PHYSICAL REVIEW 0 VOLUME 22, NUMBER 4 15 AUGUST 1980

Kahler geometry and the renormalization of suyersymmetric cr models

Luis Alvarez-Gaume and Daniel Z, FreedmanInstitute for Theoretical Physics, State University ofNew York at Stony Brook, Stony Brook, New York 11794

(Received 10 March 1980)

It is shown using general arguments based on Kahler geometry that supersymmetric o. models defined onmanifolds which are (A) Ricci-flat or (8) locally symmetric have the following ultraviolet properties. Class Atheories are finite at one- and two-loop order. Class B theories, which include the O(n) and CP" ' models, (i) areone-loop divergent, (ii) but have no two-loop divergences. The geometrical argument can be extended to higherorder with new complications which are discussed. It seems probable that class B theories have no higher-loopultraviolet divergences and quite possible that class A theories are entirely ultraviolet finite.

The nonlinear o model in two space-time dimen-sions has been widely studied. The geometric na-ture of the bosonic model is evident from the ac-tion

I = p d xg]g

where the fields P'g), i =1, .. . , n, are maps fromQat Euclidean or Minkowski space to an n-dimen-sional manifold I with Riemannian metric g,&(P).One can obtain an N=1 supersymmetric extension'with n two-component Majorana spinor fields P'(x)which transform as a contravariant vector on M.There is also an important theorem due to Zuminoconcerning extended N = 2 supersymmetry. Givena complex manifold M' wit:h coordinate fields z (x)and conjugates r~(x), n, P= 1, . . . , n, and Hermit-ian metric g ~(z, r), then the bosonic model withaction

I= d'xg ~, r e„z e„zI' (2)

has an N=2 supersymmetric extension, with n

complex spinor fields, if and only if the metricsatisfies the Kahler condition

&„g ~(z, r) = s g„~(z, r) . (3)

Thus Kahler geometry enters the subject in con-nection with N =2 extended supersymmetry. It isalso possible to have N =4 supersymmetry in non-linear models, ' and it has recently been shownthat the simplest N =4 models involve Kahlermanifolds with vanishing Ricci tensor.

In this paper we show that Kahler geometry hasvery strong implications for the renormalizabilityof supersymmetric o models defined on eitherRiemannian or Kahler manifolds. We now sum-marize the results.

Strict renormalizability does not hold for generalmetric a models in two space-time dimensionsbecause there are typically an infinite number ofcounterterms allowed by invariance and power

counting. The one-loop counterterm of the bosonico model involves the Ricci tensor of the manifold'together with a noncovariant field renormalization.The latter vanishes for all Kahler manifolds, andthe former vanishes for all Ricci-flat manifoldswhether Riemannian or Kahler. One-loop bosoniccounterterms do not change when fermions areadded with supersymmetric coupling, so that su-persymmetric models on Ricci-flat manifolds arealso one-loop ultraviolet finite.

At two-loop order the situation is rather dif-ferent. All bosonic 0 models have two-loop diver-gences. ' However, using Kahler geometry one canargue without calculation that invariant diver-gences cancel in two classes of supersymmetricmodels, those defined on Ricci-flat manifolds

(R&& =0) or on locally symmetric manifolds (DP=0). The latter class includes the well-knownsupersymmetric extensions of the 7 O(n) and' CP" 'models and all other models with instanton solu-tions. ' The general argument allows one two-loopultraviolet counterterm of specific tensor formwhich is nonvanishing on manifolds with D,.R,.~c 0.Explicit two-loop calculations confirm the absenceof divergences for Ricci-Qat and locally symmetricspaces and the presence of divergences associatedwith the one allowed tensor for general manifolds.This divergence appears to be a consequence of thegeneralized renormalization-group' pole equations.

Our argument can be applied to the study of ultra-violet divergences at three-loop and higher order,but there are new complications. At present theconclusions are more speculative because theyare based on mathematical results which seemplausible but are not yet established. It appearsvery probable that all higher-loop invariant ultra-violet divergences cancel in supersymmetric 0.

models on locally symmetric spaces. This meansthat these models are field theories of a new type.They have ultraviolet divergences only at one-looporder, even though power counting and generalinvariance considerations permit higher-loop di-

22 846 1980 The American Physical Society

22 KAHLKR GEOMETRY AND THE RKÃORMALIXATION OF. . .

vergences. It is also possible that higher-loopdivergences cancel for Ricci-flat manifolds so thatsupersymmetric models on Bicci-flat spaces areentirely ultraviolet finite.

In this paper the general line of argument lead-ing to the conclusions above is developed. At oneor two points we must use detailed properties ofthe background-field expansion. These will bemore fully explained and detailed two-loop calcula-tions presented in another planned paper. "

LOCAL KAHLER GEOMETRY

The local geometry of Kahler manifolds is im-portant for our considerations. " An n-dimensionalcomplex manifold can always be regarded as a 2n-dimensional real manifold with coordinates z",where the index A runs through the n holomorphicindices a =1,2, . . . ,n and n antiholomorphic in-dices &=1,2, . .. ,n, and we have z =Z . A vectorV„ then has 2n components V and V-. All form-ulas of Riemannian geometry, such as definitionsof connection and curvature, are valid for complexmanifolds when expressed in terms of 2n-valuedindices &,&, C, ... . In particular there is a realline element given by ds'=gods"dz with g»=g». Reality is ensured by the conditions g ~

=g-t7 andg g=g~-. So far we have made no re-striction but have simply chosen to describe aneven-dimensional Riemannian manifold using com-plex coordinates.

Now we make the restriction that the manifoldis Hermitian. This means that there is a preferredclass of coordinate systems in which unmixedcomponents of the metric tensor vanish (g z=g-z= 0) leaving the line element in the Hermitian formds'=2go~dz dY~.

Finally we make the restriction to Kahler mani-folds by requiring that the Hermitian metric satis-fy (3). There are then simplifications in the stand-ard formulas for connections and curvature, andmany components vanish. For example, all com-ponents of the connection I'~~ vanish except thoseof the form I'&„=I'~-„. The nonvanishing compon-ents of the curvature tensor are of mixed formR ~„6 together with components obtained using theusual symmetries. There is also a new symmetry,Ro&~6 =B»~6, following from the usual cyclicityproperty.

We shall need the concept of a "KaMer tensor"which is a second-rank tensor T„~ with vanishingunmixed components and mixed components T ~satisfying the covariant condition &„T~g =e~T„&. AKahler tensor defines a closed two-form T=7.' ~d~ ndZ~, and is locally derivable from ascalar potential, i.e., T ~

=8 B~S(z,Z). The Rieeitensor R,~ =g "R», is a Kahler tensor; it satis-

fice R ~=O=R B-and R g=& &~lndet(g@).

GENERAL RENORMALIZABILITY CONSIDERATIONS

We consider the bosonic o model with classicalaction (1). Power counting implies that the counterterms which correspond to ultraviolet divergencesof one-particle-irreducible (1PI) Green's functionsare operators of dimension two. There are twoclasses of counterterms. The class of major in-terest in this paper are the invariant countertermswhich represent the "on-shell" divergences of thetheory and take the form

I„=-,' d'x 9, '8„&T,~ (4)

where T&& is a symmetric second-rank tensoralgebraically constructed from the curvature ten-sor and covariant derivatives of the curvaturetensor. For example, T,~(p) might involve R,, ,

R,»~,"",D,R» „D,R", or many other possi-bilities. In addition there are noncovariant counter-terms which vanish if the classical field equations

5I (Vy'+-r'„s, y's, y') =0

are used. Except at one-loop order we will notdiscuss such terms which are compensated by fieldredefinitions and are of secondary interest.

In a general Riemannian metric infinitely manyof the counterterm tensors T„- will not be of thesame functional form as the metric g,&(Q). There-fore strict renormalizability does not hold and onecannot usually give meaning to a nonlinear o modelon a given manifold M with metric g,&(Q). Thereseem to be the following possibilities for definingquantum field theories:

(a}Exceptional cases where at most finitelymany eounterterms differ in form from g,&(P).Such an exception occurs for spaces of constantcurvature where the curvature tensor takes theform R,», =c(g,„gz, -g«gz~) and covariant deriva-tives vanish. All tensors T,J algebraically con-structed from the curvature must be proportionalto g,&(P). Such is the ease for the O(n} modelswhich are well known"" to be renormalizablein the traditional sense.

(b) Cases where all but a finite number of eount-erterms vanish. This case is unlikely for bosonico models, but in this paper we show that it is verylikely for a broad class of supersymmetric o mod-els, due to cancellations of boson and fermion di-vergences required by supersymmetry.

(e) Henormalization-group interpretation. Usingdimensional regularization and minimal subtrac-tion, Friedan' has derived renormalization-groupequations of the form

848 LUIS ALVAREZ-GAUME AND DANIEL Z. FREEDMAN 22

(8)

Through two-loop order explicit calculations give

P,.~ — -Ro+~NR, ~, R "'"+0(I')

for bosonic a models. In these equations p, is therenormalization scale parameter and 0 is Planck'sconstant. Integration of these equations yields tra-jectories in the space of aIl Riemannian metrics,so that the initial geometry changes under renor-malization.

GENERAL STRATEGY

Our approach to the ultraviolet structure ofsupersymmetric a models is to combine the uni-versality of the background-field method' of cal-culation of invariant counterterms with the theo-em' connecting Kahler manifolds and N =2 models.In the background-field method applied to a bosonico model the field P'g) is split, P'g) =P,', g)+ 5 4'),into a term satisfying the classical field equations(5) and a remainder $'(r). The remainder doesnot transform simply under reparametrizations ofthe manifold, but using Riemannian normal co-ordinates it can be expressed" in terms of thequantum field g'g) which transforms as a con-travariant vector. After expansion of the action(1) as a functional Taylor series in P tr), the cal-culation of a relatively small number of Feynmangraphs will give the counterterms up to any givenorder in perturbation theory in the form

f„=-,' ~Id'x e,y's, y'(a,R,, +a@ „,„R,

+aP, ,R,.'+ ~ ~ ~ ), (8)

where the coefficients a, are each a series of polesin the dimensional-regularization parameter &

=d-2.The universality of this method is most import-

ant; to wit, the calculations can be done for themost general Riemannian manifold M with metric

g,.~(&f&) unspecified, so that the coefficientsa„a„a„... are universal and correct for allgeometries. In particular results for a generalRiemannian manifold apply immediately to o mod-els on general Kahler metrics g z (z, z) providedonly that we incorporate the algebraic and differ-ential restrictions implied by the fact that we areregarding the Riemannian case as the 2n-dimen-sional real description of an n-dimensional Kah-ler metric. For example, the counterterm (8)would become

I„= d'x B z B„~a,R z+2a2R -„~-,B&"'

+ a,R „R"g+~ ~ ~)-

with the same coefficients a„a„a„.. . as in (8).A further implication of the geometric structure

of (1) [and of (10) below) is that the tensors T,~

which can appear as counterterms at l-loop orderin perturbation theory can be characterized ascontractions of products of the curvature whichscale as T,&-A' 'To under the constant scalingof the metric g,&-A 'g, &. The reason for this isthat the l.-loop counterterm has 5' ' as a factor,and a change in k is equivalent to the scaling ofthe metric above. Thus the tensor B,, can only

appear as a one-loop counterterm, while the ten-sors quadratic in the curvature in (8) are possibleonly at two-loop order.

We now turn to supersymmetric models withclassical action

which is the supersymmetric extension' of (1) fora general Riemannian manifold. The covariantderivative is (D„g)~= & /+1"»8„$"g', and R&», isthe curvature tensor of the manifold.

The background-field method requires splittingof both Q'g) and g'(r). However, our generalargument requires only implicit knowledge of thebosonic counterterms of the form (4) becausesupersymmetry requires that these are accom-panied by fermion terms so that the whole struc-ture is a generalization of (10) with g,.&

replacedby T,&. Strictly speaking this is true only beforeauxiliary fields are eliminated.

Let us suppose that the background-field method

applied to the supersymmetric 0 model on a gener-al Riemannian manifold M gave the bosonic invar-iant counterterm

I„=-,' d'xB, B, ~ b,R,, +b,g,,R

+ b, R „,„R,"'"+ ~ ~ ~ ) . (11)

Universality then implies that upon restriction toa general Kahler manifold, one would obtain

I~t= d~x B ~ B ~ b~+~~+b2g

+2a,R.-„;,R-""+~ ~ ~ ) (12)

with the same coefficients b„b„b„.. . . Sincethe classical action would have N=2 supersymme-try, the bosonic counterterm must be accompaniedby fermion terms with overall K =2 supersymme-try. However, by Zumino's theorem' this canhappen only for tensors T ~ which are Kahlertensors. Any linear combination of the possibletensor counterterms for a given order in pertur-

22 KAHLER GEOMETRY AND THE RENORMALIXATION OF. . . 849

bation theory which does not have vanishing curlcannot occur, and its coefficient must vanish uni-versally both for general Kahler and Riemannianmanifold models. For example, the tensors R

&

and g &R are the only possible invariant one-loopcounterterms. The first has vanishing curl butthe second does not for general Kahler manifolds.Therefore the coefficient b, must vanish for allsupersymmetric o models.

Thus our general strategy is to show that theKahler condition is sufficiently restrictive thatmany possible counterterms allowed by Rieman-nian geometry cannot actually be present in super-symmetric models. However, there are severalqualifications. First, some tensors T,.J acquireunmixed components upon specialization to theKahler geometry. As discussed below some ofthese can be compensated by field redefinitions,but others cannot, causing complications in thegeneral arguments at three-loop order and beyond.Second, the interplay between Riemannian and

Kahler manifolds implied in our strategy surelyfails for Riemannian manifolds of odd dimension.However, the universality of the background-fieldmethod is so strong that the coefficient of anyparticular tensor counterterm is actually indepen-dent of the dimension of the manifold. Dimensiondependence could only arise from contractionsg'~g;& =n and these cannot occur as can be seen bycursory examination of the action obtained by thegeneral background-field expansion. "

Finally our strategy requires a regularizationprocedure which preserves N = 1 supersymmetryof Riemannian-models and automatically givesN =2 supersymmetry when restriction to a Kahlermetric is made. Dimensional' regularization can bemade consistent with N =1 supersymmetry, ' '" andN=2 supersymmetry is simply a matter of an ad-ditional SO(2) internal symmetry which is alsopreserved. Note that in general an'N =2 super-symmetric theory does not necessarily have SO(2)internal symmetry, but that this symmetry is pre-sent here because all Kahler-metric supersym-metric 0 models on two-dimensional space-timecan be obtained by reduction from four space-timedimensions. 2 It is part of our assumptions thatsupersymmetric dimensional regularization can beperformed.

ONE-LOOP ORDER

The one-loop counterterm of the general bosonica model is obtained from the background-fieldexpansion through second order in quantum fields0'g) The diver. gent graphs of the expanded La-grangian yield the one-loop counterterm for (1)and are easily calculated. The result is

~1"&= ir d'sit. ;(z, r)s„z.s,z~.2m& J

(14)

The noncovariant term cancels because g ~=0and 1"

&-0 in any Kahler manifold. This curious

result is confirmed by calculations in the O(n)model" where the field renormalization [when de-fined appropriately, see Eq. (A12) of Ref. 13]vanishes for n = 3 because the O(3) and CP' modelsare equivalent and CP' is a Kahler manifold.Actually because of the noncovariance of I'z~ thevanishing of the O(3) model field renormalizationis required only for real coordinates adapted tothe Kahler structure" such as the conformal co-ordinates of Ref. 15. The invariant term in (13)also agrees with O(n) model calculations. ""

We note also that the invariant counterterm van-ishes for any Ricci-flat geometry, whether Rie-mannian or Kahler.

We now come to the supersymmetric model (10).This action must also be expanded to second orderin quantum f ields. We are only interested in theeffect of fermions on the bosonic counterterms.Thus there are no fermionic background fields,and the P'(x) in (5) can be viewed as quantum fields.Only the quadratic term in (5) is relevant for one-loop effects. The fermion contribution is actuallyfinite which can be seen by referring the Fermifields to tangent frames on the manifold in whichcase the fermion Lagrangian becomes

& =l~0'r'[s 0'+~"(p)& p*0'] (15)

where &u (Q) is a spin connection on the manifoldand the product ~ (&f&)&~ &f&' transforms like a gaugepotential under rotations of the tangent frame.Thus one-loop fermion effects are equivalent tothose of a non-Abelian gauge field theory in twodimensions which are known to be ultraviolet fin-ite.

Therefore to one-loop order the invariant count-erterm of a Bose o model involves the Ricci tensorof the underlying manifold and is unchanged by thesupersymmetric addition of fermions. A nonco-

(13)

The invariant term has been known for some time'[and is obviously related to (7)). The noncovariantadditional term was found" using a slightly differ-ent method" so that &I"' actually gives all diver-gences, both on-shell and off-shell, of 1PI Green'sfunct'ions. Because it is proportional to the clas-sical field equation the noncovariant term repre-sents a field renormalization.

For a bosonic Kahler manifold the one-loopcounterterm obtained immediately from (13) is

850 LUIS ALVAREZ-GAUME AND DANIEL Z. FREEDMAN

variant field renormalization vanishes for Kahlermani folds.

TWO-LOOP ORDER

For a bosonic o model the two-'loop ultravioletcounterterm cannot vanish. This is obvious fromthe trace g'~P,.

&of the generalized P function (7).

For supersymmetric models the situation im-proves dramatically.

The candidate two-loop invariant bosonic count-erterm involves tensors T&& of weight one underthe scaling g,-& -A 'g, ,. After taking into accountrestrictions due to the Bianchi and Ricci identitiesone finds that the most general possible counter-term is

Tu DDP&y+2R&ayi—-R +2RlP"y

=DEED@(J+2[D,, D~]R~~. (17)

Thus the two-loop ultraviolet counterterm in asupersymmetric 0 model must be a universalmultiple of this tensor. Since T,&

vanishes forRicci-Qat and locally symmetric manifolds weconclude that these two classes of 0 models aretwo-loop ultraviolet finite. It is curious to notethat T~J coincides with the generalized I aplacianof Lichnerowicz" applied to second-rank symme-tric tensors.

EXPLICIT TWO-LOOP CALCULATIONS

Several calculations of two-loop diagrams havebeen performed to check the results of the general

+ b4B,-»&R~'+ b, B,~R&

+b~Dp )q+ b7D)D)R) (16)

which involves seven independent tensors. Thesame answer can also be obtained by inspectionof the background-field expansion of the action (10)through fourth order in quantum Bose fields g (x)and fermion fields g'(x) [and the expansion of thesupersymmetric extension of the one-loop counter-term (13) through second order in quantum fields].

We establish a lemma below which shows thatthe tensor D,.D,JV is compensated by a field re-definition and is not an on-shell divergence of thetheory. Upon restriction to a general Kahlermanifold, the first six tensors all have vanishingunmixed components. The remaining task is toexamine whether any linear combinations of thesix are Kahler tensors. The detailed calculationis presented in the Appendix. The result is thatone linear combination has vanishing curl and isan allowed counterterm. This is the tensor

argument of the previous section. There is a sim-ple form of the background-fieM method called themoving frame method" which is applicable to thebosonic O(n) models" and which can be extendedto include supersymmetrically coupled fermions.For the bosonic case one works with the I agran-gian and constraint

(16)

Given a classical solution P,', (x) one defines aset of n —1 unit vectors e,'(x) which are mutuallyorthogonal and orthogonal to P„(x). An arbitraryfield is then referred to this frame, i.e., we ex-pand P(x) =u,g„(X)+u,e,(x). The constraint im-plies uo ——(1 —u,u, )'~', which can be expanded toprovide a simple expansion in quantum fields u, (x).The calculation further simplifies due to consid-erations of SO(n —1) gauge invariance and powercounting. By this method we have reproduced theresults of Ref. 12 for the two-loop coupling-con-stant renormalization of the bosonic model andhave then shown that fermion effects cause thetwo-loop divergence to cancel.

We have also completed the two-loop background-field calculation of ultraviolet divergences for 0.

models on general Riemannian manifolds. We firstmake the simplification to a Ricci-Qat metric.Here we reproduce the result of Ref. 6 for thebosonic case and then show that fermions cancelthe divergence found there. For a general super-symmetric o model we find an additional diver-gence with tensor form given by (17) and a coefficient which is a second-order pole in &.In a standard renormalizable field theory two-loopdouble poles in counterterms are determined fromone-loop single poles by renormalization-grouppole equations. " One would also expect such poleequations in the generalized renormalizableframework' for nonlinear o models. Indeed a sim-plified analysis suggests that the tensor T,.& is theexact two-loop divergence predicted from the one-loop counterterm (13) by the generalized poleequations. We hope to discuss this situation fur-ther in Ref. 10.

A LEMMA

We now show that any invariant counterterm ofthe form

where Vz(P) is a vector field on the manifold canbe canceled by a field redefinition of the formQ" (x) = Q '(x) —(1/2&) V'(P (x)). The infinitesimaleffect of this redefinition on the initial bosonicaction (1) is simply

22 KAH I.ER GEOM ETR Y AN D THE RENORMALIZATION OF. . . 851

fr =-—~"d'~(B„V's„y~g„+-,'s„y ~s„y~v's, g„)

d2x 9 Q'8 P~D VI

which cancels (19). We have not studied the ef-fect of this field redefinition on fermionic termsof the supersymmetric action. However, it seemsclear from the superspace formulation' that afermionic field redefinition of the form P"(x)= P 4) —(&/2&)DP'($(r))g~(x) will also be required.'Terms quadratic in V and effects of the field re-definition on the one-loop counterterm (13) mustalso be considered. It appears that these inducehigher-order poles in f which must cancel consis-tently due to the renormalizable structure. Thislemma is easily interpreted in the language of therenormalization group. We are interested' onlyin changes of the metric g,z(Q) under renormaliza-tion which are actually changes of the geometryand therefore must exclude diffeomorphisms ofg,&(Q). A field redefinition is a poor man's diffeo-morphism.

Note that this lemma applies to any countertermof the form D,D&S where S is a scalar. When re-stricted to Kahler manifoMs such tensors haveboth mixed components 8 8~S and unmixed com-ponents D,BP. A Kahler tensor has mixed com-ponents which can be represented as 8 8&S butunmixed components vanish. Such a tensor cannotbe canceled by a field redefinition.

HIGHER-LOOP ORDER

Let us consider tensors T,z which are algebrai-cally constructed from products of curvature ten-sors with no derivatives. On a Kahler manifoldthe curvature tensor has an equal number of barredand unbarred indices, so that tensors T» con-structed from them have vanishing unmixed com-ponents. Tensors with nonvanishing unmixed com-ponents can appear when derivatives of curvatureare included and in general they cannot be com-pensated by field redefinitions. For example, thetensor D,R» „D&Rk™has this property, and is acandidate three-loop counterterm for nonlineara models. At present such tensors cannot be ex-cluded by N =2 supersymmetry since Zumino'stheorem requires a Hermitian metric by hypothe-sis. However, such tensors could be excludedby a stronger form of the theorem to the effectthat N=2 supersymmetry is not possible in a mani-fold which is genuinely non-Hermitian, i.e., notdiffeomorphic to a Kahler manifold. We think thatthis extension is plausible and hope to pursue itelsewhere. Independently of this, however, thetensors in question vanish for locally symmetric

spaces and it may be possible to establish higher-loop ultraviolet finiteness for this class of super-symmetric a model without confronting the prob-lem of~on-Hermiticity.

The key problem is still the question of Kahlertensors, that is, tensors of the form

—(R ~ ~ I Dft e ~ e fte ~ ~ ) (21)

CONCLUSIONS

We have demonstrated that there is a powerfulconnection between the ultraviolet properties ofsupersymmetric o models and Kahler geometry.This connection is not exhausted by present re-sults but can probably be pushed further to derivestronger results on higher-loop finiteness. Infour-dimensional field theories it is well knownthat there is improvement of the ultraviolet diver-gence structure in global supersymmetry and sup-ergravity. What is discussed here is the analog

with all indices but two contracted, and with T~=o and 8„T~& =8oT„~. The discussion at the two-loop level suggests that the vanishing curl condi-tion is very restrictive but does not eliminate allpossible tensors. At least those required by therenormalization-group pole equations are al-lowed. It is also clear that the proof by exhaustionwhich eliminated all but one two-loop tensor mustbe replaced by a more general argument. Towardthis end it appears useful to discuss the Kahlertensor question in terms of the potential S forwhich T z

—8 8&S. If S is not a true scalar or onlylocally defined on the manifold, as in the case ofthe Ricci tensor 8 ~

=& &~ lndetg„-, or the metricg ~

=8 8~+, where I" is the Kahler potential, thetensor T ~ defines a 2-form which is closed butnot exact. It may be possible to use cohomologytheory to show that the Ricci and metric tensorsare the only closed inexact 2-forms algebraicallyconstructed from curvature on a general Kahlermanifold. If so, one can restrict one's attentionto tensors for which S is a true scalar constructedfrom curvature as in the case of the two-looptensor T,~ for which T~„=o and TI~=8O8~R. Allsuch tensors vanish for locally symmetric spaces.One might also hope that further geometrical argu-ment can establish that the only such tensors which

appear are those required by the renormalization-group pole equations. In this case the tensors willvanish on Ricci-Qat manifolds.

The discussion of this section is speculative, butwe would summarize it by saying that it appearsprobable that higher-loop counterterms can beexcluded by general argument for supersymmetrico models on locally symmetric manifolds and pos-sibly excluded for Ricci-flat manifolds also.

852 LUIS ALVAH, EZ-GAUME AND DANIEL Z. FREEDMAN 22

in two-dimensional nonlinear theories. It is sur-prising that there is no improvement due to super-symmetry in one-loop order, but remarkable con-sequences in higher order.

this tensor alone, independent of linear combina-tions with others. Taking the trace of the curl andusing the Bianchi identity, we find

ACKNOWLEDGMENTS

We are very grateful to D. Friedan for severaldiscussions of his work prior to publication andfor useful suggestions and also grateful to S. Mukhifor help with the two-loop calculations. We thankDr. P. Breitenlohner and Dr. T. Curtright whopointed out an error in the earlier version ofthis manuscript. This work was supported in partby NSF Contract No. PHY 79-06376.

APPENDIX

In this appendix we will show in detail that thetensor T,.z of (1V) is the only linear combinationof the first six tensors in the two-loop counterterm(16) which is a Kahler tensor. We start with themanifest Kahler tensor

T g=e 8BR

=28 8~g"'8„88 inde@.

We move the 8 and 8B derivatives to the rightusing the relations

1~ usa

+1+u ~ u ' (1+u u)"u uauyI' ~-6 Buy+6 uB- 1+u u'

(As)

(As)The second term is nonvanishing for general Ricci-flat Kahler manifolds, so that T"' is not a Kahlertensor.

The most straightforward way to show that thereare no curl-free linear combinations among theremaining four tensors in (A4) is to compute thesetensors for one explicit Kahler metric. With badluck one might choose a particular metric wherethese are allowed linear combinations but this doesnot turn out to be the case. A suitable metric isdefined by the Kahler potential &(u, u) =e"'~ withu u=Z, u u™,n, P=1, . . . , N andu =8„zu~, u~= 5+u~. For this manifold

g ~=e"'"(8 e+u,u~),

uuBg oB eu I g+B

1+u u

BBg = gy5 ye 6

e68By g BgyB &

eg =-g I (ge(A2) By6 B y6+ y B6

which are correct for general Kahler manifolds. "After using R z

-—8 8& lndetg, we find

(As)

which is the restriction of (1V) to a Kahler mani-fold.

To show that T,&of (17) is the only curl-free

linear combination, we study the five independenttensors

1+u u ~5 "+ "' ~ (1+u u)' '

Each tensor of (A4) hah the generic form

where x =u u, and curls take the form

C(t)( ) f(f)( ) ('&( )dx

(A7)

(AS)

(1) y6g™T B=R y6-RB

T ~B, ROBR

(3)T ~B ~gai BR

T 'B =R -y6BR",

T~5)B

=R -Ry-

(A4)

obtained by restriction of the tensors of (16) to aKahler manifold. Since only T' ' contributes forRicci-flat Kahler manifolds (R ~ =0) we can study

Using (A6) we explicitly compute the functionsf'"(x) andg'"(x') for i=2, 3, 4, 5 and then computeC"'(t') which appears in the curl. We find thateach C'"(x) can be written as C'"(x') =e "P'"(y)where the P"'(p) are polynomials in the variabley = (1+x) ', three polynomials being of 5th orderand one of 7th order. One readi'. y tests that thesepolynomials are linearly independent which showsthat no linear combination of T' ', T'", T' ', orT"' is a Kahler tensor.

22 KAHLER GEOMETRY AND THE RENORMALIZATION OF. . . 853

D. Z. Freedman and P. K. Townsend, Stony Brook Re-port No. ITP-SB-80-25 (unpublished).B. Zumino, Phys. Lett. 87B, 203 (1979).

3T. Curtright and D. Z, . Freedman, Phys. Lett. 90B, 71(1980).

4L. Avarez-Gaumeand D. Z. Freedman, Phys. Lett. B(to be published).

~J. Honerkamp, Nucl. Phys. B36, 130 (1972); G. Eckerand J.Honerkamp ibid. B35, 481 (1971).

D. Friedan (private communication).P. di Vecchia and S. Ferrara, Nucl. Phys. B130, 93(1977); E. Witten, Phys. Rev. D 16, 2991 (1977).

E. Cremmer and J. Scherk, Phys. Lett. 74B, 341 (1978);A. D'Adda, P. di Vecchia, and M. Luscher, Nucl.Phys. B152, 125 {1979).

9A. M. Perelomov, Commun. Math. Phys. 63, 237 (1978).L. Alvarez-Gaumy, S. Mukhi, and D. Z. Freedman(in preparation).

~~See Chapter IX of S. Kobayashi and K. Nomizu, Eoun-dations of Differential Geometry (Interscience, NewYork, 1969),Vol. II; see also section 1 of M. Ko, E. T.Newman, and R. Penrose, J.Math. Phys. 18, 58(1977).E. Brezin and J. Zinn-Justin, Phys. Rev. Lett. 36, 691(1976); Phys. Rev. B 14, 3110 (1976).W. A. Bardeen, B, W. Lee, and R. E. Shrock, Phys.

Rev. D 14, 985 (1976).W. Siegel, Phys. Lett. 84B, 193 (1979); M. T. Grisaru,M. Rocek, and W. Siegel, Nucl. Phys. B159, 429(1979}.

'

~~D. M. Capper, D. R. T. Jones, and P. van Nieuwen-huizen, NucI. . Phys. B (to be published).

~ G. 't Hooft, Nucl. Phys. B62, 444 (1973); G. 't Hooftand M. Veltman, Ann. Inst. H. Poincarb 20, 69 (1974).

YIf an even-dimensional Rieme»iari manifold is reallya Kahler manifold in disguise, then there is a pre-ferred class of real coordinate charts P, . .., @".,P"', .. . , P ", such that when complex coordinates aredefined by z =Q +i/"' and s0'= P -iP"' for 0'=1,... , n, the transformed metric g~g ls Hermitian andsatisfies (3). We say that such coordinate charts areadapted to the Kahler structure.A. Lichnerowicz, Inst. des Hautes Etudes Scientifique,Publications Mathematiques 10 {1961). See alsoG. Gibbons and M. J.Perry, Nucl. Phys. B146, 90(1971).

~9A. D'Adda, M. Luscher, and P, di Vecchia, lecture- atthe Winter Advanced Study Institute in Les Houches,France, 1978 (unpublished).A. M. Polyakov (private communication).

~~G. 't Hooft, Nucl. Phys. B61, 455 (1973).