Volume 85B, number 4 PHYSICS LETTERS 27 August 1979

GREEN'S FUNCTIONS FOR GRAVITATIONAL MULTI .1NSTANTONS

Don N. PAGE Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, England

Received 7 May 1979

The Green's functions for scalar fields propagating on the self-dual gravitational multi-instantons and multi-Taub-NUT metrics are given explicitly in closed form. The special cases for flat space, Taub-NUT and the Eguehi-Hanson instanton are listed. A construction is described for obtaining the Green's functions for fields of arbitrary spin.

In the path-integral approach to quantum gravity [1], one would expect a central role to be played by gravitational instantons, which are complete nonsin- gular positive-definite metrics that are solutions of the classical Einstein equations. Two important families of gravitational instantons have been discovered which have self-dual curvature that falls off at large distances. One family is the multi-Taub-NUT metrics [2], which are asymptotically flat in three, dimensions but are periodic (with an asymptotically constant period) in the fourth dimension. The other family is the gravita- tional multi-instantons [3,4]~ which are asymptotically locally euclidean (ALE), approaching the flat metric on R 4 identified under a discrete subgroup of SO(4). The first family may contribute to the partition func- tion of the thermal canonical ensemble for the gravita- tional field [5], and the second family may contribute to various transition amplitudes or S-matrix elements [6,7].

In order to evaluate the effects of quantum fluctua- tions about these classical field configurations, one would like the Green's functions for various fields propagating on the instanton backgrounds. Among other things, the Green's functions can be used to cal- culate the contributions of closed loops to the parti- tion function and to the effective stress-energy tensor, and they can be used to calculate S-matrix elements be- tween initial and final states containing various par- ticles [6,7]. In this paper the Green's functions are ex- plicitly given in closed form for scalar fields on these

two families of instantons, and it is shown how to con- struct the Green's functions for minimally coupled fields of arbitrary spin.

The metrics for the two families of instantons have the self-dual stationary form [2--4,8]

ds 2 = v - l ( d r + to. dx) 2 + Vdx. dx , (1)

where

v x to = vv, (2)

in the flat three-space metric dx. dx. For a complete nonsingular metric, V is generated by s mass points at

spatial position x n ,

S

V = V 0 + ~ rn I , (3) n=l

where r n = I x - x n I is the fiat three-space distance to the mass point and V 0 is a constant that is nonzero for the multi-Taub-NUT instantons but zero for the ALE multi-instantons. The time coordinate r is given a p e r iod 4~r to remove singularities along the Dirac strings of the vector gauge potential w, which also removes the apparent singularities at the mass points.

I shall choose a particular gauge in which

s

w" dx = ~ cos O n d~Pn , (4) n = l

where ( r n , O n , ~ n ) are spherical polar coordinates cen-

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Volume 85B, number 4 PHYSICS LETTERS 27 August 1979

tered on the mass point at x n. One can easily see that s = 1, V 0 = 0 gives flat four-

space in hyperspherical polar coordinates, or

ds 2 = dX 2 + dY 2 + dZ 2 + dW 2, (5)

if four-dimensional rectangular coordinates are defined by

X + iY = 2r 1/2 cos ½0 e ± (i/2)(r+~o) (6)

Z + iW= 2r 1/2 sin½0 e ±(i /2)(r-¢) . (7)

Also, s = 0, V 0 4 :0 gives flat four-space periodically identified. Thus the instanton metrics are formed by combining flat four-space metrics in a particular way, and it turns out that the Green's functions can also be formed by combining fiat four-space parts in a par- ticular way.

The scalar Green's function G (x, x ') is the solution of

tZ a(x ,x ' )= -8(x ,x ' ) , []'C(x,x') = -~(x,x'), (8)

that dies off at large distances in the positive-def'mite metric. In terms of the 3 + 1 split of the coordinates x = (x, r) the first equation becomes

V -1 [V2a 2 + ( V - w ~ ) r ) ' ( V - It)~r) ] G ( x , r ; x ' , r ' )

= - V-163 (x - x ' ) 6 ( r - - r ' ) , (9)

and the second equation simply has x replaced by x ' in the laplacian.

If one complexities the coordinates, one would ex- pect G(x, x ') to be singular when x and x ' are null separated, i.e. when

S + ( x , x ' ) = O or S ( x , x ' ) = O , (10)

where S÷ and S are functions o f x which vanish, re- spectively, on the past and future null cones o f x '. Such functions can be found from the Penrose non- linear graviton technique of ref. [4]. However, they may more simply be obtained as solutions of the Hamil ton-Jacobi equation

(VS_+) 2 = V(a~.S+) 2

+ V - I ( v s _ . - tO~)rS±)'(VS ± - W~)rS+)= 0, (11)

since the normals VS+_ of the surfaces (10) are the null generators of these null cones. The appropriate solu- tions which are unchanged when r is increased by its

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period of 47r can be written as

x

S±(X,X')=I--exP2 f [(dr" +co . d x " ) x

-+ iV(dx" "dx") 1/2] , 0 2 )

where the path of integration is chosen to give an ex- tremum. Because of (2)~ it can be shown that the inte- gration over x " may be deformed to a straight fine with- out changing the value of the integral. Then one can easily evaluate the Hami l ton-Jacobi functions as

S , ( x , x ' ) = 1 - e x p i [ T ( x , x ' ) + i V ( x , x ' ) ] , (13)

T ( x , x t) = 1 (14) ~ ( r - r ' )

[COS~(0 n +On) 1 + t an-1 - - i - - - -2-77,, t a n ~(~0 n - tpn) ,

n=l Lcos ~(o n - On)

$ t 1 2 p 1 {r_ n +r__ n, +&-.~ (15) U ( x , x ' ) = ~ Vo A + lnkr n + r n - A ] '

A = ! x - x ' l = (r 2 + rn 2 - 2rnr n [cos O n cos O n

+ sin O n sin O n cos(~0 n - ~On) ] }1/2 . (16)

One may directly verify that this is a solution of (11) , since VS_+ is proportional to a sum of coefficients times one null vector plus another sum of coefficients times an orthogonal null vector. The surfaces S_+ = 0 include the point x = x ' and have normals which are a full set of generators of the null cones, so S± are the desired functions which vanish on the null cones.

Now the ansatz is that the Green's function for fixed spatial coordinates x and x ' only has simple poles in S± at S+ = 0. The spatial dependence can be deter- mined by integrating the Green's function over a time cycle, which by (9) should give the fiat three-space Green's function 4¢r~ (x, x '), where

qb = (161r2A) -1 . (17)

By choosing the relative contribution of the two poles to give the appropriate boundary conditions, one finds

- - I - (167r2A) - l s i nh U (18) a ( x ' x ' ) = ' b ( s j l - ~ - ) - c-o~ U---~6s ~ "

To verify that this is the scalar Green's funct ion , one can first easily check that it is (i) symmetric in x and

Volume 85B, number 4 PHYSICS LETTERS 27 August 1979

x ' , and on the euclidean section (i.e. for real coordi- nate values where the metric is positive definite), G(x, x ') is (ii) real, (iii) asymptotically zero, and (iv) singular only when x = x ' . Near the singularity it has the correct form

G(x, X ') ~ (4rr2) -1

X (V A2 + V -1 [(7- - 7-') + o ' ( x - x ' ) ] 2 ) -1

"" (47r 2 Ix - x ' ! 2) -1 , (19)

where Ix - - x ' l is the distance between the two near- by points in the full four-space metric (1).

All that remains is to check the laplactan for x :~ x ', which is

E]G = 2(S+ 1 - S ~ 1 ) I--]d~

+ 2¢ [S+ 3 (VS+) 2 -S_-3(VS_) 2 ]

-- S+ 2 CI(dPS+) + S_-2[-q(¢S_). (20)

The first term vanishes because of (17) for A =~ 0 and because S+ = S_ for A = 0. The second term vanishes by (11). The last two terms can be shown to vanish by using (2) in the straight-line integral (12) or alterna- tively by using (11) and (17) to reduce them to expres- sions proportional to rT(q~T) and IT(~U) and then differentiating the explicit forms (14)-(16) . Indeed, one finds that any expression of the form • [f+(S+) - f _ ( S _ ) ] , where f± are arbitrary functions of S±, has zero laplacian (except at A = 0 unless/'+ = f _ for S+ = S_). The choice that gives the correct boundary condi- tions (previous paragraph) is (18), which is thus the Green's function for scalar fields on the self-dual multi- Taub-NUT and ALE multi-instantons.

Now let us see what the Green's function becomes in some simple examples of these instantons. In flat four-space (s = 1, V 0 = 0) with rectangular coordinates defined by (6) and (7), one gets the correct expression

G = (47r2) -1 [(X - X ' ) 2 + ( r - - y , )2

+ (Z -- Z ' ) 2 + (I4/ - 14/') 2] -1 . (21)

In flat four-space identified with period 47rV~ -1/2 (s = 0), (18) becomes

(16rr2A)-lsinh ½ V0A G = (22)

t V o a cos l ( r - r ') cosh ~ -

= ~ (4zr2) -1 [ V~ -I (7-- 7" + 4/rk) 2 + VoA2 ] -1

Theself-dual Taub-NUT metric

ds 2 = [4M2R/(R + 2M)] (d$ + cos 0 d~p) 2

+ (1 + 2M/R) (dR 2 +R2d02 +R2sin20d~02), (23)

is obtained from (1) with one mass point (s = 1) by set- ting

V 0=(2M) -2 , 7 = f t . r 1 =2MR,

Then if one further defines

' V 0 A = ( 4 M ) - I (R 2 + R ,2 y = $

01 =0, ~o 1 =~0. (24)

(25)

- 2RR'[cos 0 cos 0 '+ sin 0 sin 0' cos(~0 - ~0')] }1/2,

1 1 P 1 ~/I p = cos ~ 0 cos ~0 cos~(~ + ~o . . . . ~0')

1 t 1 ~ r + s i n { 0 s i n ~ 0 cos~(~0 -~o- + 9 ' ) , (26)

the Green's function is

G - (1281r2M2y)-I [4My coshy + (R + R ' )s inhy]

(R + R ' ) coshy + 4My sinhy - 2(RR') 1/2/d

(27)

The Eguchi -Hanson metric [9]

ds 2 = (1 - a 4 / r 4 ) -1 dr 2 + ¼r 2 [(d0 2 + sin20 d~02)

+ (1 - aa/r 4) (d~ + cos 0 d~0) 2 ] , (28)

is obtained from (1) with two mass points (s = 2) at separation ¼ a 2 by setting

le 0 = 0 , r =2~o, r 1 = ~ ( r 2 + a 2 c o s 0 ) ,

r 2 = ~(r 2 . - a 2cos 0),

~01 =tP 2 = ~, rl cos 01 = ~(r2cos 0 +a2) ,

r 2 cos 02 = ~ (r 2 cos 0 -- a2 ) . (29)

Then the Green's function becomes

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Volume 85B, number 4 PHYSICS LETTERS 27 August 1979

G = (21r2)-1 (r 2 + r '2)

× [(r 2 + r,2)2 _ (r2r,2 + a4)× _ p20,2~2 ] -1 ,(30)

where

p 2 = ( r 4 - a 4 ) l / 2 , p ' 2 = ( r ' 4 - a 4 ) 1/2 , (31)

× = 1 + cos 0 cos 0' + sin 0 sin 0' cos(~o - ~0'), (32)

~2 = [(1 + cos 0 cos 0') co s (~ - ~ ' )

+ sin 0 sin 0'] cos($ - ~k ') (33)

+ [ - ( c o s 0 +cos 0') sin@ - ~0')] sin(~ - ~k ') .

It can be checked that this expression has the same U(2) symmetry as the metric and that it reduces to the Green's function for fiat four-space with opposite points identified when a ~ 0.

Since all of these instantons have covariantly con- stant spinors which generate global supersymmetry transformations between eigenfunctions for different spins [6], one may generate the Green's functions for different spins from the scalar Green's function. For a Green's function of the (j, k) irreducible representa- tion of the rotation group, one simply multiplies the scalar Green's function at both x and x ' by 2j + 2k covariantly constant spinors to get the right number of two-component spinor indices and then contracts on 2] of these indices at both points with covariant derivative operators to convert the right number of

dotted spinor indices to undotted indices. The result will be the Green's function or inverse of the appro- priate second-order differential operator (e.g. that given in ref. [10] ) with the zero modes projected out. The zero modes (which occur for j = 1, higher values of ] being forbidden by consistency requirements [10] ) can also be obtained explicitly from the known (1,0) zero modes by multiplying by the covariantly constant spinors [6].

I wish to thank S.W. Hawking for spirited encour- agement without which this work would not have been accomplished, and G.W. Gibbons, C.N. Pope and M. Ro~ek for help in simplifying and checking the results.

References

[1] S.W. Hawking, Phys. Rev. D18 (1978) 1747. [2] S.W. Hawking, Phys. Lett. 60A (1977) 81. [3] G.W. Gibbons and S.W. Hawking, Phys. Lett. 78B (1978)

430. [4] N.J. Hitchin, Math. Proc. Camb. Phil. Soc., to be pub-

I/shed. [5] G.W. Gibbons and S.W. Hawkings, Phys. Rev. D15 (197.7)

2752. [6] S.W. Hawking and C.N. Pope, Nuel. Phys. B146 (1978)

381. [7] M.J. Perry, TP-inversion in quantum gravity, Princeton

preprint. [8] K.P. Tod and R.S. Ward, Self-dual metrics with self-dual

Killing vectors, Oxford preprint. [9] T. Eguchi and A.J. Hanson, Phys. Lett. 74B (1978) 249.

[10] S.M. Christensen and M.J. Duff, New gravitational index theorems and super theorems, Harvard preprint.

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