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Enrico Fermi Institute, University of Chicago

arXiv:hep-th/9108020v1 26 Aug 1991EFI-91-30STRING THEORY AND THEDONALDSON POLYNOMIALJeffrey A. HarveyEnrico Fermi Institute, University of Chicago5640 Ellis Avenue, Chicago, IL 60637Internet: harvey@curie.uchicago.eduAndrew StromingerDepartment of Physics, University of CaliforniaSanta Barbara, CA 93106Bitnet: andy@voodooABSTRACT: It is shown that the scattering of spacetime axions with fivebrane solitonsof heterotic string theory at zero momentum is proportional to the Donaldson polynomial.July, 1991

I. INTRODUCTIONA p-brane (i.e. an extended object with a p + 1-dimensional worldvolume) naturallyacts as a source of a p + 2 form field strength F via the relation∇MFMN1· · ·Np+1 = q∆N1···Np+1(div)where ∆is the p-brane volume-form times a transverse δ-function on the p-brane.

In ddimensions they can therefore carry a chargeq =ZΣd−p−2 ∗F. (qcharge)where the integral is over a d−p−2 dimensional hypersurface at spatial infinity.

The dualchargeg =ZΣp+2F(gcharge)can be carried by a d −p −4 brane. A straightforward generalization [1] of Dirac’s orig-inal argument implies that quantum mechanically the charges must obey a quantizationcondition of the formqg = n,(quant)just as for the special case of electric and magnetic charges in d = 4.

In particular, stringsin ten dimensions are dual, in the Dirac sense, to fivebranes.Thus fivebranes are themagnetic monopoles of string theory.In [2,3] it was shown that heterotic string theory admits exact fivebrane soliton so-lutions. The core of the fivebrane consists of an ordinary Yang-Mills instanton.Thusheterotic strings are dual, in the Dirac sense, to Yang-Mills instantons.This simple connection between Yang-Mills instantons and heterotic string theoryraises many possibilities.

On the one hand, heterotic string theory might be used as a2

tool to study the rich mathematical structure of Yang-Mills instantons, or to suggest inter-esting generalizations. On the other hand, the mathematical results of Donaldson [4] andothers on the construction of new smooth invariants for four-manifolds may have directimplications for non-perturbative semi-classical heterotic string theory.In this paper this connection is elucidated as follows.

We consider N parallel fivebraneson the manifold M6×X where M6 is six-dimensional Minkowski space and X is the four-manifold transverse to the fivebranes. (The consistency of string theory requires c1(X)≥0,though this condition might conceivably be relaxed in the present context by allowingsingularities along the divisor of c1.) The quantum ground states of this system are foundto be cohomology classes on the N-instanton moduli space MN(X).

Transitions amongthese ground states may be induced by scattering with a zero-momentum axion. Suchaxions are characterized by a harmonic two-form or an element of H2(X), and the S-matrix then defines a map H2(X)→H2(MN).

This map turns out to be precisely theDonaldson map.The fact that the scattering is a map between cohomology classes isultimately a consequence of zero-momentum spacetime supersymmetry. Multiple axionscattering is given by the intersection numbers on MN of these elements of H2(MN),which is the Donaldson polynomial.This representation of the Donaldson map as a string S-matrix element leads to anapparently new geometrical interpretation of Donaldson theory.

For any K¨ahler manifoldX there is a K¨ahler geometry on H2(X, C) × MN(X) with K¨ahler potential defined byK = 12ZXE ∧J −lnZXJ ∧J(why)where J is the K¨ahler form on X and E is a solution oftrF ∧F = i∂¯∂E. (ffe)E is an analog of the Chern-Simons form for K¨ahler geometry.

The Donaldson map is3

then given by a mixed component of the Christoffel connection, computed as the thirdderivative of K.It is noteworthy that the final expressions we derive for the Donaldson map and poly-nomial are similar to those given by Witten [5]. Indeed, the embedding of four-dimensionalYang-Mills instantons into ten-dimensional string theory given by fivebrane solitons seemsto produce a structure of zero-momentum fields and symmetries similar, if not identical,to that of Witten’s topological Yang-Mills theory.Though we have not done so, it ispossible that the complete structure of topological Yang-Mills theory can be derived fromzero-momentum string theory in the soliton sector.

This is perhaps in contrast to theusual notion [5] that topological field theory is relevant to a short-distance phase of stringtheory.We work only to leading order in α′ in this paper. An interesting question, whichwe do not address, is whether or not higher-order or non-perturbative corrections providea natural deformation of the Donaldson polynomial analogous to the deformation of thecohomology ring provided by string theory.This paper is organized as follows.

In Section II we establish our notation and reviewsome properties of instanton moduli space. The collective coordinate expansion leading tothe low-energy effective action is derived in some detail, and is then used to characterize theN-fivebrane ground states.

In Section III.A the collective coordinate expansion is continuedto reveal the Donaldson map as a subleading term in the low-energy effective action.Section III.B gives an alternate derivation of the Donaldson map using supersymmetryand K¨ahler geometry, and derives (why). In Section III.C we discuss the representationof the Donaldson map as a period of the second Chern class which may be relevant inthe present context.

In Section III.D the string scattering amplitude which measures theDonaldson polynomial is described. We conclude with discussion in Section IV.4

II. INSTANTON MODULI SPACE AND THECOLLECTIVE COORDINATE EXPANSIONThe derivation of the Donaldson map and polynomial from ten-dimensional stringtheory is straightforward though somewhat involved.

We begin with the action describingthe low-energy limit of heterotic string theory:S10 =1α′ 4Zd10x√−g e−2φR + 4∇Mφ∇Mφ−13HMNPHMNP −α′trFMNF MN−¯ψM ΓMNP ∇NψP + 2¯λΓMN∇MψN + ¯λ ΓM∇Mλ −2α′ tr¯χ ΓMDMχ−α′tr ¯χ ΓMΓNP FNP (ψM + 16ΓMλ)+2 ¯ψN ΓMΓNλ∇Mφ −2 ¯ψMΓMψN∇Nφ+ 112HMNP(2α′tr¯χΓMNP χ −¯λΓMNP λ + ¯ψS Γ[SΓMNP ΓT ]ψT+ 2 ¯ψS ΓSMNPλ) + · · ·(sten)where “+· · ·” indicates four-fermi as well as higher-order α′ corrections, H = dB+α′ω3L−α′ω3Y , ω3L and ω3Y are the Lorentz and Yang-Mills Chern Simons three-forms respectively,and “tr” is 130 the trace in the adjoint representation of E8 × E8 or SO(32). A supersym-metric solution of the equations of motion following from (sten) is one for which thereexists at least one Majorana-Weyl spinor η obeyingδψM = ∇Mη −14HMNPΓNP η = 0δλ = 16HMNP ΓMNPη −∇MφΓMη = 0δχ = −14FMNΓMNη = 0.

(susy)The general solution of this form on X×M6, where X is a K¨ahler manifold with c1≥0 andM6 is flat six dimensional Minkowski space, was found in [6]. The gauge field may be anyself-dual connection on X:Fµν = 12ǫµν ρσ Fρσ(sdual)where µ, ν are indices tangent to X.

Let ˆg be a Ricci flat K¨ahler metric on X. Then the5

dilaton is the solution ofˆ e2φ = α′(tr ˆRµν ˆRµν −tr Fµν F µν)(deom)and the metric and axion aregµν =e2φ ˆgµνHµνλ = −ǫµνλρ∇ρφgab =ηab(metax)where a, b are tangent to M6. Special cases of this general solution are discussed in [2,7,3].In [8] it was argued for X = R4 that such solutions are in one to one correspondence withexact solutions of heterotic string theory.

For c2(F) = N, this may be viewed as a config-uration of N fivebranes transverse to X. (It may also be viewed as a “compactification”to six dimensions, though X need not be compact.) Since (given the metric on X) there isone unique solution for every self-dual Yang-Mills connection (sdual), the space of staticN fivebrane solutions is identical to the moduli space MN of N-instanton configurationson X.For c1(X) ≥0 and c2(R) ≥c2(F), the metrics g of (metax) are geodesicaly complete,but may be non-compact.

If c1(X) > 0, there are geodesically complete but non-compactRicci-flat metrics with bounded curvature [9]. This may be viewed as a singular metricon X or a non-singular metric on X minus the divisor of c1.

There appears to be nospecial difficulty in defining string propagation on such geometries (though they wouldnot be suitable for Kaluza-Klein compactification). Singularities may also arise in solving(deom).

If c2(R) > c2(F), the singularities are of the type studied in [3] and again produceno difficulties.On the other hand if c1 < 0 or c2(R) < c2(F), the metric in (metax) may have realcurvature singularities, which could potentially render string theory ill-defined. More work6

must be done before our methods can be used to directly study these cases, but the validityof our final formulae for all K¨ahler X suggests that it may be possible to do so. Possibleapproaches would be to consider the more general case of time-dependent metrics, or toconsider the (eight-dimensional) cotangent bundle of X which has c1 = 0.The solutions of (sdual),(deom) have bosonic zero modes tangent to MN.

To leadingorder in α′ these zero modes involve only the gauge field and will be denoted δiAµ(x),where i = 1,...,m, m ≡dim(MN) and x is a coordinate on X. The zero modes obey thelinearized self-duality equationD[µδiAν] = 12ǫµν ρσ DρδiAσ.

(aeom)For gauge groups larger than SU(2) or for metrics on X which are not “generic” thegauge connection will in general be reducible (there exist non-trivial solutions of Dφ = 0).This leads to orbifold singularities in MN. In what follows we will restrict ourselves toirreducible connections and ignore such subtleties.If Zi is a coordinate on MN, and A0µ(x, Z) a family of self-dual connections, the zeromodes are given byδiAµ = ∂iA0µ −Dµǫi(zmode)where ǫi(x, Z) are arbitrary gauge parameters and ∂i =∂∂Zi.

It is convenient to fix ǫi byrequiringDµδiAµ = 0(choice)so that the zero modes are orthogonal to fluctuations of the gauge field obtained by gaugetransformations. The gauge parameter ǫi then defines a natural gauge connection on MNwith covariant derivativesi = ∂i + [ǫi,](sconn)which has the property[si, Dµ] = δiAµ.

(sd)7

The Jacobi identity for si, Dµ and Dν impliessiFµν = 2D[µδiAν]. (sdd)The Jacobi identity for si, sj and Dµ impliesDµφij = −2s[iδj]Aµ(ssd)whereφij = [si, sj](phi)is the curvature associated to si.

These relations will be useful shortly.A natural metric G on MN is induced from the metric g on X:Gij =ZXd4x√g e−2φgµνtr(δiAµδjAν). (modmet)In addition there is a complex structure J on MN induced from the complex structure Jon X:Ji j =ZXd4x√g e−2φJµ ν tr(δiAλδkAν)gµλGkj.

(modj)It is easily seen that the zero modes are related byJi j δjAµ = −Jµ ν δiAν. (aa)In addition to bosonic zero modes, there are fermionic zero modes of the superpartnerχ of AM.

These zero modes are paired with the bosonic zero modes by the unbrokensupersymmetry [10] and are given byχi = δiAµΓµǫ′(fmode)where ǫ′ is the four-dimensional chiral spinor obeyingJµν = −i ǫ′†Γµν ǫ′,JµνΓνǫ′ = iΓµǫ′,ǫ†ǫ = 1. (ee)It is easy to check, using (choice) and (aeom), that ΓµDµχi = 0.8

Equation (fmode) would appear to give m zero modes, where m is the dimension ofMN, but we know from the index theorem that these are not linearly independent. Using(ee) and (aa) one findsJ ji χj = iχi.

(xx)This gives m2 independent zero modes, as implied by the index theorem.The low-energy dynamics of N fivebranes in X×M6 is best described by an effectiveaction Seff. This action can be derived by a (super) collective coordinate expansion whichbeginsAµ(x, σ) = A0µ(x, Z(σ)) + · · ·χ(x, σ) = λi(σ)χi(x, Z(σ)) + · · ·(exp)where (x, σ) is a coordinate on X×M6 and the bosonic (fermionic) collective coordinatesZi (λi) are dynamical fields on the soliton worldbrane.

λi = λi+ + λi−is a doublet ofsix-dimensional Weyl fermions obeying λj = iJkjλk. Under SO(5, 1) worldbrane Lorentztransformations, the λi’s transform into one another.

It is possible to assemble the λi’sinto m2 six-dimensional symplectic Majorana-Weyl spinors transforming covariantly underSO(5, 1). However SO(5,1) covariance is not necessary for our purposes, and the super-symmetric SO(5, 1) covariant formulation introduces a number of extraneous complicationswhich obscure the connection with Donaldson theory.

Our expressions will have manifestinvariance under two-dimensional super-Poincare transformations which we take to act inthe σ0, σ1 plane. The subscripts on λi± denote the corresponding two-dimensional chiral-ity.

In accord with this and as a further simplification, we henceforth restrict λi and Zi todepend only on σ0 and σ1.The effective action Seffcan be expanded in powers of inverse length. Taking Zi tobe dimensionless and λi to have dimensions of (length)−12 this is an expansion in the9

parameter n = n∂+ nf/2 with n∂the number of σ derivatives and nf the number offermion fields. The expansion (exp) solves the spacetime equations of motion to ordern = 0, while the leading terms in Seffare n = 2.

To have a consistent action we must stillsolve the spacetime equations to order n = 1. This requires that the component of thegauge field tangent to the worldbrane acquires the termAa = ∇aZiǫi −12φij¯λiΓaλj(atan)with φij given by (phi).The leading order worldbrane action may now be derived by substitution of the expan-sion (exp) (atan) of A and χ into the ten dimensional action (sten) and integration overX, the transverse space.

UsingFaµ = ∇aZiδiAµ −s[iδj]Aµ¯λiΓaλj(fza)one has the bosonic termSbeff= −2α′3Zd4x√ge−2φZd6σ tr (δiAµδjAν) gµν∇aZi∇aZj= −2α′3Zd6σGij∇aZi∇aZj. (sbwb)Including the fermionic terms gives the d = 6 supersymmetric sigma model with targetspace MN:Seff= −2α′3Zd6σGij∇aZi∇aZj + 2¯λiΓa(∇aλj + ∇aZkΓjklλl)+ (fermi)4.

(swb)Because we have maintained only an SO(1, 1) subgroup of SO(5, 1) as a manifest symmetryof (swb), only two of the eight supersymmetries are manifest.For values of Zi corresponding to N widely separated instantons, (swb) is approximatedby N separate terms describing the dynamics of each of the N fivebranes. The full action(swb) includes additional fivebrane interaction terms.10

Classically, there is one static ground state for each point ZiǫMN. However quantummechanical groundstates involve a superposition over Zi eigenstates.

As explained by Wit-ten [11] the supercharges of the supersymmetric sigma model (swb) act at zero momentumas the exterior derivative on the target space MN, and the general supersymmetric groundstate can be written in the form|Os⟩= Os|0⟩Os = Osi1· · ·ip(Z)ψi1· · ·ψip(gstate)where Osi1· · ·ip is a harmonic form on MN, ψi = Reλi+ +iReλi−and the state |0⟩is chosenso that(ψi)∗|0⟩= 0. (vacuum)In summary, the low-energy dynamics of N-fivebranes is described by a supersymmetricsigma-model with target space MN and the ground states of this system are cohomologyclasses on MN.III.THE DONALDSON MAP AND POLYNOMIALIn addition to the leading term (swb) in Seff, there are a number of terms representinginteractions between spacetime fields which are not localized on the fivebrane and thelocalized worldbrane fields appearing in Seff.

This corresponds to the fact that the state ofthe fivebrane can be perturbed by scattering with spacetime fields. For the special limitingcase of zero-momentum spacetime fields, energy conservation implies that scattering canonly induce transitions among the groundstates.

Zero momentum spacetime axions arecharacterized by harmonic forms on X, so axion scattering is a map involving H(X) andH(MN(X)). This strongly suggests that the scattering should be given by the Donaldson11

map. In the following two subsections we demonstrate that this is indeed the case bytwo separate methods.Subsection (A) contains a straightforward continuation of thecollective coordinate expansion.

In subsection (B), it is observed that the Donaldson maphas a geometric interpretation as a certain connection coefficient derivable from a K¨ahlerpotential. It’s form is then deduced in a few lines using supersymmetry.A.

Derivation by Collective Coordinate ExpansionConsider the interaction of a low-momentum spacetime axion with the worldbranefermions. Other spacetime fields can be treated in an analogous fashion.

Spacetime axionsare described by the potentialBµν = Y (σ)Tµν(axion)where T is a harmonic two form on X, and Y depends only on σ0, σ1. The ten-dimensionalcouplingL′int =12α′3e−2φ∂MBNP tr¯χΓMNP χ(hxx)appearing in (sten) descends to a coupling in Seffbetween one spacetime axion and twoworldbrane fermions.

Substituting (axion) and (exp) into (hxx) and integrating out thezero mode wave function one findsL′int = 12α′3Zd4x√ge−2φTµν tr (¯λiχ†i ΓaΓµνλjχj)∇aY= 2α′3O′ij¯λiΓaλj∇aY(what)whereO′ij(T)≡14Zd4x√g e−2φχ†iΓµνTµνχj. (mapone)The lambda bilinear appearing in (what) can be seen, using (vacuum), to be equivalentto ψiψj when acting on a vacuum state.

Substituting the formula (fmode) for χi one has,12

after some algebraO′ij(T) =ZXtr(δiA∧δjA)∧T+,(aat)where T+ is the self-dual part of T. It is easily checked that O′ij is not closed and so doesnot represent a cohomology class on MN.This is remedied by the observation that (hxx) is not the only term in S10 which givesrise to the coupling of a spacetime axion to worldbrane fermions. Because of the bilinearterm in the expansion (atan) for Aa, such couplings also arise from the ten-dimensionalterm2α′3e−2φ∂MBNPωMNP3Y.

(bomega)From the expansion for Aa, the relevant term in ω3Y isω3Yaµν = −12tr (φij¯λiΓaλjFµν). (phif)This formula may then be used to reduce (bomega) to a coupling in Seff.

The result may beadded to (what) to give the total coupling of a single spacetime axion of the form (axion)to two worldbrane fermions:Lint = 2α′3Oij¯λiΓaλj∇aY(coup)whereOij(T) =ZXtr(δiA∧δjA −φijF)∧T+(mfinal)(mfinal) has several important properties. The first is that O is closed on MN:∂[iOjk] = −ZXtr(Dφ[ij∧δk]A + φ[ijsk]F)∧T+ = 0(dm)upon integration by parts on X. Secondly, if T+ is trivial in H2(X) so that T+ = dU onehasOij(dU) =ZXtr(Dδ[iA∧δj]A + s[iδj]A∧F)∧U13

= −2∂[iZXtr (δj]A∧F)∧U(tdu)i.e. the image of an exact form on X is an exact form on MN.

Thus (mfinal) gives a mapfrom the cohomology of X into the cohomology of MN. Using Poincare duality (mfinal)may be written:Oij(Σ) =ZΣtr (δiA∧δjA −φijF)(pdual)where Σ is the surface Poincare dual to T+.

This is a standard expression [4] for theDonaldson map from H2(X)→H2(MN(X)) in terms of differential forms, and is identicalto that derived in the context of topological quantum field theory by Witten [5].B. Derivation from K¨ahler Geometry.In this subsection we will provide an alternate derivation of (coup) which is less direct,but shorter and provides some geometrical insight.

For these purposes it is convenient toview the solution (sdual)–(metax) not as N fivebranes on X, but as a “compactification”from ten to six dimensions. The low-energy action then contains, in addition to Zi, complexmassless moduli fields Y α that parameterize the complexified K¨ahler cone ( a subset ofH2(X, C)).

The imaginary part of Y α is the axion associated to the harmonic two formTα on X (The α index was suppressed in the previous subsection. ).Six-dimensionalsupersymmetry then implies that the metric appearing in the kinetic term for the modulifields is K¨ahler, or equivalently in complex coordinatesJI ¯J = i∂I∂¯JK.

(jform)To give an expression for K, we note that on a K¨ahler manifold a closed (p, p) formHp,p is locally the curl of a 2p −1 form:Hp,p = dG2p−1 = (∂+ ¯∂)(Gp,p−1 + Gp−1,p). (gcurl)14

Since the left-hand side of (gcurl) is of type (p, p),∂Gp,p−1 = ¯∂Gp−1,p = 0,(igiveup)so that locally Gp,p−1 = ∂Fp−1,p−1. We conclude that locally a closed (p, p) form canalways be written in the formHp,p = i∂¯∂Fp−1,p−1.

(ddbar)F is real if H is, and is determined up to a closed (p −1, p −1) form.K is then given byK = 12ZXE ∧J −lnZXJ ∧J(whytwo)where J is the K¨ahler form on X and E is a solution oftrF ∧F = i∂¯∂E. (ffetwo)E is related to the two-dimensional WZW action and can not be simply expressed as afunction of A.

A formula for K as a conformal field theory correlation function is given in[12].The second variation of K can be computed by noting that∂I∂¯JtrF ∧F = 2¯∂∂tr(δIA∧δ ¯JA −φI ¯JF). (ddbarff)This determines the second variation of E up to a closed two-form on X times a closedtwo-form on MN.

The ambiguity in the definition of E may thus be fixed so thati∂I∂¯JK = −ZXtr(δIA∧δ ¯JA −φI ¯JF)∧J=Z √g Jµν tr δIAµδ ¯JAν=JI ¯J(dk)15

where in the last line we have used J·F = 0.The coupling of Y to two λ′’s is thendetermined by supersymmetry to be proportional to the mixed Christoffel connection (asin (swb) with an index lowered) on H2(X, C) × MN:Lint = −2iα′3 ∇aY α ¯λ′ ¯JΓaλ′IΓ ¯JIα(lint)In K¨ahler geometry, the Christoffel connection is given byΓ ¯JIα = K, ¯JIα . (ok)Differentiating (dk) one more time and using ∂αJ = Tα we easily recover the formula(mfinal) of the previous section−iΓ ¯JIα =ZXtr (δIA∧δ ¯JA −φI ¯JF)∧Tα = O ¯JIα(recov)except for the absence of a projection on to the self-dual part of Tα.

This difference canbe accounted for if λ′ is related to λ of the previous section by the field redefinitionλ′ = e−iXαY αλ(fielddef)where Xα ≡RTα ∧J/RJ ∧J.C. The Donaldson Map as the Second Chern ClassIt is known [4, see also 13,14] that O can be written as integrals of trF2 for a certaincurvature F. The fact that O couples to axions then strongly suggests that the observationsin this paper are connected with the structure of anomalies in string theory.

While we donot understand this connection, in the hope that it might be understood later we recordhere this representation of O. Introduce a connection D (on the universal bundle overMN) byD = dZisi + dxµDµ.

(bigd)Then the associated curvature F = D2 has componentsFij =φij16

Fiµ =δiAµFµν =Fµν. (bigf)Now consider the integral c2(Σ) of the second Chern class of F over a four surface Σ inMN × Xc2(Σ) =18π2ZΣtr F∧F(chern)If Σ is a product of a two surface ΣM in MN with a two surface ΣX in X one findsc2(ΣM×ΣX) = −18π2ZΣMOij(ΣX)dZi∧dZj.(cm)i.e.

the Donaldson map H2(X) →H2(MN) is a period of the second Chern class. Asimilar result holds for the maps Hα(X)→H4−α(MN).D.

The Donaldson PolynomialA physical realization of the Donaldson polynomial may be obtained by consideringmultiple axion scattering. Let |m⟩be the ground state corresponding to the top rank formon MN:|m⟩= ǫi1· · ·im λi1· · ·λim|0⟩.

(top)The amplitude for scattering p axions associated to the classes T1· · ·Tp offthe state |0⟩and winding up in the state |m⟩is proportional toA(T1, · · ·Tp) = ⟨m|OT1· · ·OTp|0⟩. (multi)It is easily seen that this reduces toA(T1, · · ·, Tp) =ZMNO(T1)∧· · ·∧O(Tp)(poly)which is the Donaldson polynomial.While our derivation from string theory of (mfinal,poly) was only valid for c1(X) ≥0,it is known [4] that (mfinal) and (poly) are representations of the Donaldson map andpolynomial for any algebraic X.

It would be interesting to try to extend our derivation tothe more general case.17

IV.CONCLUSIONWe have shown that the Donaldson map appears explicitly as a coupling in the low-energy action for heterotic string theory in the soliton sector. This implies the Donaldsonpolynomial can be measured by scattering massless fields and solitons.

This realizationleads to concrete formulae for the Donaldson map and polynomial which are equivalentto, and provide a new perspective on, formulae derived by Witten in the framework oftopological Yang-Mills theory. It also led to an interpretation of the Donaldson map as aK¨ahler-Christoffel connection on H2(X, C) × MN(X).The fact that this scattering is a map between cohomology classes was insured byzero-momentum worldbrane supersymmetry, which acts like the exterior derivative onMN.

This should be contrasted with topological Yang-Mills theory where the exteriorderivative on the instanton moduli space is constructed in terms of a BRST operator.Our work suggests a number of generalizations and applications. Perhaps this connec-tion can provide new insights into, or stringy interpretations of, the various theorems onthe structure of four-manifolds which follow from Donaldson’s work.

Alternately, the re-markable properties of the Donaldson polynomial may translate into interesting propertiesof the fivebrane-axion S-matrix, or even have implications for the closely related problemof instanton-induced supersymmetry breaking in string theory.ACKNOWLEDGMENTSThis work was supported in part by DOE Grant DE-AT03-76ER70023 and NSF GrantPHY90-00386. We are grateful to P. Bowcock, R. Gregory, D. Freed, S. Giddings, D.Moore, C. Vafa and S. T. Yau for useful discussions.18

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