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Superstring Compactification and Target Space Duality

arXiv:hep-th/9108022v1 26 Aug 1991CALT-68-1740DOE RESEARCH ANDDEVELOPMENT REPORTSuperstring Compactification and Target Space Duality⋆John H. SchwarzCalifornia Institute of Technology, Pasadena, CA 91125ABSTRACTThis review talk focusses on some of the interesting developments in the areaof superstring compactification that have occurred in the last couple of years.These include the discovery that “mirror symmetric” pairs of Calabi–Yau spaces,with completely distinct geometries and topologies, correspond to a single (2,2)conformal field theory. Also, the concept of target-space duality, originally dis-covered for toroidal compactification, is being extended to Calabi–Yau spaces.

Italso associates sets of geometrically distinct manifolds to a single conformal fieldtheory.A couple of other topics are presented very briefly. One concerns concep-tual challenges in reconciling gravity and quantum mechanics.

It is suggestedthat certain “distasteful allegations” associated with quantum gravity such asloss of quantum coherence, unpredictability of fundamental parameters of parti-cle physics, and paradoxical features of black holes are likely to be circumventedby string theory. Finally there is a brief discussion of the importance of supersym-metry at the TeV scale, both from a practical point of view and as a potentiallysignificant prediction of string theory.Presented at Strings and Symmetries 1991May 1991⋆Work supported in part by the U.S. Department of Energy under Contract No.

DE-AC0381-ER40050

1. IntroductionThe conference organizers have asked me to give a review survey of significantdevelopments in superstring compactification that have occurred in the last coupleof years since the last review papers that I wrote on this subject.1 A great deal ofimpressive progress has been made, and it will only be possible to survey a portion ofit.

The choice of topics is based mostly on what has caught my attention, and what Ihave been able to digest. There are undoubtedly many important developments thatwill not be mentioned.The two main topics to be discussed are Calabi–Yau compactification and target-space duality.

Two important developments in Calabi–Yau compactification will bestressed. The first is the existence of holomorphic prepotentials that determine theK¨ahler potentials that describe the moduli spaces M21 and M11, associated withcomplex structure deformations and K¨ahler form deformations, respectively.

The sec-ond is the remarkable mirror symmetry that associates a pair of Calabi–Yau spaces tothe same (2,2) conformal field theory. They are related to one another by interchangeof the moduli spaces M21 and M11.The most famous example of target–space duality is the R →1/R symmetry as-sociated with compactification on a circle of radius R. As with mirror symmetry thistransformation relates distinct geometries that are associated with the same confor-mal field theory.

Generalizations appropriate to toroidal compactifications have beenknown for some time and will be reviewed very briefly. The little bit that is knownabout such symmetries in the case of K3 and Calabi–Yau space compactification willalso be discussed.

An interesting proposal has been made to restrict the possibili-ties for low-energy effective actions that incorporate nonperturbative supersymmetrybreaking by the duality symmetries.Recently, my interest in target-space duality was reactivated by the realizationthat it can be viewed as a discrete symmetry group that is a subgroup of sponta-neously broken continuous gauge symmetries—what has been referred to as ‘localgauge symmetry.’ This led me to propose that it could have a bearing on resolving1

certain deep problems associated with quantum gravity.2 After reviewing some of thedisturbing allegations that are made about the inevitable consequences of reconcilinggeneral relativity and quantum mechanics, I will discuss possible ways in which theymay be circumvented in string theory.The concluding section makes a plea to demonstrate to our experimental col-leagues that our work is relevant. It is suggested that a strong case could be made forsupersymmetry at the TeV scale as an almost inevitable feature of any quasi-realisticstring vacuum.

This being so, perhaps it would not be inappropriate for us to stickour necks out a bit and call this a ‘prediction of string theory.’2. Progress in Calabi–Yau CompactificationLet me begin by recalling a few basic facts.

By definition, a Calabi–Yau spaceis a K¨ahler manifold of three (complex) dimensions and vanishing first Chern class.The complex (Dolbeault) cohomology groups Hp,q have dimensions bpq given by theHodge diamond1000b1101b21b2110b110001One generator of H1,1 is the K¨ahler form J = igµ¯νdxµ ∧dx¯ν. Thus b11 ≥1.A Calabi–Yau space, with specified K¨ahler class, admits a unique Ricci-flat metric.Also, there exists a covariantly constant spinor λ, in terms of which the holomorphicthree-form is Ωµνρ = λγµνρλ.In the context of heterotic string compactification, the existence of λ is responsiblefor the fact that the 4D low-energy theory has N=1 supersymmetry.

Altogether, themassless spectrum contains the following N=1 supermultiplets:2

a) N=1 supergravity (graviton and gravitino)b) Yang–Mills supermultiplets (adjoint vectors and spinors for E6 × E8 × . .

. )c) Various chiral supermultiplets (Weyl spinor and a scalar)The chiral supermultiplets include matter and moduli multiplets.

The mattermultiplets consist of b11 generations (27 of E6) and b21 antigenerations (27 of E6).Which of these one chooses to call ‘generations’ and ‘antigenerations’ is purely amatter of convention, of course. The moduli consist of an ‘S field’ and ‘T fields.’ TheS field contains the dilaton φ and the axion θ.

The vev of the dilaton gives the stringcoupling constant (< φ >∼1/g2), and θ is the 4D dual of the antisymmetric tensorfield Bµν. Locally, the vevs of φ and θ parametrize the coset manifold SU(1, 1)/U(1).The T fields consist of b11 E6 singlets whose vevs parametrize the moduli spaceof K¨ahler form deformations, M11, and b21 E6 singlets whose vevs parametrize thespace of complex structure deformations, M21.

Altogether, the Calabi–Yau modulispace is the tensor product MCY = M11 × M21. Locally, this is the same thing asthe space of (2,2) conformal field theories, though the K¨ahler geometry of M21 differsin the two cases by the effects of world sheet instantons.

However, as we will discuss,they differ globally by duality symmetries, so that the moduli space of (2,2) theoriesis given by M(2,2) = MCY /G, where G is a discrete group. This is the space thatclassifies inequivalent string compactifications.Remarkably, the moduli spaces M11 and M21 are themselves K¨ahler manifolds.Here I will simply state the salient facts without attempting to give the proofs, whichcan be found in the literature.3Let zα, α = 1, 2, .

. .

, b21 be local complex coordinates for M21. Then linearlyindependent generators of H2,1 are given byχακλ¯µ = −12Ωκλρgρ¯η ∂∂zα g¯µ¯η,α = 1, 2, .

. .

, b21.Out of χα = χακλ¯µdxκ ∧dxλ ∧d¯x¯µ and ¯χ¯β one constructs the K¨ahler metric for M213

by the formulaGα¯β = −RM χα ∧¯χ¯βRM Ω∧¯Ω= −∂∂zα∂∂¯z ¯β logiZMΩ∧¯Ω.From this it follows that K = −logiRM Ω∧¯Ωis the K¨ahler potential.Moreover, M21 is a K¨ahler manifold with a holomorphic prepotential. (SuchK¨ahler manifolds are sometimes said to be of ‘restricted type.’) The formula is givenmost succinctly using projective coordinates (in Pb21) za, a = 1, 2, .

. .

, b21 +1, definedwith respect to a canonical homology basis of H3. The basic cycles Aa and Bb arearranged to intersect in a manner analogous to the A and B cycles of a Riemannsurface.The coordinates za are given by za =RAa Ωand the derivatives of theholomorphic prepotential G(za) are given by∂∂zaG =RBa Ω.

In terms of G and itscomplex conjugate ¯G(¯za) the K¨ahler potential is given bye−K = −iza ∂¯G∂¯za −¯za ∂G∂za.The prepotential also encodes the Yukawa couplings of the antigenerations, which aregiven by its third derivatives. (I am confident that more details of this constructionwill be presented by other speakers.

)The moduli space M11 has a very similar description. In terms of local complexcoordinates wA, with A = 1, 2, .

. .

, b11.Its metric is given in terms of a K¨ahlerpotential by GA ¯B =∂∂wA∂∂¯w ¯B K, as usual. The K¨ahler potential is given by K =−log κ(J, J, J), where κ(J, J, J) = 43RM J ∧J ∧J and J is the K¨ahler form.Interms of projective coordinates wi, i = 1, 2, .

. .

, b11 + 1, there is again a holomorphicprepotential F(wi) withe−K = −iwi ∂¯F∂¯wi −¯wi ∂F∂wi.As before, the third derivatives of F give the Yukawa couplings of the generations.The significant asymmetry between F and G in the case of the geometric limit is4

that F is a cubic function so that the Yukawa couplings are constants, whereas G isa complicated nonpolynomial expression, so that those Yukawa couplings depend onthe complex structure deformation moduli.A Calabi–Yau space corresponds to a geometric limit of a (2,2) superconformalfield theory. This geometric limit includes effects to every order in α′ in the associatedsigma model, but it does not include the nonperturbative effects associated withworld-sheet instantons.

It turns out that these world-sheet instantons contribute tothe moduli space of K¨ahler form deformations M11 but not to the moduli space ofcomplex structure deformations M21. As a result the prepotential G can be computedexactly in the geometric limit, but the prepotential F cannot.

The earlier assertionthat F is a cubic function referred to the geometric limit.When the instantoncontributions are included it is no longer cubic. It is the latter expression that isrelevant to string theory.

This modification of Calabi–Yau geometry implied by thecorresponding conformal field theory is sometimes referred to as “quantum geometry”in the recent literature.We now turn to the mirror symmetry conjecture. The similarity in the descriptionof the two moduli spaces M11 and M21 suggests that to every Calabi–Yau space M(with b21 > 0) there is a mirror partner Calabi–Yau space ˜M such that the two modulispaces are interchanged:˜M11 = M21and˜M21 = M11.Clearly, in view of the remarks made above, this is only possible if we include theinstanton corrections in the description of the moduli spaces.

Thus the precise state-ment of the conjecture is that there is a a mirror partner Calabi–Yau space suchthat the two spaces correspond to distinct geometric limits of the same (2,2) confor-mal field theory. This a remarkable conjecture from a mathematical point of view,since the CY spaces have completely different geometries and topologies.

It is alsoof some practical importance from the physical point of view, since computationsof the prepotentials are significantly easier to carry out in the geometric limit (us-ing topological formulas) than for the exact CFT in general. By computing moduli5

spaces of complex-structure deformations for a pair of mirror CY spaces one deduces(by the equations above) the exact geometry of the moduli spaces of K¨ahler formdeformations, including the effects of world-sheet instantons.A considerable amount of evidence in support of the mirror symmetry conjecturehas been amassed. I am not completely sure of the history, but I believe the ideaoriginated when Dixon noticed the symmetrical way in which generations and anti-generations are treated in Calabi-Yau compactification of the heterotic string `a laGepner, and when Lerche, Vafa, and Warner noticed the symmetrical appearanceof the (c,c) and (a,c) rings in N=2 Landau–Ginzburg theory.4 Candelas, Lynker,and Schimmrigk computed the Hodge numbers b11 and b21 for several thousand CYspaces that can be described as intersections of weighted complex projective spacesand observed that there was an almost perfect correspondence between pairs of CYspaces with these numbers interchanged.5 Some unpaired examples are to be expectedin their list, since it is certainly not complete.

Given that fact, it is remarkable howfew of them there are. Also, one class of spaces is certainly special.

There existCalabi–Yau spaces with b21 = 0. The mirror of such a space should have b11 = 0, butsuch a space cannot be a Calabi–Yau space, since they always have the K¨ahler formitself as at least one generator of H1,1.

The significance of this class of exceptions isbeing investigated by the experts. It may be of some practical importance if we hopeto eventually find an example that gives three generations and no antigenerations, soas to avoid the problem of understanding how extra generations and anti-generationspair up to acquire a large mass.

Such a space with b11 = 3 and b21 = 0 might notexist, however.Further evidence in support of the mirror symmetry conjecture and insight intovarious structural details have been obtained from a variety of additional studies.These include analysis of explicit examples based on (2,2) orbifolds6 and additionalstudies of the Landau-Ginzburg connection.7 Another approach utilizes Gepner’s cor-respondence between Calabi–Yau compactification and compactification of Type IIsuperstring theories. The constructions based on Type IIA and Type IIB theoriesturn out to be related by mirror symmetry.

This fact provides a rather powerful tool6

for detailed studies, since it benefits from the restrictions implied by the additionalsupersymmetry of the Type II theories.8 Finally, I should mention the detailed inves-tigation of a specific mirror pair of CY spaces by Candelas et al.9 They compute theprepotential F for the one-dimensional moduli space of the K¨ahler class for the CYspace given by a quintic polynomial in P4 and compare it to the prepotential G forthe mirror space (given by orbifolding the original space by a suitable discrete sym-metry group). By comparing the two expressions they are able to explicitly identifythe instanton contributions and infer the number of “holomorphic curves” of varioussorts—deep results in algebraic geometry.3.

Target Space DualityThe notion of target space duality is becoming a more and more prevalent themein string theory, with a wide range of applications and implications. Target-spacedualities are discrete symmetries of compactified string theories, whose existence sug-gests a breakdown of geometric concepts at the Planck scale.

The simplest exampleis given by closed bosonic strings with one dimension of space taken to form a circleof radius R. In this case the momentum component of a string corresponding to thisdimension is quantized: p = n/R, n ∈Z. This is a general consequence of quantummechanics and is not special to strings, of course.

What is special for strings is theexistence of winding modes. A closed string can wrap m times around the circulardimension.

A string state with momentum and winding quantum numbers n and m,respectively, receives a zero-mode contribution to its mass-squared given by10M20 = (n/R)2 + (mR/α′)2,where α′ is the usual Regge slope parameter. It is evident that the simultaneousinterchanges R ↔α′/R and m ↔n leaves the mass formula invariant.11 In fact, theentire physics of the interacting theory is left unaltered provided that one simultane-ously rescales the dilaton field (whose expectation value controls the string couplingconstant) according to φ →φ −ln(R/√α′).12 The basic idea is that the radii R7

and α′/R both correspond to the same (c = 1) conformal field theory, and it is theconformal field theory that determines the physics. This is the simplest example ofthe general phenomenon called “target-space duality.”The circular compactification described above has been generalized to the caseof a d-dimensional torus, characterized by d2 constants Gab and Bab.13 The symmet-ric matrix G is the metric of the torus, while B is an antisymmetric matrix.

Theseparameters describe the moduli space of toroidal compactification and can be inter-preted as the vacuum expectation values of d2 massless scalar fields. The dynamicsof the toroidal string coordinates Xa is described by the world sheet actionS =Zd2σ[Gab∂αXa∂αXb + ǫαβBab∂αXa∂βXb]In this case we can introduce d-component vectors of integers ma and na to describethe winding modes and discrete momenta, respectively.

A straightforward calculationthen gives the zero-mode contributionsM20 = Gabmamb + Gab(na −Bacmc)(nb −Bbdmd)where Gab represents the inverse of the matrix Gab.The one-dimensional case isrecovered by setting G11 = R2/α′ and B = 0. The generalization of the dualitysymmetry becomes G + B →(G + B)−1 and φ →φ −12ln det(G + B).The Bterm in the world-sheet action is topological.

The parameters Bab are analogous tothe θ parameter in QCD, and the quantum theory is invariant under integer shiftsBab →Bab +Nab. Combined with the inversion symmetry, these generate the infinitediscrete group O(d, d; Z).14 (The analogous group in the case of the heterotic stringis O(d + 16, d; Z).) Thus, whereas the moduli Gab and Bab locally parametrize thecoset space O(d, d)/[O(d) × O(d)], points in this space related by O(d, d; Z) trans-formations correspond to the same conformal field theory and should be identified.It is conjectured that when nonperturbative effects break the flatness of the effectivepotential, so that the scalar fields that correspond to the moduli can acquire mass,the discrete duality symmetries of the theory are preserved.158

The O(d, d; Z) target space duality symmetries are discrete remnants of sponta-neously broken gauge symmetries.16 Specifically, for values of the moduli correspond-ing to a fixed point of a subgroup of this discrete group, the corresponding stringbackground has enhanced gauge symmetry. (This is most easily demonstrated byshowing that there are additional massless vector string states in the spectrum.) Atsuch a point some of the duality symmetry transformations coincide with finite gaugetransformations.

By considering all possible such fixed points it is possible to identifyan infinite number of distinct gauge symmetries, with all but a finite number of themspontaneously broken for any particular choice of the moduli.One may wonder whether the occurrence of target-space dualities is special totoroidal compactification or whether it occurs generically for curved compactifica-tion spaces such as Calabi–Yau manifolds.The four-dimensional analog, namelyK3 compactification, has been analyzed in some detail.17 In that case (applied tothe heterotic string) the moduli space is O(20, 4)/[O(20) ×O(4)], parametrized by 80massless scalar fields and the duality group is O(20, 4; Z). Remarkably, this is exactlythe same manifold and duality group that arises in the case of toroidal compactifica-tion of the heterotic string to six dimensions.

One might be tempted to speculate thatthe two compactifications are equivalent, but that cannot be correct since K3 com-pactification breaks half of the supersymmetry while toroidal compactification doesnot break any.18 By modding out certain symmetries of the torus, it is possible toform an orbifold for which half the supersymmetry is broken and the moduli space isstill essentially the same. This orbifold seems likely to correspond (at least locally) tothe same conformal field theory as K3.

Having spaces of distinct topology correspondto identical conformal field theories goes beyond what we learned from tori. (Therevarious different geometries all having the same topology were identified.) However,as we have seen, such identifications do exist for Calabi–Yau spaces, which occur inmirror pairs of opposite Euler number.The moduli space of a Calabi–Yau compactification factorizes into the manifoldM21 that describes complex-structure deformations times the manifold M11 thatdescribes K¨ahler form deformations.

There are duality symmetry transformations9

that act on each of these spaces separately. The two classes of transformations wouldseem to have very different interpretations from a geometrical point of view.

However,the two factors are interchanged for the mirror manifold, so if one associates themwith a mirror pair of Calabi–Yau spaces, then they appear on an equivalent footing.The problem is to determine the target-space duality group that acts on each of thesemoduli spaces. A natural action of the discrete group Sp(2 + 2b21, Z) can be definedon the third cohomology groupH3 = H(3,0) ⊕H(2,1) ⊕H(1,2) ⊕H(0,3)analogous to the symplectic modular group for Riemann surfaces.

This symplecticgroup contains the possible discrete symmetries of M21. The mirror symmetry impliesa corresponding action of Sp(2 + 2b11, Z) on the spaceH(0,0) ⊕H(1,1) ⊕H(2,2) ⊕H(3,3)describing possible discrete symmetries of M11.

Thus altogether the target space du-ality of the (quantum corrected) Calabi–Yau space should be given by some subgroupGT D ⊆Sp(2 + 2b21, Z) × Sp(2 + 2b11, Z).Examples have been worked out in special cases.A potentially important application of the duality symmetries has been proposedin connection with the construction of low-energy effective actions. The idea is thatthese should be exact symmetries of the complete quantum theory and should still bepresent even after nonperturbative effects (such as those that break supersymmetry)are taken into account and after heavy fields are integrated out.

This means that, interms of a low-energy effective action in four dimensions with N=1 supersymmetry,the duality symmetries should be realized on the superpotential, which is thereforerestricted to be a suitable automorphic function. This is a very significant restriction10

on the characterization of the low-energy theory. Therefore there is some hope forsaying quite a bit about nonperturbative effects without solving the difficult problemof computing them from first principles.

There is some evidence that the combina-tion of gluino condensation and duality symmetry are sufficient to remove all flatdirections from the potential.19 This means that the size of the compact space, whichis one of the moduli, is dynamically determined and all the other parameters thatdetermine the vacuum configuration are also determined. Supersymmetry is brokenand the cosmological constant typically comes out negative (corresponding to anti deSitter space).

However, there are some examples for which the cosmological constantvanishes.20 Recently, there have been studies of automorphic prepotentials for general(2,2) compactifications.214. Conceptual Challenges in Reconciling Gravity and Quantum MechanicsThere are a variety of technical and conceptual obstacles that need to be overcomeif a satisfactory understanding of the reconciliation of general relativity and quantummechanics is to be achieved.These can be divided into two categories—amazingrequirements and distasteful allegations.

If string theory is the correct approach toconstructing a fully consistent unification of all fundamental forces, then it shouldhold the keys to the right answers.In this case, our job is to discern the clevertricks that string theory employs. This may sound like a strange way to approachthe problem, but string theory has proved to be a fruitful source of inspiration in thepast.

Examples range from the discovery of supersymmetry to unexpected anomalyand divergence cancellation mechanisms and much more. It could hold many moresurprises in store for us.One “amazing requirement” is perturbative finiteness (or renormalizability).

Thisis not yet fully established, but there is considerable evidence that this is achievedin string theory, even though it is apparently impossible for any point-particle fieldtheory that incorporates general relativity (in four dimensions). A second requirementis that causality have a precise meaning when the space-time metric is a dynamicalquantum field.

This undoubtedly happens in string theory, but it would be nice to11

understand in detail just what is involved. Third, the theory should be applicableto the entire universe, perhaps describing it by a single wave function.

An obviousquestion in this connection is whether string theory suggests some special choice ofboundary condition, such as that proposed by Hartle and Hawking, and whether thiscould provide a rationale for selecting a particular vacuum configuration.The second category of conceptual issues consists of certain “distasteful allega-tions”, which string theory might cleverly evade. The first of these is the claim thateffects associated with virtual black holes cause pure quantum states to evolve intomixed states.22 If this were true, it would mean that the entire mathematical frame-work of quantum mechanics is inadequate.

It seems reasonable to explore whetherstring theory could avoid allowing pure states to evolve into mixed states. To the ex-tent that string theory can be consistently formulated as an S matrix theory, it seemsalmost inevitable that this should work out.

A second distasteful allegation is thatwormhole contributions to the Euclidean path integral23 render the parameters of par-ticle physics stochastic.24 In a previous paper,2 I referred to this phenomenon as ‘thecurse of the wormhole,’ since it would imply that even when the correct microscopictheory is known, it will still not be possible to compute experimental parameters suchas coupling constants, mass ratios, and mixing angles from first principles.A third issue concerns the classification of black holes.According to the “nohair” theorems, in classical general relativity black holes are fully characterized bymass, electric charge, and angular momentum. On the other hand, they have a (large)entropy that is proportional to the area of the event horizon.25 This amount of entropycorresponds to a number of degrees of freedom that is roughly what one would get froma vibrating membrane just above the horizon.26 In fact, ‘t Hooft has tried to makesense of such a physical picture, interpreting the membrane as a string world sheet.27Alternatively, the black hole degrees of freedom might be accounted for in stringtheory in more subtle ways that utilize possibilities for evading the classical no-hairtheorems by quantum effects.

Recent studies have shown that black holes can have‘quantum hair,’ which is observable (in principle) by generalized Bohm–Aharonovinterference measurements.28,29 Charges that can characterize quantum hair for black12

holes are precisely the same ones whose conservation cannot be destroyed by wormholeeffects.Thus the nicest outcome might be for the correct fundamental theory toprovide so many different types of quantum hair as to produce precisely the numberof degrees of freedom that is required to account for the entropy of black holes. Ina theory with enough distinct degrees of freedom to account for black hole entropyand to protect quantum coherence, there should be no deleterious effects due towormholes.A mechanism that has been proposed as an origin for quantum hair is for a contin-uous gauge symmetry to break spontaneously leaving a discrete subgroup unbroken.As we have discussed, string theory has a large group of discrete symmetries that canbe understood as remnants of spontaneously broken gauge symmetries, namely thetarget space dualities.

This fact led me to propose that these are the relevant symme-tries for understanding quantum hair in string theory.2 ⋆However, following furtherstudies and discussions with others, it has become clear that this suggestion has se-rious problems, mostly stemming from the fact that these symmetries are almost allbroken for any particular choice of vacuum configuration.The first proposal for “quantum hair” of black holes, detectable only by Bohm–Aharonov-type interference effects, was put forward a few years ago by Bowick et al.28As initially formulated, the analysis only applied to theories containing a massless‘axion.’However, a subsequent paper demonstrated that this restriction was notessential and that a suitable massive axion could do the same job.32Stripping away all interactions, the basic idea can be explained quite simply.Assume four-dimensional space-time and let Aµ be a U(1) vector field and Bµν anantisymmetric tensor gauge field (called the axion). In the language of forms, theassociated field strengths are given by H = dB and F = dA.

The action consistsof the usual kinetic terms, schematically given by Skin = 12Rd4x(H2 + F 2), anda topological mass term of the form Smass = mRB ∧F. The equations of motion,⋆This idea has been proposed independently in Ref.

[30], though the emphasis there is on‘duality of the S field.’13

d∗H = mF and d∗F = −mH, imply that both fields have mass m. What happens isthat the antisymmetric tensor eats the vector to become massive. In four dimensionsit is equivalent to say that the vector eats the scalar (which is dual to Bµν) to becomemassive.The axion charge in a region of three-dimensional space V with boundary ∂V isdefined byQaxion =ZVH =Z∂VB.For a space-time with nontrivial second homology, such as Schwarzschild space-time(whose topology is S2×R2), it is possible to obtain nonzero axion charge while havingthe H field vanish outside some central region.

In this case the B field on the enclosingtwo-surface is proportional to a two-form that is closed but not exact (i.e., belongsto the second cohomology group). In a theory with axions there are strings (cosmicor fundamental) that contribute a term to the action proportional toRΣ Bµνdxµ ∧dxν.As a result, a world sheet enclosing a black hole with axionic charge givesa Bohm–Aharonov phase exp[2πiQaxion].

This makes the charge observable throughinterference effects (modulo unity). If there were nontrivial third homology, the axioncharge itself would be quantized and nothing would be observable.33 However, this isnot the case for a Schwarzschild black hole.In the string theory context, the formula for the field strength H is embellishedby various Chern–Simons terms that were omitted in the discussion above.

Also,one-loop effects in ten dimensions give contributions to the effective action of theformRB ∧tr(F 4), which play a crucial role in anomaly cancellation.34 Upon com-pactification it can happen that the ten-dimensional gauge fields acquire expectationvalues that result in a nonvanishing effective term of the formRB ∧F, where F isa U(1) gauge field in four dimensions, as required to give mass to the axion. Thishappens when the associated U(1) gauge symmetry in four dimensions appears to beanomalous by the usual criteria based on triangle diagrams.

However, as in ten di-mensions, Bµν has nontrivial gauge transformation properties that give compensating14

contributions and render the quantum theory consistent.Axion charge appears to be a good candidate for quantum hair in string theory.Of course, if this particular charge were the only type of quantum hair in stringtheory, we would still be very far from achieving the goal of finding enough quantumdegrees of freedom to account for all the entropy of black holes and overcoming theother problems in quantum gravity that we have discussed.Fortunately, string theory seems to allow various categories of generalizations ofaxion charge that could provide many more kinds of quantum hair. For example,the field Bµν(x, y) is defined in ten dimensions.

(Here x refers to four-dimensionalspace-time and y to six compactified dimensions.) In the usual Kaluza–Klein fashion,this represents an infinite family of four-dimensional fields B(n)µν (x) corresponding toan expansion in harmonics of the compact space of the form P Cn(y)B(n)µν (x).

Theanalysis above only utilized the axion corresponding to the leading term in this seriesfor which C(y) is a constant. The other terms describe fields that naturally havemasses of the order of the compactification scale.

It seems plausible that they couldprovide additional types of quantum hair. (When the analysis is done carefully,target-space duality may yet prove to be important!) Even this infinite collectionof charges may not be the end of the story.

The massive string spectrum containsan infinite number of gauge fields of every possible tensor structure. The particulargauge field Bµν is special by virtue of its coupling to the string world sheet, whichplayed a crucial role in the reasoning above.

Other gauge fields enter the world sheetaction with couplings given by their associated vertex operators. For fields that arenot massless in ten dimensions these give nonrenormalizable couplings in the sigmamodel, and are therefore difficult to analyze.

Still, from a more general string fieldtheory point of view, they are not really very different, and so there may be manymore possibilities for quantum hair associated with the massive string spectrum.In addition to string symmetries altering some consequences of general relativityat the quantum level, it is also possible that special features of string theory play animportant role at the classical level. One indication of this appears in a recent study15

of charged black holes,35 where effects of the dilaton field make qualitative changesalready at zeroth order in α′. Specifically, whereas a Reissner–Nordstrom black holeof mass M and charge Q has its horizon at the radius RH = M +pM2 −Q2, thecorresponding string solution has RH = 2Mp1 −Q2/2M2.

Also, as the charge ofthe black hole approaches its maximum allowed value, the entropy S →4πM2 andthe temperature T →0 in the Reissner–Nordstrom case. In the string case one findsS →0 and T →8πM.

(All these results are to leading order in α′ and ¯h.) As onemight expect, the thermodynamic description breaks down in either of these extremelimits.365.

The Importance of SupersymmetryIn order to gain the attention and respect of our experimental colleagues it isimportant to make predictions that bear on near-term experimental possibilities. Itis pretty clear that the best prospect in this regard is supersymmetry.

It would benice if we could honestly assert that supersymmetry (broken at the weak scale) isan inevitable feature of any quasi-realistic string model and thus a necessity if stringtheory is the correct basis of unification. Experimentalists could then be in a positionto demonstrate that “string theory is false” or to discover important evidence in itssupport.

But can we honestly make such an assertion?Certainly no string models that are remotely realistic have been constructed with-out low-energy supersymmetry. Also, in the context of string theory any alternativemechanism for dealing with the hierarchy problem seems very unlikely.

Still, if asis generally assumed, string theory has a clever way of preventing a cosmologicalconstant from arising once supersymmetry is broken, then maybe it could also havea clever stringy alternative for preventing Higgs particles from acquiring unificationscale masses through radiative corrections.No plausible alternative to supersym-metry is known, but how sure can we be that one doesn’t exist? There is alwaysthe possibility that some mechanism, not yet considered, could be important, butthat shouldn’t completely prevent from us ever sticking our necks out a bit.

I don’tthink it would be dishonest for string theorists to assert that according to our present16

understanding, supersymmetry broken at the weak scale is required by string theory.In fact, the experimental prospects are beginning to look up. The requirementthat the three couplings of the standard model should become equal at a unificationscale fails badly without supersymmetry.

On the other hand, for a susy scale rangingfrom 100 GeV to 10 TeV they merge very nicely at about 1016 GeV.37 While this is farfrom conclusive, it is a very impressive bit of evidence. Given the present experimentalsituation, together with various theoretical and astrophysical considerations, it seemsquite plausible that the lightest supersymmetry particles are at the low end of thisrange.

Others, such as squarks and gluinos, maybe be around a TeV or so. If this iscorrect, it is unlikely that any of these particles will be produced and detected beforethe LHC or SSC comes on line.

However, supersymmetry has important implicationsfor the Higgs sector that could be confirmed sooner.Low-energy supersymmetry has two Higgs doublets, which after symmetry break-ing result in a charged particle H±, and three neutral particles h, H, and A. Theminimal supersymmetric standard model (MSSM) requires, at tree level, that h isthe lightest of these and that its mass not exceed MZ. When radiative correctionsare taken into account, it can be somewhat heavier, depending on the mass of thetop quark.

For example, if the top quark mass does not exceed 160 GeV then the hmass should not exceed 120 GeV. For a top quark mass below 130 GeV the boundis lowered to 100 GeV.

These bounds need not be saturated, so there is a reasonablechance for a mass in the range 50-100 GeV, making it open to discovery at LEP2. Another interesting possibility is that the top quark could decay into H+ plus abottom quark.

If it is kinematically allowed, this could be a significant branchingfraction. (The precise prediction depends on the parameter tanβ = v2/v1.) I amoptimistic that some of these particles will turn up during this decade and that thiswill open up an exciting era for string theorists (as well as all particle physicists).17

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