Wormholes and effective actions in two dimensions

Nuclear Physics B333 (1990) 279-295 North-Holland

WORMHOLES

A N D E F F E C T I V E A C T I O N S IN T W O D I M E N S I O N S Alex LYONS

Department of Applied Mathematics and Theoretical Physics, Universi(v of Cambridge, SilL,er Street, Cambridge CB3 9EW, UK

Received 1 September 1989

We discuss the replacement of wormholes by effective interactions in a simple two-dimensional scalar model. In the region of moduli space where the wormholes are small this leads to a bilocal effective addition to the action on the complex plane. The connection with string theory is described.

1. Introduction Bosonic string theory diverges at the one-loop level, i.e. when the world sheet is a torus. This divergence is related to the existence of a massless particle, the dilaton, in the spectrum. This can be seen in the following way. Consider the one-loop scattering amplitude for a number of particles. If the vertex operators for the particles are close together, the world sheet can be conformally deformed so that it looks like a sphere joined by a thin tube to a torus. Fig. 1 shows this world sheet e m b e d d e d in a flat space-time background. A dilaton can propagate down the tube and disappear into the vacuum (via the torus). The point is that the one-loop dilaton expectation value (or one-point function) does not vanish, even though at tree level it does. This dilaton has vacuum q u a n t u m numbers and carries zero m o m e n t u m , so the divergence comes from the propagator, 1 / p 2, as p ~ 0. The divergence contains three factors (shown in fig. 2): (1) The tree amplitude for the original process plus a dilaton at zero m o m e n t u m , (2) The p r o p a g a t o r for the dilaton at zero m o m e n t u m , (which diverges), (3) The one-loop dilaton expectation value*. This way of looking at things makes it look like an infrared divergence, occurring as one of the m o m e n t a goes to zero. This interpretation of the divergence has been k n o w n for some time [1], and it was shown by Fischler and Susskind [2] that it could be cancelled b y a world-sheet sigma model divergence, provided the background metric satisfies the Einstein equations with a cosmological constant. Thus the * This third factor is proportional to the one-loop cosmological constant, or vacuum energy, as computed by Polchinski [3] and Rohm [4]. 0550-3213/90/$03.50.t)Elsevier Science Publishers B.V. (North-Holland)

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@

Fig. 1. The dilaton propagator divergence.

non-trivial topology of the string world sheet seems to modify the flat space-time background, and makes it appear to be curved. In the sigma model approach the background fields, such as the space-time metric, are interpreted as coupling constants in a two-dimensional world-sheet field theory, and these coupling constants are modified by string loops. This whole process is reminiscent of the wormholes of Hawking [5] and Coleman [6], which give rise to effective interactions in space-time, except the wormhole is now on a world sheet. The baby universes have zero energy and momentum since they satisfy the Wheeler-DeWitt equation and momentum constraints [5]. They are also singlets of any local symmetries of the theory. They are thus analogous to on-shell dilaton and graviton states in string theory. Baby universes shift effective space-time coupling constants [6], while dilatons and gravitons are responsible for a similar shift in the background metric g,~(X), to be thought of as an infinite set of coupling constants in a non-linear sigma model [2]. The idea is to see if this resemblance is more than just token. We shall calculate the Green function for a free scalar theory on the complex plane with two discs removed and their boundaries identified. This space is topologically a torus with a puncture. The reason for working in this space is to attempt to find the wormhole effective interactions on a world sheet which is closer in spirit to the asymptotically flat space-time used by Hawking [5] for discussing space-time wormholes. In the limit where the circles are extremely small and close together, it should be possible to replace them by a single interaction at one point. If they become small and far apart, then we get the situation most analogous to space-time wormholes, and it should be possible to replace them by a sum of bilocal operators, at each end. This is the form usually

external lines toru s

dilaton

Fig. 2. Factorizationof the divergence.

0

A. l~vons/ Wormholes

281

assumed in studies of wormholes and baby universes [5-7], and it would support this assumption to derive it explicitly in a simple model.

2. Fischler-Susskind effect in string theory In order to calculate scattering amplitudes in bosonic string theory, one needs to know the scalar Green function for free bosonic fields on the world sheet. For the one-loop contribution the world sheet is a torus, T. One can treat the problem using conformal field theory on the torus [8]. The string coordinates are then thought of as quantum fields living on T. It is convenient to use the representation of T as C / L , where L is the (lattice) group generated by translations by the complex numbers 1 and ~', and I m T > 0. T is the complex modular parameter distinguishing conformal structures on T. We also introduce coordinates v on the fundamental parallelogram with vertices at the points 0, 1, ~ and 1 + ~-. More information about the conformal structure of T is provided in the appendix and ref. [9]. The Green function satisfies the equation

1

~A

G(v, v') = 8 2 ( r - v') - l / A ,

where A is the area of T. The 1/A term is necessary since the torus is compact and so the integral of Aft(u, ~') over T vanishes [8]. The boundary term vanishes upon integration by parts, since there is no boundary. However, the integral of 8 2 ( ~ - v') equals 1, so a balancing term is needed on the right-hand side. One can also see this in the following way. There is a zero mode of the laplacian on T, namely any constant. Thus the laplacian is not invertible over the space of all functions on T, but only over those which are orthogonal to the zero mode. One must therefore subtract the constant zero mode of the laplacian on T. This is the 1/A term. Other balancing terms could be used, provided that they integrate to 1, but this is the standard choice since it is the simplest and preserves world-sheet translation invariance. On-shell scattering amplitudes are independent of the choice of balancing term, and we shall later exploit this fact to use a different choice. The unique symmetric Green function is then found by using the transformation properties of Jacobi 0 functions under v ---, ~, + 1 and ~, --* v + ~-. These functions are analytic and are pseudo-periodic under these translations. This means that they are doubly periodic up to multiplication by some complex function. Explicitly, the periodicity relations for 01 (henceforth referred to simply as 0) are 0(v + 1,~-) = - 0 ( ~ , ~'),

0(v +-r, r ) = -exp[-iTrr-2iTrv]O(v,r).

0 also has a first-order zero at the points v = m + n ~', for integer values of m and n, and no other zeros on the p-plane. A prime will denote a derivative with respect to

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l,, i.e. 0'(~) - 0'(p, ~-) = 8,O(u, "r). Note that we shall also often leave out the second argument, ~-, in the 0 function when it is obvious from the context. Further properties, including explicit formulae, are collected in appendix A. The Green function can now be written down in terms of 0 functions as

G(u,u')

= ln(x(u-

u')},

where ( I m v ) ] O(u) X(e) = 2~rexp ~ (Imp-) j] o,(o) The normalization is such that X - 27flY] for small ]v]. The area of T is A = Imp-. This Green function can now be used to calculate a one-loop string amplitude. The idea is to isolate the divergences discussed briefly in the introduction and interpret them as certain divergent propagators multiplying vertex operator insertions. In order to see the divergence appearing, we expand it for small v since this corresponds to the region of integration where the vertex operators are all close together. For small v we have X - 27r]v](1 + ¢r(Im v)Z/(Im 7)), after averaging over the phase of v. This is inserted into the expression for string amplitudes (e.g. for M-tachyon scattering):

A = fFd2ZC('r)(Im~)-2F('r). We shall now explain what these various symbols mean. The function C(~-) is defined by C ( T ) = ( l I m T ) - 1 2 e4~vim~r[f(e2Cri,r) [ 48

where f(e2"i~) = f i (1 - e2~in~) = {(0'(0, ~-))/2~re=i'/4) 1/3. n~ 1 The one-loop cosmological constant or vacuum energy [3, 4], is given by A = ~ d 2 ~ C ( z ) ( I m ~ - ) -2 This factor C(~) would be present in any one-loop (or torus) amplitude. It can be thought of as the measure in the Polyakov path integral over metrics on the torus T. The ~--integral is over a fundamental region F of the modular group (see appendix A). This is what is left of the path integral over metrics, once the Weyl rescalings and diffeomorphisms have been factored out.

A. Lyons / Wormholes

283

The important part of the scattering amplitude A is the factor F ( r ) given by

F('r)=4(,c~)MlmrfFI2d2vrQX~;
where X rs = X (Vrs) and v~s = vr - vs. Here the r th external tachyon has m o m e n t u m kr, and • is the coupling constant associated with the string vertex. F ( r ) is independent of v1, which can be put back in as an integration variable by writing the factor I m z = f d 2 v l . The v variables are all integrated over the fundamental parallelogram. The factor F ( r ) comes from the path integral over string coordinates X" (the matter fields on the torus). The path integral can be done exactly, as X~ are free fields on T. The integrand then comes from contractions of exp[ik~. X(Vr)], the vertex operators for the external tachyons, integrated over their positions v,. To make the appearance of the divergence a bit clearer let us rescale variables, following refs. [9,10]. Define new variables ,/,, e, q~ (e and q, to be taken real and r = 3 . . . . . M ) by C"1~2= ¢ el4' = v2 -- ?]1 •

£'l~r = Pr -- Pl ,

The jacobian for this transformation of variables is M H d2/-'r = i¢2M r=2

M

3 d~

dO I--[ d2~, "" r=3

The integrand of F ( r ) is now expanded in a power series in c. There are two divergences in the integral over E as t + 0. The leading divergence in the amplitude is proportional to [9] d2r f07Td0

1-I d2"qr r=3

H ] n r - - ~ls] k ' ' * ' / 2 [ C('r) l<~r
defining ~1 = 0 and using k 2 = 8, the mass-shell condition for the tachyon. (The units of ref. [9] have been chosen, so that the open string slope is a ' = *2" ) This d e / e 3 divergence can be thought of as a divergent tachyon propagator. This interpretation comes from the parametric form of the propagator in the operator formalism. The string propagator can be represented as 1 H

fo~dte_m

where H = L 0 + L o - 2 is the hamiltonian and t is the proper time. Changing variable to e = e - ' this becomes

/o'- exp [(Lo+

:)Ind.

A. I~vons / Wormholes

284

One can then insert a complete set of string states Eilq~i)(~b,I between the propagator 1/H and the vacuum 10) in the operator form of the scattering amplitude. The states all have k = 0 and so only the massless states are on shell. Also, only states with vacuum quantum numbers (i.e. scalars) will contribute. The states of lowest m 2 dominate the integral over the propagator as c --, 0. The tachyon state has L 0 = ['0 = 0, so the divergence is of the form fodE/E 3. The dilaton has L 0 = / ' 0 = 1 so the divergence is then of the form fodE/c. Out of the massless states the dilaton is the only scalar, and the graviton and antisymmetric tensor cannot couple to the vacuum. It can also be seen from this that the massive states in the spectrum (i.e. states with positive H eigenvalue) will not give any divergences, as they have larger L 0 eigenvalues and so the integral over c will converge. The factors multiplying the divergent propagator 1/c 3 have the following interpretation. The q-integration is an ( M + 1)-point tachyonic tree amplitude (i.e. on the sphere), where the ( M + 1)th tachyon has zero momentum and the rest of them are on shell. This integral is proportional (with a divergent constant of proportionality) to the M-tachyon amplitude, and may be treated [11] as contributing to the renormalization of the slope cd and coupling constant •. We shall not worry unduly about these issues here. The r-integration may be thought of as a zero-momentum tachyon coupling to the vacuum via the torus. The integrand differs from the one-loop cosmological constant ( v a c u u m - v a c u u m torus amplitude) by a factor of I m ~'. This integral diverges because of the tachyon, as I m ' r ~ oe. This shows us that the vacuum is made unstable by the tachyons. We shall not treat the troublesome tachyon exchange term any further. The next term in the expansion of the integrand of F ( r ) in powers of e factorizes in a similar way. There is a divergent factor lode~E, corresponding to the propagation of a massless dilaton, rather than a tachyon, down the long tube of fig. 1. The ~--integral multiplying this divergence is

f F d 2,r C ( ' r ) ( i m ,r)2 "

This is proportional to the coupling of a zero-momentum dilaton to the vacuum, via a world sheet with the topology of a torus. This is also proportional to A, the one-loop cosmological constant*. The other factor is the Tt-integral which is

',fdO

Hd2~r r=3

)

]7 I~r--~s'] kr'k~/2 ~ I~r--]Jsl2kr'ks , l<~r
using translational and rotational invariance to turn [ I m ( ~ r - ~s)] 2 into 5]1,~r- "~s I2. * Of course the cosmological constant is also divergent, due to the presence of the tachyon, but this will not concern us.

A. Lyons / Wormholes

285

This is proportional to the tree (i.e. sphere) amplitude for scattering of M tachyons, plus one dilaton at zero momentum, as is shown explicitly in appendix B. The higher terms in the expansion in powers of e correspond to zero-momentum massive off-shell scalar states propagating from the M-tachyon tree amplitude to the vacuum, via the torus. The propagators no longer diverge as ~ ~ 0. The sum over all such states reproduces the effect of the handle on the world sheet, and one may define a "handle operator" by 1

The sum runs over a complete set of string states. The propagator 1/H is at zero momentum. The subscript T means evaluation on the torus. }0> on the 1.h.s. refers to the usual SL(2,C)-invariant vacuum state. One-loop (i.e. torus) amplitudes may then be evaluated, simply by inserting the operator ~ into the amplitude on the sphere. This approach is described in more detail in ref. [12]. We have exhibited the factorization of the divergences associated with the integrals over the positions of vertex operators on the torus. They have been associated with massless or tachyonic string states propagating from a sphere to a torus, down a long throat. The effect of these divergences can be absorbed by inserting a vertex operator (carrying zero momentum) at the points where the external particles converge. Fischler and Susskind have suggested [2] that the divergences could then be cancelled by an effective non-flat background. The argument runs as follows. A string propagating in a curved space-time background can be described by a non-linear sigma model. The sigma model is finite at tree level if the B-functions for conformal invariance vanish. This gives a set of equations that the background fields such as the metric must satisfy. To lowest order in the slope c~', the equations for the background metric are just the vacuum Einstein equations. This is the easiest way to see how general relativity emerges as the classical limit of string theory. One now wants to find a way of incorporating the effects of world sheets with non-trivial topology. These are incorporated in the Fischler-Susskind approach by moving the sigma model away from conformal invariance and vanishing B-functions. There are then divergences which can be regulated by a world-sheet cut-off. These are of the same type as the torus divergences described earlier, and the Fischler-Susskind proposal is that the new condition on the sigma model (to replace vanishing B-functions) is that these divergences cancel each other. This gives new equations for the background fields to satisfy. For the metric these are the vacuum Einstein equations including a cosmological constant A. This is just the one-loop cosmological constant given earlier and computed in refs. [3, 4]. This effect is thus an example of the non-trivial topology of the string world sheet making the background metric appear to be curved. Since the background metric can be thought of as an infinite set of coupling constants, there is an analogy with the Hawking-Coleman mechanism [5,6] for the changing of the effective values of

286

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Wormholes

coupling constants by wormholes in space-time. The analogy will be discussed in greater detail in sect. 3.

3. Small handles on the complex plane The first step in making the previous effect look more like wormholes in space-time is to decompactify, so that there is some region at infinity. The above picture may be mapped to the domain ~J consisting of the complex plane with two discs removed and their S 1 boundaries identified, by the following conformal (or analytic) transformation (setting t = tanh(Tr Im ~')): z = z o + ~tz I

tanh[i~r(~, - ~'/2)] .

The midpoint between the two discs is z 0, while the phase and modulus of z~ are the orientation and distance between the centres of the discs respectively. The ratio of ~ = {Zll to the radius a of the discs is 2cosh(~r Imp-) and the twist angle in the identification is 27r Re • (see fig. 3). We can find the Green function in Z either by the method of images or, equivalently, by using the properties of 0 functions. In the 0-function approach, we solve for the symmetric Green function satisfying:

1 '

=

_

_

82(

_

%),

Z0

Fig. 3. The --picture of the punctured torus. The points P are identified and identifications follow the arrows. Alsoz I = A BandX= tz~l.

287

A, Lyons / Wormholes

where vo = ~_(1 + ~') is the point on the torus which gets mapped to infinity by the conformal transformation z = z(v). We are here exploiting the fact that the amplitudes are unchanged when the balancing term in the Green function is changed, so as to use a 8 function as the balancing term. The new Green function is G,o(V, v') = G(v, v ' ) - G(v, Vo)- G(v', Vo) in terms of the old one. The function F(r) which appears in the scattering amplitude does not depend on the balancing term in the Green function equation (i.e. it is independent of vo, the position of the puncture). This can be seen by writing it as

F('r) = 4( i~.ir) MS I-~Ird2#.'~exp , [ r~sG%(#.'r,, ) kr • ks/4] , and using m o m e n t u m conservation to get rid of the terms }2rsG(vr, vo)k r" ks~4 which occur in the exponential. The regularized value of G(v, v) can be absorbed in the renormalization of K (or cancels the arbitrary constant which can always be added to G). Thus the scattering amplitudes are conformally invariant, even though the Green functions used to compute them are not. The new Green function can be expressed in terms of 0 functions as

G.o(V, v') = ln(X.o(V, v ' ) } , where

, [ Im(..o>tm(.',]

X~,,(v, v ' ) = ~-~-exp 2~ Here we are writing N is given by

(Imr)

O(v) = O(v, r) for ease of notanon. Then the Green function on

G(z,z') = where

0(.-.'>01(0>

O(vTfvo)O~(v~-Vo)

Q0(.(z),.'(z')),

v(z) is the inverse transformation, i.e. 1

gg71n g77- 1'

¢(z)-

tzt 2(z__ Zo ) -

This is because the &function sink at v0 gets mapped to infinity by z = z(v), and the rest of the equation is conformally invariant with weight zero. This function can be expanded for ?t = IZa I --* 0 and takes the form

I-Iz-z'l ] + -

G ( z , z') ~ ln[ ~ - - t

t2

2~r I m r

Ret Zo)Re( )+,z

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A. Lyons /

Wormholes

ignoring terms which disappear upon averaging over the phase of z~ or which are higher order in X. The term which concerns us in this discussion is the O(X2) term, which, upon averaging over the phase of z~ is )k2t 2

D( z' z' ; z°)

Z o ) ( Z - z0)* } ]( z' - zo)( Z - Zo)] 2 '

Re{(z'-

4rr I m r

where * denotes complex conjugation. This term comes from the expansion of the exponential in the expression for X- It is essentially a dipole term in the method of images, coming from + charges located at z + = z o +_ 5zlt. These are the fixed points of the image map J and its inverse: z ( v ) - - + J ± l z = z ( v + _ r ) . J maps the whole region S., exterior to both discs, into the interior of one of the discs, while J - ~ maps 2; into the interior of the other disc. In the method of images approach, one gets a sum of images of strength + 1 at % = Jnz', and strength - 1 at vn = J%o. The charges do not decay in strength in this two-dimensional problem. (They would decay in strength in more than two dimensions.) What happens instead is that the infinite sum of charges still converges to a well-defined Green function on ,Y, because the _+ charges tend to cancel as n--+ _+ 00 where wn, v,,--+ z +. However, there are also extra charges of strengths _+Q at z +. Q must be chosen so that the Green function is periodic when you go from one circle to the other. It is these + Q charges which give the dipole term referred to above. In symbols, the term in the method of images which gives the dipole is

1 [zzj] f z z ] 2~r I m r

In

[ z - z+[

In

[[z'

z+[

.

If this is expanded for small X = ]Zl] and averaged over the phase of z t, the leading term in powers of X is D(z, z'; Zo). This term can be thought of as arising from an interaction at the point z 0. We have

D( z, z'; Zo) = {ep( z ' ) V ( zo)ep(z )>, where the vacuum expectation value is evaluated on the complex plane, and the interaction is V ( z ) = (X2tz/8rr I m r ) : cO.0(gz.q,:. In string theory with a flat spacetime background, one would replace (/) by X ~'. The Green function G(z, z') is then multiplied by ~/,,. The interaction operator :(gzep0z.0: would be replaced by ~ :OzX" Oz. X ~ :, which is the vertex operator for a zero-momentum dilaton. Thus small topological fixtures are equivalent to operator insertions on the world sheet. If one adds the contributions from many small handles, these should exponentiate. The contribution from n small handles comes with a factor 1 / n !, as the n handles m a y be permuted among themselves. It also has a vertex operator raised to the

A. Lyons /

289

Wormholes

power n. Thus the final result is a shift in the coupling constants multiplying 0z¢ 0z. ¢ in the action. In other words for string theory, for which we have 26 fields X ~ replacing ¢, and the action is } f d 2z Oz X ~ 8 z . X ~ g . . ( X ) , the background metric g . . ( X o ) is no longer the flat metric of Minkowski space, but appears to be curved. This is the same conclusion as was reached by Fischler and Susskind [2] using a somewhat different approach. We have exhibited a wormhole-like phenomenon on the world sheet, where the two ends of the wormhole join at the same point. We now consider a wormhole connecting two regions far apart, and attempt the same replacement of topological fixtures by vertex operators in that case.

4. Long handles and the four-point function This model may be used to find the leading interactions associated with wormholes connecting two points which are far apart on the complex plane. To do this we shall construct the four-point function where two of the points are near each end. We shall find that the wormhole may be replaced by a sum of operators at each end, including the tachyon, dilaton and graviton vertex operators. It provides a new way of looking at the factorization of string amplitudes and justifies the bilocal interaction ansatz [5-7] in a simple case. We therefore need to look at the Green function in the region z0, z 1 ~ ~ while z = z o - t z l / 2 is held fixed, z is the position of one of the ends. We also need the radii of the circles, a, to become small in this limit. Then define 8 = 1 - t, so that 6z21 ~ O. Also set z + = z o + t z l / 2 , so that z+ is the position of the other end. We shall be interested in the region where each of the points in the Green function is an O(1) distance from some wormhole end. The asymptotic limit is thus a ~ 0 and X ~ ~ . This is in effect the dilute wormhole approximation [5, 6], as the two ends are far apart in the asymptotically flat space and have vanishingly small radii. We then have

1

and similarly for u'. Also z_z

[z-z_\

(a2)]

and

(o4)}

+0~

,

A. Lyons / Wormholes

290

up to multiplication by a constant, which may be dependent on ~-. Thus, up to addition of a constant,

z')

] In z - z 2~Imr z-z+

In

z'-z

~

+ l n [ z - z'[

z,l)

(z-z)(z'-z+) - Re{e 2 ~

(z-

+

z_)(z' : Z )

+ •...

We are here neglecting O(a 4) terms which will disappear when the four-point function is averaged over Re ~. Also neglected are O ( a 8) and O ( a 2 / ~ ) terms. The last term in the expression above is only relevant when the two points in the Green function are near different ends, and even then only one of the terms in brackets will contribute. The four-point function G4(z, w, z', w') can be obtained from sums of products of G(z, z'), since this is just free field theory. G4 is then averaged over the parameter Re ~- which specifies the twist angle in the identification of the two circles. This is done since there is an integral f ~ 2 / 2 d ( R e ~ ) in the expression for scattering amplitudes. The averaging is analogous to the SO(4) average performed for wormholes in four dimensions [5,13]. The averaging procedure ensures U(1)-invariant interactions on the world sheet. In refs. [5,13] the averaging procedure ensured SO(4) invariance in euclidean space-time. The final expression for the four-point function, up to O(a 4) terms, and suitably averaged over Re ~', is G4 ( z, w, z', w') = G4c ( z, w, z', w') + sum of products of logarithms

+la4Re

/ (z-z~)(z'-z

1

)(w-z+)*(w'-z

)* + ( z ' ~ w ' )

),

where G4c denotes the four-point function on the complex plane for a free scalar theory. We here assume that unprimed variables are near z+, and primed variables are near z . The logarithms which occur are logs of [z - z+ I, ]w - z+ ], ]z' - z ], [w' - z I, and the modulus of the difference between any of z, w, z', w' with any other.

5. Biiocal operators: discussion and conclusions The four-point function of sect. 4 can now be thought of as if we were on the complex plane, but with vertex operators of a particular type inserted at z and z +. The logarithms correspond to even polynomials in ~ of up to second order at each point, while the last term is exactly what we would obtain from the vertex operator

A. Lyons / Wormholes

291

l a 2 : 0 ~ 0 " ~ : inserted at each end. A process, similar to that in sect. 3, of replacing q~ by X ", results in the effective string theory vertex operators at each end of the wormhole. One finds not only the dilaton, but also a sum over all polarizations of graviton vertex operators, at the points z +. One may think of this as a sum of all string states being created at one end, and then "propagating down the tube" to the other. These dilaton and graviton propagators do not lead to divergences appearing in string amplitudes as they do not propagate with zero space-time momentum. They are not coupled to the vacuum so they do not need to have vacuum quantum numbers, unlike the situation in sect. 3. This leads to the appearance of the graviton vertex operators. The various logarithms correspond to tachyon vertex operators exp{ik- X} inserted at each end. We will only pick up the 1 and ( X ) 2 parts of this operator in an expansion in powers of X. An n-point function would pick out terms up to (X)". By comparing the coefficients of 1 and ( X ) 2, one should be able to see the relation between the size of the wormhole and the momentum k flowing through. The momentum is analogous to axion charge in the Giddings-Strominger ( G - S ) wormhole [14], which can be thought of as a solution to the field equations for a massless Goldstone boson coupled to Einstein gravity [15]. Here the field X" is analogous to the Goldstone boson, and k" is the charge associated with the symmetry X ~ ~ X" + C ". For the G - S wormhole there is a simple relation between the size of the wormhole and the charge flowing through it. In the dilute wormhole limit it is just Q - a 2. There might be a similar relation in the example given in sect. 4. There are no cross terms between the polynomials in q~ and the 10q, I2 terms. This is because the tachyon states are orthogonal to the dilaton and graviton states in string theory. Thus the interpretation in terms of string states propagating down the wormhole is self-consistent. As in the case where the two ends coalesce, the integrated vertex operators are exponentiated when one adds the contribution from any number of handles. The wormholes therefore contribute a bilocal interaction term to the effective action in flat space, verifying the bilocal interaction ansatz of many papers on wormholes [5-7]. It is clear that this description is only valid in a certain region of moduli space where it makes sense to regard the handles as long thin tubes. Also, the use of bilocal action has only been verified for the calculation of Green functions in the asymptotically flat region, and it may well be wrong to substitute it into a path integral. This point has recently been made by Hawking [16]. The phenomenon of bilocal interactions replacing wormholes presumably persists to higher order in a 2, and in the calculation of higher n-point functions. One would expect to generate any terms which are rotationally invariant, but there are no other obvious restrictions. These would correspond in string theory to vertex operators for massive string states. It would be interesting to see what the a-states (discussed by Coleman [6]) are in this model and relate them to string states. They would be some superposition of number states containing different numbers of strings. This model can also be

A. Lyons / Wormholes

292

extended in a straightforward way to higher-dimensional wormholes such as the Misner wormhole [17] which is topologically S 1 × S n and is constructed using the method of images from flat space. Misner's original paper [17] takes n = 2 as he was interested in initial data surfaces in general relativity, but the general case is no harder. These metrics have R = 0, so it is easy to construct the scalar Green functions using conformal invariance, recently used by Hawking in the case of the T o l m a n wormhole [16]. A similar replacement of the handle by a bilocal action in flat space should again be possible. I would like to thank Gary Gibbons, Peter Goddard and Stephen Hawking for useful discussions during the writing of this paper. I am particularly indebted to Stephen Hawking for suggesting this problem and assisting me with the solution. I would also like to thank everyone at King's College and DAMTP, Cambridge, who have provided a good working environment, and SERC for providing financial support.

Appendix A Our conventions on the modula parameter • and 0 functions are those of Green, Schwarz and Witten [9]. Namely, the torus is described by a fundamental parallelogram in the v-plane with vertices at the points 0,1 and ~, such that Im ~- > 0. Points in the v-plane are identified under translations by 1 or ~-. We must restrict any integration over • to a fundamental region of the modular group, e.g. F in the ~-plane defined by - ~ ~< Re ~-~< ½ and ]~'r > 1. This is to avoid counting diffeomorphism-related geometries more than once in the path integral. It is important to divide out by the full diffeomorphism group (not just those which are connected to the identity) to avoid an infinite result in the integral over ~-. The first Jacobi 0 function appears in the text. It is an analytic function, defined on the complex plane as follows:

0i(u,'r)=2f(q2)ql/4sin~rufi (1 - 2q2" cos27r~, + q4"), n=l

where q = e i~ and

f(q2)_ f i (l_q2,). n=l

We write O(u) -- 01(u, ~-), and a prime will denote a derivative with respect to ~ at constant ~-. The 0 function satisfies the following periodicity relations: 0(.+

1) = - 8 ( . ) ,

O(v+T) = -exp[-i~-

2i~ulO(u),

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Wormholes

293

together with having a first-order zero at the points u = m + n ~', for integer values of m and n, and no other zeros on the u-plane. These are the only properties required in the text, although the 0 functions also possess remarkable transformation properties under modular transformations [9] which are important in string theory applications.

Appendix B We shall show in a simple calculation that the factor coming from the torus amplitude for M-tachyon scattering really is the tree amplitude for scattering of M tachyons plus a dilaton at zero momentum, as asserted in sect. 2. The tree amplitude involves the correlation function (on the sphere) of M vertex operators for the tachyons] VT(k i, z i ) = :exp(ik~. X(zi)): and the dilaton vertex operator, which in conformally flat coordinates takes the form V D ( k , z ) = % / O X " O X " e x p ( i k • X)" Here X-- X ( z , 5) and I have not always shown the z, 5 dependence explicitly. The polarization tensor for the dilaton is %~. The amplitude is given by 1

where the correlation functions are calculated using the Green function on the sphere, which, after a stereographic projection to the complex plane, takes the simple form ( X ~ ( z ) X ~ ( w ) ) = In klz - w I. k is an infrared cut-off which does not appear in any scattering amplitudes. The tachyon vertex operators must be normal ordered so that no contractions occur between two X operators at the same point. These can all be absorbed in the normalization of the tachyon coupling constant. The dilaton would not need to be normal ordered if %~ were a symmetric tensor satisfying k~%, = 0. However, for our purposes we can put %~ = ~ , . This is because the tree amplitude for a single external longitudinal dilaton vanishes automatically, by gauge invariance. The integrand is SL(2, C) invariant. That is, it is invariant under M~Sbius transformations az+b Z ----~ Z p - -

cz + d '

where a d - bc = 1. This is because there are conformal Killing vectors on the sphere. The SL(2, C) invariance is what remains of Weyl and diffeomorphism invariance of the path integral once the conformally flat gauge has been chosen. This gauge only partially fixes Weyl and diffeomorphism invariance. The integral is therefore divergent, and must be divided by Vol(SL(2,C)) to obtain a finite result.

294

A. I:vons /

Wormholes

The standard way of dealing with this is to gauge-fix SL(2,C) by putting three of the external particles at standard points on the sphere, say z A, z~, z c. The usual choice of points is 0,1, oo. The residual integrand then exhibits a factor of Vol(SL(2,C)) which cancels the denominator. There is also a F a d d e e v - P o p o v determinant from the SL(2, C) gauge fixing which is IzA- zBI21z~- Zcl2[Zc- 2A] 2The correlation functions can now be evaluated using standard methods. The formula ( F I i : e A ' : ) = exp[Zi
[

VoI(SL(2,C)) j d 2

zl~d 2 z,l~]z_

,

i
Z]]k.k,/2 i ~ ] z - zi[ k k j 2

i

~

k,. kj

~.(z-z,)(z-zj)*"

One can show explicitly that this is invariant under SL(2,C). As an exercise we shall now do this. First, note that the apparent divergence of the integral as z -~ oo is due to the coordinate singularity of the stereographic projection of the sphere down to the complex plane. We see that in fact the integral over z does not diverge if the double sum over i and j is performed first, since k 2 = 0 for an on-shell dilaton. This we shall now assume. Under SL(2,C) transformations we have d2z

z i - zj

d2z ' -

Ic:+dl

4,

' - zf = z, j

and we use the mass-shell conditions: k 2 = 8 for the tachyons and k 2 = 0 for the dilaton, and momentum conservation k + ~ , i k ~ = O . In the double sum we write cz, + d = (cz + d ) - c ( z - zi) are similarly for the j terms. The terms like k. ~,,k,(~z + d)/(zz,) give zero upon integration, because the rest of the integrand is invariant under rotation of the points z, around z. The various factors of cz + d and cz, + d cancel, and we see that the integrand is indeed invariant, as it should be. Since the amplitude is SL(2,C) invariant we can eliminate the z-integration by gauge fixing z ~ ~ , being careful to multiply by the Faddeev-Popov determinant [Z -- Z A [ 2 [ Z -- ZB] 2 before taking the limit. The terms which diverge as z ~ oo are removed by momentum conservation, the mass-shell condition for the dilaton, and the rotational invariance of the rest of the integrand as the points z~ rotate around z = oo. The rest of the gauge fixing may be done by taking any two tachyons to be at the points z A = 0, z B = 1. The result of doing this is exactly the expression given in the text, as, in the limit k --, 0, we have Y ~ k , . k j z i z ? = - ~2~'~ki " k j l z , i, i i, j

zyl 2 ,

A. Lyons / Wormholes

295

using momentum conservation to get rid of the S,i, jk i. kj[zil 2 terms. The integral over ff in the expression in the text just gives a factor of 2~r as the integrand is already rotationally invariant under rotations of all points zi around the point

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