examined by Gibbons and Hawking

the asymptotic and in fact

=

behavior the entire

of the metric action,

[41.

is drastically

including

the surface

term,

now vanishes:

SE+-)In1 = 0. This case therefore in the path integral.

will

probably

dominate over the

(4.10)

E = 1 solutions

-37-

As before,

we need the periodicity

(4.5)

for

a regular

manifold,

but now one finds

(

E = 0, n=l

The boundary at

~0 for

2.

one finds

For

tensor.

n=2, In this

s3/z2

n = 1

foundf

a two-nut

thus it

= 0,n

E

change of frame.

the boundary

00 consists

of the cyclic

We conclude that with

E

solution

=

0

strongly

[ 251 .

in the description

of

metric

space boundary

with nonzero Riemann

~0 is the same lens space (2.26)

and also the curvature that

= the metric For higher

(2.26)

values

S3 with points

(4.12)

of

n,

identified

under

group of order n. in general parallels

In particular, of the

The topological

(4.11)

the Euclidean

seems certain

= 2

up to a singular at

space metric.

is just

or our metric

are the same;

the action

flat

case the boundary at

= P3(m)

invariants

=

>

the multicenter

metric

the Jackiw-Nohl-Rebbi there

are

"n-instanton"

invariants

(n + 1)

(4.1-4.4)

multi-instanton

positions

appearing

solution.

for the

E

= 0

metric

are [211

X = n T

= f (n - 1)

*l/2

= O

I

= 2T=*+(n-1)

3/2

(4.13)

.

There is in fact vanishes

for

a general

asymptotically

led to conclude that the spin l/2 the roles

theorem showing that

axial

flat

for gravity,

metrics

a spin 3/2

and Yang-Mills

[ 131 I

axial

anomaly induced by Yang-Mills

of gravitational

breaking

self-dual

the spin lL2 index We are

anomaly replaces instantons.

instantons

Thus )

in symmetry

may be summarized as ,follows: Yang-Mills Einstein

solution,

Chern class

signature

solution,

k +

Dirac

index

? + Rarita-Schwinger

= k

index = 2-r. (4.14)

We conclude with the remark that self-dual

Maxwell fields

for the metrics

for the multicenter

in previous

A

= V-'(dY+;

F

=

we can write

Sections.

metrics

One such field

down natural just

as we did

is

d;)

l

. d.A = Vm2 aiV(eo n e1 k--E ' 2 ijk

e j A ek). (4.15)

C.

More General Metrics

We have now seen the natural spaces of Hitchin sa.I.formsof boundaries

S3 in the multicenter [5]

appearance of higher self-dual

Einstein

order lens metrics

has examined the known complete classification S3 and has found regular corresponding

a unique self-dual

metric

using the Penrose twistor is not completely

complex algebraic

to each sphericalform. can be obtained construction

understood

at this

(4.1). of spheri-

manifolds

with

It is conjecturedthat

for each of these manifolds [271.

time,

let

Although

this

us at least

subject list

the

-3% of S3 corresponding

sphericalforms locally -

Euclidean

according

to their -q

Series

Dk : T

:

tetrahedral

0

:

octahedral

I

:

icosahedral

(4.1)

(2.26)

solutions

essentially

solved.

structure

groups as follows

[28] :

group group

self-dual

x

--cubic group

group x

corresponds

corresponds

spherical

zero-action

forms areclassified

-group of order k (= lens spaces L(k + Ll)) dihedral group of order k

If we could derive each of these

discrete

asymptotically

cyclic

our metric

metric

The spherical

metric.

associated

Series

We note that n-center

self-dual

to each possible

to

dodecahedral to

while

.

the general

A n-l'

metrics

for the manifolds

forms as boundaries, of the Euclidean

Al,

group

the problem of finding

Einstein

We would then have a better

of the vacuum in quantum gravity.

having

equations

would be

understanding

of the

-4o-

VJ The discovery

4

Yang-Mlls

theory

CONCLUSIONS

of the. self-dual

suggested the possibility

to the Euclidean

Einstein

equations

gravity.

Here we have discussed

Euclidean

gravity

centrated

particularly

similar (1)

(2.26)

solutions

ized in Euclidean (2)

Their metrics Euclidean

(3)

solutions

to

Euclidean

have properties

metrics,

nontrivial

which are

instanton

excitations

solutions:

which are local-

space-time.

approach an asymptotically

vacuum metric

They have nontrivial

However, there

in quantum

is the simplest

gravitational

solutions

We have con-

locally

to those of the Yang-Mills

They describe

to Euclidean

analogous

properties.

on the asymptotically

These gravity

strikingly

that

might be important

their

solution

solutions

a number of self-dual

and indicated

of which the authors' example.

instanton

locally

at infinity.

topological

are also some important

quantum numbers.

distinctions

between the two

sets of solutions: (1)

The gravity axial

solutions

anomaly, while

to the spin l/2 (2)

The gravity Yang-fills

The pairing and the pairing

axial

solutions solutions

contribute

the Yang-Mills

with

solutions

contribute

anomaly. have zero action, have finite

of the Yang-Mills

of gravity

only to the spin j/2

field

while

the

action. with

the spin j/2

the spin

l/2

anomaly

anomaly are very likely

-4l-

due to the existence whether

of supersymmetry.

supersymmetry

It would be interesting

gives any further

insight

into

to see

the structure

these systems. As is well-known, implies tally

the suppression inequivalent

gravity action that

the finite

there

of the transition

sectors

of the theory.

instanton

amplitude

locally

in the path integral

vacuum metric.

Euclidean

between topologi-

The vanishing

self-dual

they have the same weight

Thus these solutions

in understanding

will

action

Gn the other hand, in

appears to be no such suppression.

of asymptotically

importance

Yang-Mills

metrics

implies

as the flat

presumably be of central

quantum gravity. Acknowledgments

We are grateful work was done, for its S. W. Hawking, informative us with

to the Aspen Center for'physics, hospitality.

Q. Hitchin,

discussions,

preprints

of their

R. Jackiw,

We are indebted M. Perry and I.

where this to J. Hartle, Singer

for

and to G. Gibbons and C. Pope for providing work.

of

-42REFERENCES 1.

A, A. Belavin,

A. M. Polyakov,

Phys. Lett.

59B,(1975)

A. S. Schwarz and Yu. S. Tyupkin,

85.

2.

T. Eguchi and A, J. Hanson, Phys. Lett.

3.

V. A. Belinskii,

G. W. Gibbons,

Phys. Lett. 4.

76B (1978)

74B (1978)

249.

D. N. Page and C. N. Pope,

433.

G. W. Gibbons and S. W. Hawking,

"Gravitational

Multi-Instantons,"

in preparation. 5.

N. Hitchin,

6.

A. A. Belavin

private

communication.

and D. E. Burlankov,

(1976)

7; W. Marciano,

(1977)

1044.

H. Pagels and Z. Parsa,

7.

T. Eguchi and P. G. 0. Freund,

8.

N. Hitchin,

J. Diff.

Proc.

Acad. Sci.

9.

Natl.

Phys. Lett.

Phys. Rev. Lett.

Geom. -9 (1975)

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USA -74 (1977) 1798.

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10.

S. W. Hawking, Phys. Lett.

11.

A. Trautman,

Inter.

Jour.

60A (1977)

of Theor. Phys. -16 (1977)

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Differential

1962);

2752.

81.

S. W. Hawking and C. N. Pope, Phys. Lett.

12.

58A

561;

73B (1978)

Phys. Lett.

77B (1978)

42;

A.Back,

181.

Geometry and Symmetric Spaces (Academic

H. Flanders,

Differential

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1963). 13.

G. W. Gibbons and C. N. Pope, "The Positive and Asymptotically (DAMTP preprint).

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Metrics

Action

Conjecture

in Quantum Gravity"

-4314.

S. S. Chern, Ann. Math. -46 (1945) 674.

15.

T. Eguchi, B.B.Gilkey

and A. J. Hanson, Phys. Rev. D17 (19'78)

423; H. Rsmer and B. Schroer, C. Pope, Nucl. 16.

71B (1977) 182;

Phys. Bl41 (1978) 432.

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17.

Phys. Lett.

M. F. Atiyah,

-78 (1976)

334.

V. K. Patodi Proc.

Sot. -5 (1973) 229;

48; P. B. Gilkey,

and I. M. Singer, Camb. Philos.

Bull.

London Math.

Sot. -77 (1975)

43;

405; -79 (1976) 71.

18.

M. F. Atiyah,

19.

N. K. Nielsen, "Approaches

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M. T. Grisaru,

H. RGmer and P. von Nieuwenhuizen,

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A. J. Hanson and H. Rb;mer, "Gravitational' to Spin 3/Z Axial

21.

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22.

M. Perry,

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23.

G. W. Gibbons and S. W. Hawking,

24.

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25.

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private

Phys. Rev. D15 (1977) 26.

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239.

4

27.

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and Grav. 2 (1976)

31.

p. 225 (McGraw-Hill,

Curvature,

Spaces

1967)

FIGURE CAPTIONS Fig.

1:

The manifold For fixed

T*(P,(C)) S2

jR2 x s2.

topology

of

P,(p).

the canonical

to Fig.

2:

(0 ,a),

coordinates

topology

equivalently,

described

Constant At

radius

r -+ a

(2.26).

the manifold

has local

hypersurfaces

03, the metric

P3(rR) metric. as

by the metric

As

have the

on the boundary is

u -+ 0 [Eq. (2.36)]

[Eq. (2:,26)],

the manifold

or shrinks

s2 = Pi(Q).

Relations

among the manifolds

self-dual

Taub-NUT metric

(3.13)

or (3.15)

on P2(@).

(3.9)

of our metric

(2.26),

and the Fubini-Study

the metric

-45Comparison of the Euclidean

TABLE I:

Yang-Mills

and Einstein

equations. 4

Yang-Mills

Property

__----

Metric Structure Connection

A = Aa -ia dxl-I l.Jlz

Curvature

F=dA+A^A = -$ FzV $

Dual Curvature

or

ds2 =

f (ea)" a=0 0 = dea + wa A eb b a b a 1-I u~=-w~=w~~~x.

a-----

equation

Rab

Cyclic

Euler

identity

a $a Z--E 1 pv 2 pxxBFa~

%a 1 e R bed= 7 'abefR fed

3 = 3$$

%ab = ixabcd ecA ed %a c w 5, b= 2 abed w d

dxuA dxv

dF+A,,F-F^A=O-dR+wAR-RAw=O b RabA e = 0

-----

identity

equation

d? + A A$-$

AA=O

e =0 -t 'abcdR bed $a b =0 Rb*e -f

Automatic First

solution

= duab + uac A uCb d = 2 Ra eCA e 2 bed

dxuA dxv

Connection:

Bianchi

Einstein

F=t$

Rabcb = 0 Rab

=+p

b

order

automatic

solution

Basic function

F=t$ AaU( x >

wab eaJd

=+w %a b

I -46-

Table II:

Properties metrics

of the fundamental compared with

triplet

the self-dual

of self-dual metric

of the

IC3 manifold.

Metric

:q. (2.26)

Taub-NUT

Fubini-Study

Eq. (3.9)

Eq. (3.13)

X3 explicit form LlIlkTlOWn

Cosmological Constant Manifold

A=

0

A=

0

I\=0

A#0

r*c pl(@ >

- Origin

;'(Bolt)

Point

- Infinity

‘3m)

distorted

S3 S2(Bolt)

- Boundary

?3(Fo

distorted

S3 none

Euler Characteristic Hirzebruch Signature Dirac Spin-$ Index Maxwell Field Strength RaritaSchwinger Spin 3/2 Index

(Nut)

Point(Nut)

none

2

1

3

24

-1

0

1

-16

0

0

%l/r4

sl/r2

-2

?

(no spinors)

1

(no spinors

2

?

-42

P3 (IR)

q

Boundary

at u = CO

.. ..

P3 (IR) = Boundary

at u =CO

XBL 7810-

Fig. 1

6582

-

-e--B-_

---

_

--Nz

40

0

RH

--.

\ I / ‘I ‘- I /) A’ I /+ -A --__ _ -___ -- C: Taub - NUT Metric t Boundary

I’ )\ \ ‘+-\ \ \ \

\

at

Co

\ I Ii Metric

I \ I I

’ \\ \ \ \

\

I I

\

1. \ \ \ \ \

(PO

I

I / ,/=I’ :\ . .

.\

I

----

\

Our Metric

\ \

------____

I

--N .\

$ P$R)

Boundary

-Z ---_

-_____-

at a -- --

+- /A’

XBL7810-6583

Fig. 2

0/