Superstring modifications of Einstein's equations

Nuclear Physics B277 (1986) 1-10 North-Holland, Amsterdam

SUPERSTRING MODIFICATIONS OF EINSTEIN'S EQUATIONS* David J. GROSS and Edward WITTEN Joseph Hen~ Laboratories, Princeton University, Princeton, New Jersey 08544, USA

Received 13 March 1986

The modifications of the classical equations of motion of the gravitational field in type II string theory are derived by studying tree-level gravitational scattering amplitudes. The effective gravitational action is determined through quartic order in the Riemann tensor. It is shown that generic Ricci-flat manifolds do not solve the modified equations, unless in addition the manifolds are K~hhler (2N-dimensional manifolds of SU(N) holonomy). Translated into sigma model language, this calculation would indicate that the N = 1 supersymmetric sigma model in 2 dimensions, with a Ricci-flat target space, is not conformally invariant, but has a nonzero beta function at four-loop order.

1. Introduction T h e N = 1 supersymmetric nonlinear o model in 1 + 1 dimensions, with a target space that is an arbitrary Ricci-flat manifold, is k n o w n to have a vanishing beta f u n c t i o n in one-, two-, and three-loop order [1]. At first sight, it is tempting to believe that this result holds to all orders. If so [2], an arbitrary Ricci-flat ten-manifold M x° would give a solution of the classical equations of motion derived from type II strings, or for heterotic strings with the spin connection embedded in the gauge group. The Virasaro anomaly coefficient c would automatically have the correct value for Ricci-flat space-times, if these all had zero beta function, since it follows f r o m the Bianchi identity that it is a c-number and thus space-time i n d e p e n d e n t ; no local polynomial in the R i e m a n n tensor and its derivatives has this p r o p e r t y on a generic Ricci-flat manifold. A t first sight, this conclusion m a y seem reasonable, perhaps even attractive. However, consider in uncompactified ten-dimensional space-time a classical scattering process with initial data asymptotic in the far past to an incoming gravitational wave. A c c o r d i n g to classical general relativity, this wave would undergo self-interactions governed by the Einstein equation R , , = 0. If an arbitrary Ricci-flat space-time gave a sigma model of zero beta function, and hence a solution of type II string theory, then the gravitational wave scattering process would evolve in this theory * Supported in part by NSF Grant PHY80-19754. 0550-3213/86/$03.50@ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

2

D.J. Gross, E. Witten / Einstein's equations

exactly as it does in general relativity. This contradicts the fact that (tree-level) graviton-graviton scattering is totally different in string theory from what it is in general relativity. Turning this argument around, it must be that if we study the tree-level scattering amplitudes of the type II theory, we can find evidence for a nonzero beta function of the N = 1 supersymmetric sigma model with generic Ricci-flat target space. Indeed an alternate approach to the derivation of the classical string theory equations of motion is via the effective lagrangian of the massless fields. This lagrangian can be systematically constructed from a knowledge of the tree-level scattering amplitudes. One first constructs a lagrangian, ~0, which possesses all the ten-dimensional symmetries of the string theory and reproduces the spectrum and cubic vertices of the massless particles (the graviton, anti-symmetric tensor, dilatons, etc.). One then considers the scattering amplitudes. Unitarity guarantees that the massless poles will be generated by the tree graphs of £~°0. What remains can have no singularities for vanishing external momenta, and can be expanded in a power series in a,p2, where P is a typical momentum of the external massless particles. One then adds new, 4-point, 5-point, etc., local vertices to ~0, order by order in a', to obtain a new effective lagrangian. This procedure can be repeated to yield, in principle, the effective lagrangian to all orders. The effective lagrangian so constructed will not be unique, since a local redefinition of the fields will not affect the S-matrix elements. However this does not affect the equations of motion, which are unchanged by a field redefinition. These should coincide with those derived by demanding conformal invariance of the sigma model. The spectrum and 3-point couplings of the type II string theory are those of standard D --- 10, N = 2 supergravity, with no additional interactions. Focusing on the gravitational field alone the equations of motion, to this order, are Einstein's. Thus we conclude that in the generic Ricci-flat case, the beta function is zero at the one-, two- and three-loop orders, in agreement with [1]. However, since the graviton scattering amplitudes contain, in addition to massless poles, terms arising from the exchange of massive string states, we will find new contributions to the effective action, starting at fourth order. These will alter the equations of motion, and imply that the four-loop beta function does not vanish for the generic Ricci-flat case. The situation for Ricci-flat K~hler manifolds is more subtle and will be discussed later, along with a comparison of our results to recent work on a models [3]. The restriction to the K~hler case removes the paradox about scattering amplitudes because with lorentzian signature ( - + + + -.- +), there are no scattering processes described by K~hler manifolds. The real coordinates of a K~hler manifold come in pairs because of the complex structure, so if the metric of a K~hler manifold has one negative eigenvalue, it has at least two. Ricci-flat K~hler metrics are not directly relevant to scattering processes, but they can be relevant to vacuum configurations (such as M 4 x K, K having SU(3) holonomy) or conceivably to instantons (positive signature manifolds of SU(5) holonomy).

D.J. Gross, E. Witten / Einstein's equations

For heterotic strings, the N = 1 supersymmetric sigma model only becomes relevant upon embedding the spin connection in the gauge group. This is never possible for scattering processes, which (in ten dimensions) always correspond to metrics of SO(l, 9) holonomy; SO(l, 9) is a noncompact group which cannot be embedded in E 8 x E 8 or SO(32). However, embedding the spin connection in the gauge group is certainly possible for vacuum configurations or instantons. When this is done one gets the same sigma models that arise from the type II theories, so our results are relevant to that case. The general structure of the effective interactions is more complicated in the heterotic theory than for type II strings; it will be explored elsewhere [6]. In classical general relativity, we can consider a process with an incident gravitational wave with initial conditions such that it forms a black hole. If the generic Ricci-flat metric gave a solution of type II string theory, then in this way black hole singularities could form from non-singular initial data in classical type II string theory as well. The resolution of the singularity problem would therefore have to arise quantum mechanically. Conversely, our result means that classical type II string theory perhaps does not form black hole singularities. The same may well be true for all other string theories. A plausible conjecture about the fate of isolated collapsing stars in nature is that they form an object which from the outside looks like a macroscopic general relativistic black hole but actually because of string theoretic effects has no horizon or singularity even in the classical limit. This object will slowly shrink by emitting Hawking radiation until at the compactification scale it becomes effectively ten dimensional. If the compactification radius is somewhat larger than the Planck length, the final state of the "black hole" may be determined by classical string physics rather than quantum mechanics. It is tempting to conjecture that dilaton radiation in classical string physics will play roughly the role of Hawking radiation in quantum mechanics.

2. String modification of the Yang-Mills equations Turning now to our study of scattering amplitudes, we will first look at the slightly simpler case of the scattering of massless gauge bosons in type I superstring theory. Denoting the Yang-Mills field strength as F, the low-energy limit involves the standard - ~TrF~F ~ ~ interaction. There will be corrections to this coming from additional interactions induced by massive string modes. There is no F 3 term, since it would have shown up in the three-point function but does not [7]. To find an F 4 term, we will study the four-point scattering amplitude; from it we will extract a nonderivative F 4 coupling. The four-point function also gives an infinite series of higher order terms such as F2(V'F) 2. In a low-energy expansion, those terms are no more important than, say F 5 terms that could be inferred from the five-point function. We will study neither the F2(XTF) 2 nor F 5 terms here.

D.J. Gross, E. Witten / Einstein's equations In the scattering of four external gauge bosons of momenta Pto and polarization e(j), a certain kinematical factor K ( q o , Po)) appears which is common to both the tree and one-loop diagrams [7], and will appear also in closed string amplitudes. At the tree level the derivation of this factor is rather untransparent; it emerges by summing a variety of terms. At the one-loop level the derivation of the kinematical factor is much more transparent. Since the specific form of K(e(o, P(j)) is important in what follows, we will for completeness review how it appears in one-loop diagrams. The one-loop diagram for open strings can be regarded as a path integral on a cylinder; in light-cone gauge, there are eight fermionic zero modes ~b~ (arbitrary constant spinors of SO(1,9) with positive chirality and f'+t~-= 0; equivalently, arbitrary positive chirality spinors of SO(8)). To get a nonzero amplitude requires that external vertex operators couple to all of these zero modes. The vertex operator for a photon of momentum k,, polarization e~, and field strength FJ~) = - t(k~," (i)ev(i) - - k ( i ) e ( k ),V. ' l f couples to at most two zero modes, if we assume that e (o and k (i) have only transverse components. The coupling is F¢~)~/oF""+0eik"~x. Coupling to all eight zero modes requires at least four external massless bosons, and we will consider this case to extract a n F (4) coupling. The one-loop integral therefore contains a factor

K(e(i), p{i)) =

4

f

H {or"" oe2'

i=l

= f d#;ex

o

o

,

(1)

i=

which is the kinematic factor common to tree- and one-loop amplitudes, though its appearance in tree amplitudes is much less transparent. The tree-level scattering amplitude is in fact [7]

- :½s)F(A(t~'i), k(J)) = - ½ g 2 F (-~( ~ 2 s ~ ½t)½t) K( e¢'~, k(J)) Str T~T2T3T4-

(2)

Here s = (k 1 + k 2 ) 2 and t = (k I - k3) 2 are the standard Mandelstam variables; StrTIT2T3T4 represents the symmetrized trace of the group theory matrices Ta, T2... T4 for external gauge hosons. (We will suppress this factor in what follows.) With F(1 + x) = 1 + xF'(1) + ½xZV"(1) + . . . , F(x) = 1/x + V'(1) + ½xF"(1) + - . . , the product of F matrices in (2) reduces to r(- ~,)r(-

~t)

r(1-½s-½t)

4 -

st

+(r,(1)~_r,,(1))+

...

4 -

st

~(2)

+

..-

(3)

D.J. Gross, E. Witten / Einstein's equations

(~(z) is the Riemann zeta function, ~(2) = ~r2.) When inserted back in (2) the term 4/st gives the scattering amplitude of Yang-Mills field theory. While this can be checked explicitly, it is sufficient to note that the term proportional to 4/st is gauge invariant and dimensionless, like the field theory amplitude, and is determined up to normalization by these properties. The leading string correction to the Yang-Mills amplitude comes from the constant term - ~ ( 2 ) and is

A A = + gZ~(2)Str(T1T2... T4)K(e ~i), P~J)).

(4)

The leading string theory contribution to the low-energy lagrangian (beyond the field theory term tr F/,) is the local operator whose tree-level matrix element is the polynomial K(e ~), P~J)). This is easily seen to be, using (1), 1

X = - - , / d e t F /LuF 4! v ~ '

(5)

where F""F.~ is a matrix in the space of zero mode wave functions, or in other words a matrix in the positive chirality spinor representation of SO(8). This can alternatively be written X= ~trF 4- ~(trF2)2+ le~,...~F

F

E

E

~l/Z2 /L3/La P-5/Z6 /t7~8 "

(6)

In (6), t r F 4 and t r F 2 are traces over the Lorentz indices of the matrix F~,; the group theory factor Str TaT2T3T4 is understood in all terms. In the last term in (6), e~ ' ~ is the antisymmetric tensor of the eight transverse coordinates. This is SO(8) invariant but is not Lorentz invariant. Also it would have appeared with the opposite sign if we had imposed the opposite light-cone gauge condition F ~ = 0, in which case the zero modes ~ would be negative chirality spinors of SO(8). Actually, the last term in (6) is a total divergence, e~.... ~sF,~,2...F,7~ = 9~,1(e~'"~"A~2F~3~... F~7,~), which does not contribute to the four-particle scattering amplitude we have been studying and hence cannot be reliably extracted from it. While Lorentz invariance implies that the last term in (6) should not be present, to explicitly show this in the light-cone formalism would be tiresome. One approach is to note that after coupling to closed strings, the interaction X would multiply 2 powers of the dilaton field; the fact that the last term in (6) should be deleted would then follow from Lorentz invariance of the five-point function with one dilaton and four gauge bosons. The variation of this new interaction with respect to the dilaton field, which is simply proportional to X, or with respect to A~, modifies the Yang-Mills equations of motion. A general Yang-Mills field with D"F~ = 0 does not obey X = 0 or 8X/6A, = 0. Hence, with X included, a general solution of D"F,~ = 0 does not obey the equations of string theory. There are special cases, however, when X does

6

D.J. Gross, E. Witten / Einstein's equations

vanish. As preparation for our later study of the gravitational case, it is interesting to examine a situation in which the matrix F~, is not an arbitrary matrix but belongs to a subgroup of SO(8) (or SO(l, 9)). For instance, let F,~ be an SU(3) matrix, embedded in SO(8) so that the positive chirality spinor of SO(8) transforms under SU(3) as 8 = 3 + 3 + 1 + 1. In this case, the two SU(3) singlets are zero eigenvalues of the matrix F ~ F ~ , so d e t F ~ F ~ has a double zero and X /LV ~, = ~det F F~ has a single zero. Thus X = 0 for a gauge field of "SU(3) holonomy, but the variational equation 8X/SA, = 0 is not obeyed. Hence, Yang-Mills fields of "SU(3) holonomy," though they automatically obey D~F,~= 0, do not in general obey the string theory generalization of the Yang-Mills equations. An example where the equations are obeyed, in the present approximation, would be that in which F,~ is an SU(2) matrix, embedded in SO(8) so that the spinor of SO(8) is 8 = 2 + 2 + 1 + 1 + 1 + 1 of SU(2). (This corresponds to SU(2) being a minimal subgroup of SU(3).) In this case X = ~det F~F~ has a double zero, so X = 8X/3A~ = 0 and the equations are obeyed to this order. 3. The modified gravitational equations Now we move on to graviton scattering in type II superstring theory. We consider the four-point amplitude for gravitons of momenta k~i) and polarizations e~9 = e.u)G(i). The tree amplitude is [7]

,,2

r(-~s)r(-[t)r(-[u)

K ( ~ , , , k).

(7)

A = 12-----8r(1 + Is)F(1 + It)F(1 + lu) Here ~ is the gravitational coupling constant, and K and /£ are kinematic factors like those of the open string which arise from left- and right-moving string modes. A priori, R is the same as K for the type liB string and the parity conjugate for type IIA; the distinction is not important in the S-matrix since K and its parity conjugate differ only by the sign of the last term in (6), which does not contribute to the S-matrix. As in the case of open strings, we must expand the product of gamma functions in (7), getting 29

r(1 + ~s)r(1 + i t ) r(1 + ~u)

-

stu

+ [2F'(1) 3 - 3F'(1)F"(1) + F ' " ( 1 ) ] + .--

29

stu

2~(3) + .--

(8)

(with the use of the identity s + t + u = 0). The - 29/stu term, when inserted in (7), gives the tree-level scattering amplitude of D = 10 supergravity. The leading correc-

D.J. Gross, E. Witten / Einstein's equations

tion comes from the constant term in (8), and is

A A = - 1~r22~(3)K( e ~i), k(J) )K( e ~i), k(J) ) .

(9)

The factor K K is eighth order in momenta, and is the four-graviton matrix element of a certain operator Y quartic in the Riemann tensor R ~ a . This operator can be written in various ways. If we define a symbol t ~ ...~8 by saying that ~det F p,vF~ = t~ ~,F~,~, ... F~8, then our operator is up to a normalization factor

y = t~2...~,t~x~2

... V s R ~ l ~ 2 v l v z R t ~ 3 t ~ 4 v 3 v 4

•..

Rt~Tp, SvTv 8 •

Alternatively, it may be written as an integral over fermion zero modes: Y=

f dq~_ d~kRexp[ t~LF~Bt~L~bR_N~,B,~RR,~o,] 13

-a

~v

B-a'

of

B'

(11)

Now, the operator Y (added to the standard Einstein-Hilbert action I 0 = fv/-gR to give a total action I = fv/-g(R + Y), gives a theory that reproduces string scattering amplitudes up to and including terms of order R4,a~. As such it is not unique. Without changing the content of the theory to this order, one could make an arbitrary field redefinition 3g,~ = X,, (X~, being any local functional of the metric and its derivatives). After making this change of variables, I becomes i = fv/-g(R + ( R , ~ - ½g~R)X""+ Y + higher orders), and this must give a theory equivalent (to this order) to that defined by I. This means, incidentally, that one can add or subtract from i terms of the form R 2 or R,~, 2 by taking X,~ to be a combination of g ~ R and R~.. Now, in o model terms, an n-loop contribution to the beta function gives an interaction that dimensionally is of order R". This means that the interaction Y corresponds to a contribution to the beta function of four-loop order. Thus, for a space-time which is such that the one-loop beta function vanishes (which is so if R , , = 0), there will be no further contributions to the beta function in two- or three-loop order; this agrees with explicit calculation [1]. However, the occurrence of the interaction Y in the string scattering amplitude implies that the generical Ricci-flat manifold will yield a sigma model with nonzero beta function at four-loop level. The interaction Y will modify the classical equations of motion. The dilaton equation will receive a new contribution proportional to Y, the graviton equation will be modified by (3/3g~,)(v/g Y). In fact, for general Ricci-flat manifold, V~- Y is neither zero nor stationary. To prove this, it is enough to check the special case of compactification o n M 6 x K , M 6 being six-dimensional Minkowski space and K a Ricci-flat manifold of holonomy SO(4) = SU(2)L × SU(2)R" For a Ricci-flat fourmanifold, the nonzero part of the curvature tensor is the Weyl tensor, which

D.J. Gross, E. Witten / Einstein's equations

transforms as ( 2 , 0 ) + (0,2) under SU(2)L × SU(2)R. Let us denote the (2,0) and (0, 2) parts of the Weyl tensor as C and D, respectively. Then the C 4 and 0 4 terms cancel in Y, leaving only a C 2 D 2 term. The essential reason for this can be best seen by considering first the Yang-Mills case. For an SO(4) matrix F,, with a decomposition F = U + V where U and V transform as (1,0) and (0,1), respectively, (6) reduces to X = ~ t r U 2 t r V 2, which certainly does not vanish. The gravitational interaction (11) is a product of similar structures for self-movers and right-movers, and likewise nonzero for a generic Ricci-flat 4-manifold. However if the four-manifold is K~ihler in addition to being Ricci-flat, i.e. a manifold of holonomy SU(2), then either C or D vanishes. Since Y has a double zero, both Y = 0 and 6Y/Sg,~ = 0 are satisfied, guaranteeing that both the dilaton and gravitational equations are satisfied. Next we consider compactification on M 4 )< K, with K being a six-dimensional manifold. For generic Ricci-flat K, Y will modify the equations of motion; however, we might expect that if K is Khhler, i.e. of SU(3) holonomy, these will still be satisfied. Indeed heuristically one might say that Y is a product of left- and right-handed kinematical factors, each of which vanishes for SU(3) holonomy, so that Y again has double zero. If so then for a manifold of SU(3) holonomy the dilaton equation will be obeyed, since Y = 0, and the gravitational equation will be obeyed, since 8Y/Sg ~ = O. Specifically, the double zero of Y arises because the SU(3) singlet fermions (two left-handed and two fight-handed), do not appear in the exponential in (11). This conclusion is roughly correct, but needs modification because 6Y/6g~, does not really factorize between left-movers and right-movers. Let us first discuss carefully the equation for the dilaton field qs. In the absence of the corrections we are discussing, the dilaton equation of motion is rqq, = 0, and has the solution ~ = ~0 (q'o being a constant). Our correction term replaces t~q~= 0 with t ~ = Y.

(12)

For compactification on M4)< K, with K being Ricci-flat, Y does not vanish in general so we cannot obey (12) with q~= ~o- We can try to obey (12) with q~= e~0 + 64~, 6~ a small correction of order (at) 3. We get 8,t, = Y.

(13)

If we assume that four-dimensional Poincar6 invariance should hold, then 6q~ should depend on the coordinates of K only (and n o t M4). This implies

For generical Ricci-flat K, fKY 4=O, so it is not possible to find a Poincar6 invariant solution of (12). Eq. (12) forces q~ to become time dependent.

D.J. Gross, E. Witten / Einstein's equations

9

If, however, K is Ricci-flat as well as K~ihler, then Y vanishes (the factors coming from left-movers and right-movers are both zero), so (12) is obeyed. What about the gravitational equations? It has been shown in [9] that Y coincides with the counterterm obtain in [3] from a four-loop beta function calculation and analyzed independently in [8]. The variation of Y with respect to the metric was analyzed in [9] with the following results. Expanding around a Ricci-flat K~ihler metric, the first variation of Y is 0

8Y

8Y

8g~J

8g~J,

8Y

8g,) =

OsO)X,

(15)

where X is a cubic polynomial in the Reimann tensor. (In fact, X is the polynomial whose integral over a six-manifold would equal the Euler characteristic.) From (15), we see that inclusion of Y modifies the Einstein equations to 0 = Rij =

Rij,

R U = ai~jX.

(16)

The first equation in (16) is valid for any K~ihler metric. As for the second, recall that if gi) is a K~hler metric then R U = 3~0) lndet g. Let us try g = go + 8g, where go is a Ricci-flat K~thler metric and 8g is a correction of order (a') 3. Since go is Ricci-flat, 3i3 ) lndet go = 0, and since lndet g = lndet go + TrSg + O((8g)2), the second equation in (16) becomes

agO) TrSg = O,OdX,

(17)

which has the solution TrSg=x+

c,

(18)

with c being a constant. Before accepting the solution (18), we must check it is compatible with the assumption that 8g is K~ihler. Deriving 8g from a perturbation p of the K~hler potential,

O, 3)0,

(19)

Dp = x + c,

(20)

0 = ~r-10 - £ ( X + c)

(21)

8g U = (18) gives which implies

10

D.J. Gross, E. Witten / Einstein's equations

and has a solution if and only if the right-hand side of (21) vanishes. There is a unique value of c for which this is so, so our solution is valid. There is thus a major difference between Ricci-flat K~ihler manifolds and Ricci-flat manifolds that are not K~ihler. In the latter case, the perturbation Y destabilizes the vacuum; (12) has no time-dependent solution. In the former case (Ricci-flat Kahler), the perturbation Y merely brings about a microscopic readjustment of the vacuum according to (18) and (20). In the non-K~ihler case, there is no solution of the exact equations (including Y) that preserves four-dimensional Poincar6 invariance and goes over as c~' ~ 0 to a solution of the Einstein equations. In the Ricci-flat K~ihler case, there is. This conclusion illustrates the arguments in [4]. The reasoning we have sketched above can be generalized to all orders of sigma model perturbation theory, as has been done in [10]. References [1] D.Z. Freedman and P.K. Townsend, Nucl. Phys. B177 (1981) 443; L. Alvarez-Gaum~ and D.Z. Freedman, Phys. Rev. D22 (1980) 846; Comm. Math, Phys. 80 (1981); L. Alvarez-Gaum6, D.Z. Freedman and S. Mukhi, Ann. of Phys. (NY) 134 (1981) 85 [2] C. Lovelace, Phys. Lett. 135B (1984) 75; D. Friedan and S. Shenker, talk at Aspen Summer Institute (1984); P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B258 (1985) 47; E. Fradkin and A. Tseytlin, Phys. Lett. 15BB (1985) 316; C.G. Callan, Jr., D. Friedan, E. Martinec and M. Perry, Nucl. Phys. B262 (1985) 593 [3] M. Grisaru, A. van de Ven and D. Zanon, Harvard preprint (1986). [4] E. Witten, Nucl. Phys. B268 (1986) 79 [5] P. Candelas, G. Horowitz, A. Strominger and E. Witten, ref. [3] [6] D.J. Gross, to appear [7] M.B. Green and J.H. Schwarz, Nucl. Phys. B181 (1981) 502, B198 (1982) 441; J.H. Schwarz, Phys. Reports 89 (1982) 223 [8] C.N. Pope, M.F. Sohious and K.S. Stelle, Imperial preprint ITP/85-86/16 [9] M.D. Freeman and C.N. Pope, Imperial preprint ITP/85-86/17 [10] D. Nemeschansky and A. Sen, SLAC preprint (1986)