IL NI'()VO CIMENTO VOL. 52B, N. 1 11 Luglio 1979

Einstein-Maxwell Instantons and Charge Quantization.

R. I)EIr (*)

Istituto di Fisica Teorica dell' U.~iversit5 - Trieste, Italia Istituto Nazionale di Fisica Nucleate - Sezione di Trieste, Italia

(riccvuto il 7 Fcbbraio 1979; manoscritto rcvisionato ricevuto il 2 Aprile 1979)

S u m m a r y . - - We describe a class of solutions of the source-free Einstein- Maxwell equations, satisfying two given ansiitze; these solutions represent products of two two-dimensional spaces of constant curvature containing a uniform electromagnetic field. Some of these solutions are compact and can be considered as gravitational-electromagnetic instantons. Due to the nontrivial topology of these four-dimensional ((space-times)), the fluxes of the electric and magnetic field through suitable surfaces (which we call ~(charges ~), in the sense of Wheeler) turn out to be independently quantizcd.

I n t r o d u c t i o n .

There is a g rowing interes t for topological p roblems a m o n g physicis ts , due

essential ly to the recent deve lopments in the area of gauge theories.

As a possible ou t c om e of these efforts, one m i g h t hope to be able to explain

(at least) some of the q u a n t u m num be r s observed in na tu r e in topological terms.

I n this contex t , a pape r b y DuI~F a nd ~IADOI~E (1) appea red recent ly dealing

with electric- and magne t i c - cha rge quan t i za t ion in a K e r r - ~ e w m a n d y o n

g e o m e t r y (for a different a p p r o a c h see ref. (2)). Here similar goals have been

pursued , wi thin the res t r ic ted con tex t of U1 gauge theory , in a different

(*) Fellow of the Consorzio per l 'incremento degli studi e delle ricerche degli Istituti di Fisica deWUnivcrsith, Trieste. (1) M. J. Du~'F and J. MAI)OR~: Phys. Rev. D, 18, 2788 (1978). (2) A. ASIITEKAR and A. SEN: On the role o/space-time topology in quantum phenomena: superselection o] charge and emergence o] nontrivial vacua, Enrico Fermi Institute, Chicago.

90

EI'~STEIN-MAX'~VELI, INS'I'AN'rf)NS AND ('IIARGE QUANTIZATION 91

class of sp~wes. Thl: aim of this paper is twofold: firstly, to find ~ class of

Einstein-Maxwell instantons, secondly to discuss the related charge quan- tization.

We, start from the sourceless Einstein-Maxwell equations with cosmological

term (*)

(1) R, . . . . Ag, , = 8~T,,, ,

(2a) d F = O,

(2b) d* F = O,

aud we look for all solutions satisfying two answ

1) The spume be metrically decomposable into the product of two two-

dimensional spaces. (The main properties of these spaces are reviewedin par t I ;

such a requirement has already been considered in (4) and, in fact, our results

of sect. 2 are just a slight generalization of those contained in tha t paper).

2) The Fa raday 2-form F be a linear combination with arbi t rary coef- ficients of the K/~hler forms of the factor spaces. These forms provide

geometrically ~ natural ~ electromagnetic field and automatical ly satisfy Max-

well's equations (see e.g. (~)); the reason why we do not take the K~hlcr form

of the whole four-dimensional space is tha t in this way we are able to find

non-self-dual field configurations.

The solutions of eqs. (1) and (2) satisfying these requirements are found to

be products of spaces of constant curvature, and are listed in par t I I .

Some of these spaces are compact and can be considered as Einstein-

Maxwell instantons, in the sense tha t they have finite action and nontrivial topology (e.7). They are not instantons in a stricter sense (8) since they are

neither asymptotical ly flat, nor have ~t self-dual curvature.

Electric ~nd magnetic charges can be quantized independently in these

spaces by following the method of Wu and Yang (u). This is discussed briefly

in part I I I ~nd Home specific examples ~tre given in par t IV. There are two places

(*) The convcnti(,l[.~ of Misner, Thorne and Wheeler (3) have, been adopted both for the. gravitati,mal and for the electromaguetic quantities except for the signature of ttm metric which i.~ § ~ § ~ ; Planck's units h = c = G = 1 are used throughout. The form of cq. (i) (b'rives from the tracelcsslmSS of T~v = (1/4~) [ F ~ F ~ - - �88 F~a]. (3) C~. ~V. MISNI';It, K. TI[ORNE ;tnd J'. A. I, VIIEELER: Gravitation (San Francisco, Cal., 1972). (4) B. BERTOTTI: Phys. Rec., U6, 1331 (1959). (5) K. YANk1: I)i]]ere~tiai Geometry on Comple~ a~ul Ab~ost Complex Spaces (London, 19(~5). (6) ,~. ~V. [ |AWKINt;: Phys. Lett., 60 A, 81 (1977). (7) T. EGUCIII a l ld P . G. (). FREUND: Phys. Rec. Left., 37, 1251 (1976). (s) T. I'~'GUCHI a n d A. J . HANSON: Phys. Left., 74 B, 249 (1978). (9) T. T. WE and C. N. YAN(~: Phys. Rec. D, 12, 3845 (1975).

92 R. PERCACCI

in which topological considerations come into play: in the definition of the charges (part II) and in the quantizat ion process itself. The lat ter point is discussed briefly in the conclusion.

1. - S o m e properties o f product mani fo lds .

Let us consider a four-dimensional manifold ~/g which is metrically (and topologically) the product of two two-dimensional manifolds ~//~ and - ~ (more details in (~) and references cited therein).

Due to the vanishing of the conformal tensor in two dimensions the metric can be written, without loss of generality, as

(3) ds2(j//) : ds2(jK~) 4- ds~(Jd~) :

: ~ ( x l , y~)(dx~ 4- dye) 4- g2~(x~, y~)(dx~ 4- dye),

where x~, y~ are co-ordinates on J/g~ and x~, Y2 on J1'2. curvature scalar are given by

(~)

where

The I~icci tensor and

R : R~ 4- R2,

are the curvature scalars of /a'l and Jg2. In t roducing the complex co-ordinates

za ~ xA + iyA , A = 1~ 2~

z ] = z~ = x~ - - i ya ,

the metric can be recast into the form

d s - " : 2 l~e gA~ dzA d z ~ : f2~(z')ldzl[ ~ + ~(z~)idz~[ 2"

I t is then immediate to check tha t the metric is K~hlerian with (( generating

function ~> ~(z~, z2) ---- cf~(z~) 4- cf~(z~), QA = (~/~zA)(~/~Z 7) q).4 (see e.g. chapter 3 of ref. (5) or the appendix of (~0)).

(10) E. GAVA, R. JENGO, C. 0MERe and R. PERCACCI: -~uel. Phys . , 151 B, 457 (1979).

EINSTEIN-MAXWELL IN,gTANTONS AND CtIARGE QUANTIZATION 9 3

One can easily check when Jt'~ and .~z have no boundary t ha t the Euler-

Poincar~ characterist ic of ~4/is given simply b y the p roduc t of the Euler charac- teristics of ~ and ~gr (one can use the s tandard integral formulae, or singular homology theory, see ref. (~) p.102)) and t ha t the Hirzebruch signature

vanishes:

(6) z ( ~ ) - - z(.Ar z(~fz) , T(~') = 0 .

As a consequence, the num ber of harmonic self-dual and an t i - - se l f -dua l 2-forms

are equal. Applying the Gauss-Bonnet theorem one finds also t ha t the gravi ta t ional

act ion with cosmological t e rm depends only on the areas of ~ and ~r and their topology (~)

(7) ~, - - - - i f 1 16~z (R -- 2 A ) v ~ d ' x : - - ] (Aaz(~r + A,Z(Jg,)) + A A t A , . 8~

The action for the electromagnet ic fieltI is

1 I F A * F . (~) S ..... 8~ J

I t is now widely recognized (see e.g. ref. (~3)) tha t the appropr ia te ma themat i ca l

f r amework for the t r e a t m e n t of (classical) gauge fields is tha t of fibre bundles. I n this approach the electromagnetic field F is to be in terpre ted as the curva- ture 2-form in a principal U1 bundle over J [ . These bundles are character ized b y the Chern classes c 1 ~ F, c~----0.

] t is interesting to notice t ha t the solutions we are going to find can be readily t rans la ted into solutions of a S Uz gauge field theory, if we pu t F~ pv

--~ ~,3F,~ (i ~ 1, 2, 3). These SU2 principal bundles have now a nonvanishing second ( 'hern class, whose integral

e'- f {9) ~ - : ~ o. F / ~ F

is a topological invariant .

(ll) M. J. GREENBERG: Lectures ok Algebraic Topology (New York, N.Y., 1967). (12) (~. W. GIBBONS: Phys. IJett., 6 I A , 3 (1977). (13) W. DRECHSLER and M. E. /~AYER: _Fiber Bundle Techniques in Gauge Ttwories (Berlin. 1977).

94 n. r~c~cc~

2. - Solut ions o f the E ins te in -Maxwel l equat ions .

I t is well known tha t in a K~hler manifold the Ki~hler form provides a na tura l candidate for the e lectromagnet ic field (~4). The K~hler form of +/[,

however, is a lways self-dual, so t ha t the electromagnet ic stress tensor vanishes

and one can find only v a c u u m solutions to the Einstein equations.

I n order to find non-self-duM solutions, let us t ake as F a r a d a y form a linear combinat ion with a rb i t ra ry coefficients C~ and C~ of the K~hler forms of J/f~ and J//2:

F ~ ~ 2iCA g:~.

:By re turning to real co-ordinates

The Maxwell form is then

(10b) *F z C2 D~ dx, A dy, -~: C~.Q~ dx2A dy2.

This yields thc two invariants

(11) /'~u~ Fu,. = 2(C] @ C~),

and the ene rgy -mom en t um tensor

(12) 8~T"b,, = 8~T%,,

where

*Fu~F~v = 4C 1 C2

=

is a constant . Notice tha t (~, and hence Tv~, vanish in the self-dual case C~ = C=.

Let us now define the (( charges )) associated to the electromagnetic field as the fluxes o f /7 and *F through ~ and J/~:

P1 .1/1 Jd,

] fF C2A2 1 f r CIA"

.4~12 r ll 2

(13)

(14) G. W. GIBBONS and C, N. POPE: Con~)?~,. ,]lath. Phys., 61, 239 (1978).

"E.INSTEIN-MAX'W:ELL INSTANTONS AND CItAR[;E QUANTIZATION 95

These ~ro not actual charges, since the electromagnetic field is source free, bu t can be viewed according to WHEELER (3a5) as <(lines of force t rapped in the topology of apace-t ime ~. F r o m the point of view of de R h a m ' s theorems, if ~s and ~r are compact and without boundary , then the P , ' s are just the periods of F over the two independent nonbounding 2-cycles .../t~ and ~//..

The number of P ' s and Q's one can define in this way is equal to the second

Bet t i numbe r of the manifold J [ , which might exceed two, us will be the case

in the following. Fur thermore , the exact meaning of P and Q depends on which of the co-

ordinates is chosen as (, t ime ,); for instance, if in the analyt ic cont inuat ion to a pseudo-l~iemannian metr ic it is J/~ t ha t becomes timelike, while d/.. remains spacelike, then Q~ and P~ are to be in terpreted as the electric and magnet ic charges, since they represent the fluxes of E and H through Jr'...

Note t ha t one needs ~g/, to be orientable in order to define the charges.

The Maxwell equations (2a) and (2b) are tr ivially satisfied by (10a) and

(10b). The Einstein equations (1) reduce now to

(14) I -R , - - A --:- o ,

.[ R : - - ..l ----- - - '2 ,

so t ha t jgr and J/2 have to be spaces of constant curvature . We can, therefore, choose ot[ to have any combinat ion of the following

metrics:

(15) F la t space with .Q, = a, and R , = 0 .

'2 (16) Two-sphere with .(2: -- 2a,* and R, = ~ ;

.:~ § ,r,*/- y~ ~t,

under the t ransformat ion (stereographie projection)

0~ x, -- a, ctg ~ cos q~,,

0. y , = a, e t .g~ sin ~, ,

2

the metr ic reduces to the more familiar form

d~ = a~(dO~ + sin 2 O, d ~ ) .

(17) The Bo]yai-Lobaehewski space of (.onstant negat ive curva ture :

2 a ~ o

(~4) C. ~V. M1s.~r and J. A. WHEELER: .ttltl.. P hy s . , 2, 525 (1957).

96 a. PERCACCI

which, under the t ransformation

Zi Xi x~ = a i ctgh ~- cos v i , y i = ai ctgh ~- sin ~

becomes

At this point Einstein 's equations can be used to fix up the values of a~

and a~, once the parameters C~, C~ and A are given. Alternatively, one could

prefer to fix the values of A and two of the (( charges ,~ of eq. (13), which, in

view of the following discussion, it is most natural to assume to be/)1 and / )2 .

I n order to cover also the noneompact cases, we can define a quant i ty

P i ~ - C i a ~ , which represents 4zt times the charge per unit co-ordinate surface

and is a finite constant. Clearly in the case of the sphere p~ = P d the same holds

true for fiat space if we compactify it in such a way tha t Ai = 4za~.

The results are summarized in table I.

Some remarks are in order at this point.

Firstly, since Einstein's equations do not fix the scales of the fiat com-

ponents, we shall always absorb them into the co-ordinates in such a way as to have a~ = 1.

:Note tha t globally fiat space admits at most a self-dual homogeneous electro-

magnetic fiehl, whose strength C1 = C~ is the only free parameter ; if you wonder

how a field of this kind might have a vanishing stress tensor, recall tha t in the

euclideanization the components of the electric field E~ ---- ~'o~ pick up a factor

i so tha t the energy density is actually Too-----(1/8~r)~ = (1/83r)(H 2 - E~). I n cases I I and I I I both the strength and the (~ complexion ~> (s,4) can be given, but then a solution is found only for a determined value of A.

These solutions reduce to I in the limit A -~ 0 or C1 --~ C~.

The cases IV and VI do not admit the limit A -+ 0, while V gives in this 2 2 - - 2 2 case a~ ~-- a 2 ~o 1 - - P 2 - Q2 which is just the Euclidean version of the l~o-

binson-Bertott i metric (4,~e,~7).

In the self-dual case C1 = C2, the spaces IV and VI have, respectively,

2 2 - - A -~ and thus reduce to well-known Einstein , 2 A ~ and a~ a 2 a 1 = a 2 ~ ~ - -

spaces. On the other hand, the metric of V degenerates in the self-dual limit:

this is just a pathology of the Euclidean space-time, since the l~obinson-Bertotti metric is perfectly well defined in this case.

( 1 6 ) B. CART]~R: in B l a c k Holes , edited by B. S, DE WITT and C. DE WITT (New York, N.Y., 1972). (17) I. ROmNSON: Bul l , A c a d . P o l on . Sc i . , 7, 531 (1959).

EINSTEIN-MAXVCELL IxS'rANTONS AND CHARGE QUANTIZATION

T A B L E I .

Sign Solution of (4) 8olution of (4) Range of for given for given of curvature A, C~, C.., A, px, Pz parameters

2 2 arbi t rary A = 0 I R 1 = 0 a l , a.~

R 2 = 0 C1 = P1 = = P~ = C.,

2 arbi t rary A = - - ~ > 0 I I R 1 = 0 al l

R, > 0 a ~ , = + - - 2A

I I I R1 = 0 aT arbi t rary - - A = p > 0 I

R. < 0 a~ -- 2A

1 t ] / I - -4AP1Zl A > I~1>o IV RI > 0 1 at~ = ~ t 1 + [ 1 - - 4 A P ~ J R z > 0 a ~ - - A + o ,

'{ 1 / i - 4 a e : / 1 a~ = ~ 1 + [ I - - 4 A P ~ J

1 t ] / 1 - - 4 A p ~ [ IA /< 1~1 V R I < 0 - -1 a ~ = ~ t - - l + y l _ 4 A p ~ ] R ~ > 0 a ~ - - a + 0 ,

' o.', = ' I - 1/ Te / a ~ - - A _ o . - ~ 1 1 [ 1 - - 4 A p ~ J

R 2 < 0 a ~ = A + ~'

., 1 , , ~ = ~ - - 1 - - a.; - A - e 4AP-~,]

N o t i c e t h a t al l t h e s e me t r i c s a re una f f ec t ed b y d u a l i t y r o t a t i o n s of t h e e lec t ro-

m a g n e t i c f ie ld :

I F - + F c o s h y ~ - * F s i n h 7 , (18)

l *E -~ * F cosh 7 q- F s inh y ,

o r

C, ---> C, eosh 7 q- C, s inh 7 , i C j , i , j = 1~ 2,

s ince so is 0-

7 - I l N u o r o C i m e n l o B .

98 R. P~RCACCI

3. - Quant izat ion o f the charges .

Let us now come to tile main problem and see how quantization comes

about. This results from a simple application of the Wu-Yang (9) formalism

to the present context.

Since we need the space to be compact, we shall discuss only the cases of

the sphere S 2 and of periodically identified fiat space with topology T ~. I n

the case of the sphere (~), one has the metric (16) and the field strength (for

convenience of notat ion we drop the indices i in this section)

(19) t z --_ - - Ca 2 sin 0 d0A d~ ~- P sin 0 d0/k d~ .

This derives from the potential

(20) A z - - P(cos 0 + k) d~,

where k is a gauge-fixing constant. There is no choice of k such tha t A is every-

where regular on the sphere (i.e. F is not globally exact), bu t we can cover the

sphere with two charts I and I I such tha t the potential is regular in each and

in the overlap region the potential A(1 ) is a gauge t ransform of A(~). The re- gions are

I) 0 > 0 , k---- - 7 1 ,

] I) O < u , k---- -- 1 ,

so tha t

A(u ) - - A(I ) + 2P d~ .

Now, if we couple the potential to a charged (( test ~) field with charge e, this has to transform as

~P(H) ---- ~P(I)exp [ie2Pq~] ,

so tha t periodicity in ~: ~(n)(~ + 2n) ---- W(n)(~0), requires

(21) 2 e P = integer.

Let us turn now to the fiat torus. Flat space is now divided into an infinity of cells whose points are to be identified:

(x, y) = (x + n~, y + raft), n , m E Z ,

EINSTEIN-MAX1,WEI,L INSTANTONS AND CIIARGE QUANTIZATION

and zr fl are the periods, sub jec t to the cons t ra in t xfl = 4~. The field s t r eng th

(22) F -- P dxA dy

derives f rom the potent ia l

(23) A(k ) ~ P (x -~ k) d y ,

where k has again the same role as before.

Le t us consider first the cell { ( x , y ) : O ~ x ~ o t , O<y<~fl} and fix k ~ (1: we

call A(o~ the cor responding poteut ia l . Clearly

N e x t define

A(o~(X -~ ~, y) = Ar y) -~ P z t d y .

A , ) ( x , y) z A(o~(x- :r y) -= Aco~(x, y ) - Pot dy

as the po ten t ia l hav ing in the (~ f i r s t , cell the same values t h a t A(o) has in the

(~ z e r o t h , one; this is a gauge t r ans fo rm of A(o~. Coupl ing again to a complex

scalar field, one has

~ ' , ) = ~r exp [ - - iePzty] .

:Now, if we assume t h a t v2(o~ is periodi%

~V(o~(X § ha, y § mfl) = v2(ol(x, y ) ,

and we require t h a t ~(_~) be also, we are led to

exp [ - - ieP~zfl] = 1 ,

an hence again

(34) 2eP = in teger .

The fac t t h a t this fo rmula is ident ical to (21) derives f rom our conven t ion t h a t

the area of T -~ be ~fl ~ 4z~; otherwise, one could absorb the fac to r ~fl/4~ into

a redefini t ion of e. The quan t i za t ion p rocedure we have ske tched here wou ld

work also if ~ t , were nonor ientable . For instance, one could c o m p a c t i f y the flat

2-space into a Klein bo t t l e or a pro jec t ive plane by su i tab ly iden t i fy ing points (is)

and repea t the above a r g u m e n t s step by step. The p rob lem is, however , t h a t P

could no t be in te rpre ted as a charge since in tegra t ion is no t defined on a non-

or ientable manifold .

(18) B. S. DE WITT, (!. F. HART and C. J. I~HAM: Topology and quantum field theory, University of Texas preprint (1978).

100 m PERCACCI

4 . - P a r t i c u l a r c a s e s .

Gather ing the results obta ined up to this point , we can now discuss some par t icular cuses. Before doing so, let me ment ion here a difference between the present cases and the work of Duff and Madore (~).

In (1) one had a source-free field due to the fact t ha t ~ R iemann ian i za t i on , of the Reissner-Nords t rom metr ic makes the region r ~ r+ disappear. However , continuing back to the Lorentz ian section the Eucl idean results, one has to conclude t h a t it is the s t rength of the pointl ike source in r = 0 t ha t mus t be quantized.

On the contrary , in the spaces considered in this paper , the field remains source free also af ter analyt ic cont inuat ion is per formed; therefore, it would

be perhaps more appropr ia te to talk of flux quantization t han of charge quan- tization.

For S-" • S-" the metr ic is

dS"- : a~(dO~ -~ sin s 0~ d ~ ) ~- a~(dO~ ~- sin s 0~ d~])

with al and a, given by table I , ease IV. In the self-dual case this is the Eucl idean Schwarzschild-de Sit ter metr ic in the l imit in which the mass of the black hole is M = (gA) - t (~4). The electromagnet ic field is given b y

(25) F ~ P1 sin 01 d0~A d~l ~ P~ sin 02 d02A d~2,

*F ~ Q~ sin 0~ d01A d~l ~- Q2 sin 02 d02A d~...

The quant iza t ion rules are

2PI)~. = Ot,~. ,

2eQl 2cP2 a~ a~ = n,

a~ 2cQ. 2eP1 0l'---~ = ~1

V e 2 - An~ ~ - - An~ '

~ e 2 -- iln~

-- An[ "

I f 01 and ~1 are in terpre ted as ~ radius ,> and (~ t ime ~>, then P~ and Qs are the re levant charges.

Note how the periods of F on the two spheres are quant ized in a simple way, while those of *F are quant ized in a nonlinear way.

I n the self-dual ease

P1 : QI : P2 : Q2 : - . 2e

E I N S T E I N - M A X ' W E I . L I N S ' I ' A N T O N S A N D CIL~ ,RGE Q U A N T I Z A ' ] ' I O N 101

This s o l u t i o n e~t,1 b e ('onsidel't~d as all E i n s t e i n - M a x w e l l i n s t a n t o n (~), whose

r e l e w n t p a r a m e t e r s a re l i s t ed in t a b l e 1I . I t is n o t e w o , ' t h y t h a t for a g i v e n

v a l u e of . l , V (,~n t a k e o n l y dis( ' re te va lues .

TABLE 1I.

Volume I r N, Se.ra. :' Z b2

= [ a2 8e~" P1P . , = 4 2 S z / ~h '-~ l 6 ~ z a z a~, A V < 0 ~ / n l a~ -}-

4 = ~ J 1 / e ~ - c A, ,~ 8= 2e" = 2 , q ~.,

t o

2 + [ e . , _ a u 2 + maxi , ,mm of q._ n ~ /

1/0,- 1 o,, 4 [ e " - . ' l n ~ J , t mi~,imum of

16:rt" whrn ( '1 = ( ' , l ] , inimum of - -- n,"

~12 e 2

when ~'~ = (', (0 - - 0) wht,ll ( ' ('.,

- - . -- .n~ ontu 2 0 2 16~ ~ a z = I ft.-d- tt ~ 2e ~ u~ a 'z +

T z x T" 16:'(' 11 :'t :'t 2.~(-' 0 6

2e z e"

F o r ;";-'/ f/'*

(2~)

dN-* = (l.r-' + dy-" q a~(d0 2 + s in ~ 0 d~0'-'),

F P~ d x / , dy + P~ sill 0 d0A {l~,

* F : Qx d.r,s dy -}- Q. s in 0 d0 / \ d ~ .

T h e quan t i z~ t t i on ru les ~tre

o

2 e P x ~1 , 2 e Q l ~t, 2 e " - - ~l~ '

2eP. , = n.,. , 2eQz = nl n~

I l l t h i s e~tse, as well as in t h e p r e v i o u s one , t h e r e are, t w o i n d e p e n d e n t h a r m o n i c

2 - fo rms , wh ich c a n be t a k e n to be F + * F (se l f -dual ) a n d F - - * F ( a n t i - s e l f -

dtml) .

(19) A. TRAUTMAN: list. Journ, . Theor. Phys . , 16, 561 (1977).

102 ~. PERCACCI

For T ') • T 2

dS'-' ~-- dx2, -~ dy] ~- dx~ + dye.

This manifohl has b2 ~ 6, hence there are six independent nonbounding 2-cycles (the 2-tori corresponding to the six planes x , y , , xtx~ etc.) and six independ- ent harmonic 2-forms: three of them self-dual and three anti-self-dual (this corresponds to the freedom in choosing the direction of the electric and mag- netic fiehls, which are parallel). These forms are

(27)

1,'~• ~ P~(dxlA dyl • dx2A dy.) ,

F ~ ~ P2(dx~A dx~ • dy2A dyl) ,

F3~ ~ P~(dx~A dy~ • dy~A dx~) (-~ : self-dual, - - anti-self-du~l) ;

all are solutions of Maxwell's equations with vanishing energy-momentum tensor; in part icular F~+ is the one tha t results f rom eqs. (10a), (15). Each of them has vanishing periods on all the nonbounding 2-cycles except two; taking linear combinations in such a way as to preserve (anti)self-duality yields other solntions of eqs. (1), (2) having nonzero periods on (possibly) all the cycles, bu t these periods always have to be equal in pairs.

In any c~se the quant izat ion rule is simply (24).

For the Robinson-Bertott i metric (R2• S ~-)

dS2 = (P:~- -- Q~) {dz 2 ~- sinh2 Z dv~ -~ d0~ + sin2 0 (t~'-'},

F --~ Q~ sinh Z drA d Z -~ P~ sin 0 d0A d 7 ,

*F = P2 sinh Z (lvA d Z -~ Q2 sin 0 d0A d~.

The volume, and hence the action~ are infinite so this cannot be considered an instunton. This manifold has the second Bet t i number equal to one, thus only one charge can be quantized; ill view of the fact tha t T represents time, this is necessarily P~. This is not surprising, since this space is conformal to Minkowski space with a dyon's Coulomb field (2o).

5 . - C o n c l u d i n g r e m a r k s .

From a mathemat ica l point of view, the quantizat ion of the charges cor- responds to tile fact tha t U1 principal bundles over J l are classified by the

(~o) A. ~. ],APE1)ES: .Phys. Rec. D, 17, 2556 (1978).

EINSTEIN-MAXVCELL INSTANTONS AND CHARGE QUANTIZATION 103

e l e m e n t s of t he second in t ege r c o h o m o l o g y g r o u p of Jr ' , H2(J//, Z) (21). I n t h e

cases of S -~ • S 2 a n d T ~ • T '2, H*(J//, Z) = Z �9 Z, t h u s t h e c o r r e s p o n d i n g b u n d l e s

a r e c lass i f ied b y two in tege r s (n~ a n d n~). A n a l o g o u s l y in t h e case of t h e l~o-

b i n s o n - B e r t o t t i s p a c e - t i m e H * ( R 2 x S ~, Z ) = H~(S ~, Z ) = Z, t h u s one in t ege r

( the m a g n e t i c charge) classifies a l l i n e q u i v a l e n t b u n d l e s ; f ina l ly H*(T ~ • T ~, Z) :

: 0 Z, t h u s one needs six i n t ege r s : we see t h a t (27) g ives on ly a subc lass of 1

all p o s s i b l e bund le s , s ince a f o r m wi th a r b i t r a r i l y a s s igned pe r i ods w o u l d in

gene ra l n o t be a so lu t i on of t h e E i n s t e i n - M a x w e l l equa t ions .

T h e ~( u n i v e r s a l i t y ~> of t h e r e l a t i o n b e t w e e n t h e i n v a r i a n t of eq. (9) a n d t h e

cha rges (sec t a b l e I I ) has been e x p l a i n e d on gene ra l g r o u n d s in ref. (~).

The a u t h o r wishes to t h a n k Prof . G. DENAnDO a n d 1~. IE.~GO for d iscuss ion

a n d adv ice .

Note added itt proo/s.

Suppose the (~ test field ,> W (sect. 3) is a spiuor field. One might then ask whether inequivalent-spin structures can lcad to different quantization rules. I t is easily seen tha t the answer is negative. Fi rs t notice that the problem is peculiar to multiconnected manifolds, so it does not even pose itself for S 2 • 82; in the other two cases one can has 4(S 2 x 7 '2) arid 16(TZ y, T z) inequivalent-spiu structures. The 4-spinor structures associated with a T 2 factor in dr' correspond to spinor fields tha t are periodic in some directions and antiperiodic in other 0s). Enforcing these (anti)periodicity conditions on the gauge transformed fields again to eq. (24).

In order to obtain something effect, we must give a magnetic charge g to the field. In this case (21) and (24) generalize to 2eP+ 2gQ integer.

(21) S. J. Avis and C. J. ISHAM: Quantum /ield theory and /ibre bundles in a general spacetime, Imperial College preprint (1978).

�9 R I A S S I ~ N T O

Si de,<crive una elasse di soluzioni delle equazioni di Einstein-Maxwell in assenza di sorg(,~ti, elm soddisfa due date eondizioni; queste soluzioni rappresentano prodot t i di due spazi bidinw, nsionali a curvatura eostante eontenenti un campo elet tromagnetieo uni- forme. Aleune di queste soluzioni sono eompatte e possono essere considerate is tantoni gravitazionali elettromagnetiei . A eausa della topologia non banale di questi spazi quadridinmnsionati , i flussi dei eampi elettrico e magnetieo at traverso opportune superfiei (ehe ehianfiamo ~, cariehe ,> nel senso di Wheeler) risultano quantizzati indipendentemente.

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