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Instantons and Solitons in Heterotic String Theory

arXiv:hep-th/9109052v1 26 Sep 1991PUPT-1278Instantons and Solitons in Heterotic String TheoryCurtis G. Callan, Jr.Department of Physics, Princeton UniversityPrinceton, NJ 08544Internet: cgc@pupphy.princeton.eduAbstractThis is a transcript of lectures given at the Sixth Jorge Andre Swieca Summer Schoolin Theoretical Physics. The subject of these lectures is soliton solutions of string theory.We construct a class of exact conformal field theories possessing a spacetime soliton orinstanton interpretation and present a preliminary discussion of their physical properties.June 1991

1. Introduction and MotivationFor the purposes of these lectures we are going to assume that string theory is identicalto two-dimensional conformal field theory.

As is well-known, conformal field theories of theright central charge and field content describe solutions of the classical equations of motionof string field theory. To obtain true quantum physics one has to sum over higher-genusworldsheets, an important, but difficult, task which we will not attempt here.Static solutions of the classical equations of motion of conventional field theory canrepresent either candidate vacuum states, solitons or instantons.

The soliton and instantonsolutions are distinguished from a vacuum by being non-invariant to translations andboosts and by having a finite mass (which goes to infinity as the coupling goes to zero).Vacuum solutions may have a non-zero energy density, but this must be interpreted as acosmological constant rather than a mass. In short, soliton solutions identify new physicalparticles which do not appear in the spectrum of small fluctuations about the vacuum andmay have exotic quantum numbers.

The magnetic monopole solution of spontaneouslybroken gauge theory is a classic example.An important development in the study ofordinary quantum field theory was the demonstration that solitons and their associatedconserved charges survive quantization, and are not just artifacts of the classical analysis.It is obviously important to repeat this development for string theory: essentially stringyphysics could lead to even more exotic types of soliton or it could lead to weak instabilityof the old ones. Either outcome could have important phenomenological consequences.Most of what we have learned about string theory so far comes from conformal fieldtheories with a vacuum interpretation.

In order to make contact with reality, it is necessaryto compactify six of the ten spacetime dimensions of superstring theory and the mostcarefully studied nontrivial conformal field theories have precisely this interpretation. Thegeometry of the six compactified dimensions may be very complicated, but the remainingfour dimensions are perfectly flat, and the compactification theory manifests no meaningfulenergy or mass: it is a vacuum.To address the soliton issue then, it will be necessary to find conformal field theoriesdescribing localized energy density embedded in asymptotically flat spacetime (that is, wewant to find non-compactifications!) Once we find such a solution, we need to developappropriate conformal field theory methods for studying its non-vacuum properties, mostnotably its mass.

As we shall see, these are difficult problems, but some progress has beenmade on the first issue.There are two ways to proceed in the study of this problem. The first is the ”stringsin background field” method [1,2,3], in which the usual spacetime fields (graviton, dilatonand antisymmetric tensor) appear as coupling constant functions in a worldsheet nonlinearsigma model and the spacetime equations of motion for these fields arise from the conditionthat the conformal invariance beta functions should vanish.

The beta functions are com-puted as an expansion in powers of the string tension α′ and, in the leading approximation,yield standard Einstein-Yang-Mills-like spacetime equations of motion. Not-so-standardstringy effects arise from higher-order corrections.

It is conceptually straightforward tolook for asymptotically flat solutions of these field equations and most previous attemptsto study string solitons have taken this approach. The problem is that the α′ expansion isonly valid if curvatures are everywhere small, a condition which is not met in many inter-1

esting solutions. The second way to proceed is to use purely algebraic methods to generateexact conformal field theories in the hope that it will be possible to generate some thathave a solitonic spacetime interpretation.

This approach is nonperturbative and perfectlycapable of handling cases with strong curvature, but no useful exact theories have beenfound in this way.In these lectures we will tackle the string soliton problem by a hybrid of the abovetwo approaches. We will first construct a soliton solution of the lowest-order spacetimeequations and we will then use supersymmetry arguments to show that this solution getsno corrections to any order in α′.

Finally, we will use purely algebraic methods to discussa limiting case of our general solution and to argue that no nonperturbative effects havebeen missed. At the end we will draw conclusions and make some proposals for furtherdevelopment.

The work on which these lectures are based has been done in collaborationwith J. Harvey and A. Strominger and is reported in [4,5]. The presentation given here isa pedagogical elaboration of those references and will, in places, follow them quite closely.2.

Soliton Solutions: Spacetime ApproachLet us first discuss the problem of finding string solitons via the ”strings in backgroundfields” spacetime approach. The beta functions for strings propagating in a background ofmassless fields are the equations of motion of a certain master spacetime action which canbe computed as an expansion in the string tension α′.

For the heterotic string, the leadingterms in this action are identical to the D = 10, N = 1 supergravity and super Yang-Millsaction. The bosonic part of this action readsS =12κ2Zd10x√ge−2φR + 4(∇φ)2 −13H2 −α′30TrF 2,(2.1)where the three-form antisymmetric tensor field strength is related to the two-form poten-tial by the familiar anomaly equation [6]H = d ∧B + α′ωL3 (Ω−) −130ΩY M3(A)+ .

. .

(2.2)(where ω3 is the Chern-Simons three-form) so thatd ∧H = α′(trR ∧R −130TrF ∧F). (2.3)The trace is conventionally normalized so that TrF ∧F = Pi F i ∧F i with i an adjointgauge group index.

An important, and potentially confusing, point is that the connectionΩ± appearing in (2.2) is a non-Riemannian connection related to the usual spin connectionω byΩAB±M = ω ABM± H ABM. (2.4)Since the antisymmetric tensor field plays a crucial role in all of our solutions, this subtletywill be crucial.2

Rather than directly solve the equations of motion for this action, it is much moreconvenient to look for bosonic backgrounds which are annihilated by some of the N=1supersymmetry transformations (only the vacuum is annihilated by all the the supersym-metries). This is a fairly standard trick which has been applied to the magnetic monopoleproblem and to string problems in [7,8].

It is highly nontrivial that any such solutions canbe found at all, but if they can, they are automatically solutions of the usual equations ofmotion. The Fermi field supersymmetry transformation laws which follow from (2.1) areδχ = FMNγMNǫδλ = (γM∂Mφ −16HMNP γMNP )ǫδψM = (∂M + 14ΩAB−MγAB)ǫ,(2.5)and it is apparent that to find backgrounds for which all of (2.5) vanish, it is only necessaryto solve first-order equations, rather than the more complicated second-order equationswhich follow from varying the action.

We will shortly construct a simple ansatz for thebosonic fields which does just this.First, however, we have to specify what type of soliton we are hoping to construct. Inthe standard four-dimensional context, we are acquainted with solitonic solutions of manydimensionalities: Instantons are localized at a point in Euclidean spacetime and traceout a zero-dimensional worldsheet; magnetic monopoles are localized at a point in three-dimensional space and trace out a one-dimensional worldsheet; cosmic strings are localizedon a line in three-dimensional space and trace out a two-dimensional worldsheet and so on.Generically, we call a soliton whose instantaneous time slice has p-dimensional extension ap-brane.

In string theory, since the space-time dimension is ten, we could in principle findsolutions with p anywhere between zero and nine. In these lectures, we will study solutionswith p = 5: five-branes.

They are of particular interest because, as was shown some timeago by Teitelboim and Nepomechie [9] , five-branes are dual to fundamental strings in tendimensions in much the same way that magnetic monopoles are dual to electric chargein four dimensions. Dirac’s monopole argument has only to be modified by replacing theMaxwell field Aµ by the antisymmetric tensor field Bµν and changing the dimensionalityof spacetime.

The duality argument doesn’t guarantee that the dual objects actually exist,of course, but Strominger has shown that, at least perturbatively, the heterotic string doeshave soliton solutions with five-brane structure [7]. In these lectures we will concentrate onconstructing five-brane solitons: they are in some sense maximally ”stringy” and, more tothe point, many of them are amenable to exact conformal field theory analysis.

How suchobjects would manifest themselves phenomenologically is an interesting question whichdepends on the details of compactification down to four dimensions. This question has yetto be studied in any detail, and we will have little to say about it here.Let us attempt to construct a five-brane solution to (2.5).

The supersymmetry varia-tions are determined by a positive chirality Majorana-Weyl SO(9, 1) spinor ǫ. Becauseof the five-brane structure, it is useful to note that ǫ decomposes under SO(9, 1) ⊃SO(5, 1) ⊗SO(4) as16 →(4+, 2+) ⊕(4−, 2−)(2.6)3

where the ± subscripts denote the chirality of the representations. Denote world indicesof the four-dimensional space transverse to the fivebrane by µ, ν = 6 .

. .9 and the corre-sponding tangent space indices by m, n = 6 .

. .9.

We assume that no fields depend onthe longitudinal coordinates (those with indices µ = 0 . .

.5) and that the nontrivial tensorfields in the solution have only transverse indices. Then the gamma matrix terms in (2.5)are sensitive only to the SO(4) part of ǫ and, in particular, to its SO(4) chirality.One immediately sees how to make the gaugino variation vanish (in what follows wetreat ǫ as an SO(4) spinor and let all indices be four-dimensional): As a consequence of thefour-dimensional gamma-matrix identity γmnǫ± = ∓12ǫmnrsγrsǫ± one has Fmnγmnǫ± =∓eFmnγmnǫ±, where the dual field strength is defined by ˜Fmn = 12ǫmnrsF rs.

Therefore, δχvanishes if Fmn = ± ˜Fmn and ǫ = (4±, 2±) which is to say that if the gauge field is taken tobe an instanton, then δχ vanishes for all supersymmetries with positive SO(4) chirality!To deal with the other supersymmetry variations, we must adopt an ansatz for thenon-trivial behavior of the metric and antisymmetric tensor fields in the four dimensionstransverse to the five-brane (the specific form is inspired by the work of Dabholkar andHarvey [10] on string-like solitons). For the metric tensor we writegµν = e−2φδµνµ , ν = 6 .

. .9(2.7)and for the antisymmetric tensor field strengthHµνλ = √g4ǫµνλσ∂σφ = e−2φǫµνλσ∂σφ(2.8)where φ is to be identified with the dilaton field.

With this ansatz and the rather obviousvierbein choice emµ = δmµ e−φ, we can also calculate the generalized spin connections (2.4)which appear in (2.5) and (2.2):Ω±mnµ= δmµ∂nφ −δnµ∂mφ ± ǫµmnp∂pφ . (2.9)Now consider the δλ term in (2.5).

Because of the ansatz, both terms are linear in∂φ.By standard four-dimensional gamma-matrix algebra, the relative sign of the twoterms is proportional to the SO(4) chirality of the spinor ǫ. We have chosen the sign andnormalization of the ansatz for H so that δλ vanishes for ǫ ∈(4+, 2+).

Finally, consider thegravitino variation in (2.5). A crucial fact, following from (2.9) , is that while Ω± wouldin general be an SO(4) connection, with the chosen ansatz it is actually pure SU(2).

Tobe precise,Ω±mnµ γmnǫη = (γµp∂pφ)(1 ∓η)ǫη ,(2.10)so that Ω± annihilates the (4±, 2±) spinor. Since (2.5) involves only Ω+, it suffices to takeǫ to be a constant (4+, 2+) spinor to make the gravitino variation vanish.Putting all this together, we see that if we choose the gauge field to be any instantonand fix the metric and antisymmetric tensor in terms of the dilaton according to theabove ansatz, then the state is annihilated by all supersymmetry variations generated by aspacetime constant (4+, 2+) spinor.

Thus, half of the supersymmetries are unbroken, andthe other half, by standard reasoning [11] , will be associated with fermionic zero-modesbound to the soliton.4

The one unresolved question concerns the functional form of the dilaton field. Noticethat the ansatz for the antisymmetric tensor was given in terms of its three-form fieldstrength Hmnp, rather than its two-form potential Bmn.

This is potentially inconsistent,since the curl of the field strength must satisfy the anomalous Bianchi identity (2.3) .Within the ansatz (2.8) , the curl of H is given byd ∧H = 14!ǫrstu∂r{e−2φǫstuv∂vφ} ∼e−2φ(2.11)(whereis just the flat Laplacian) and one can thus, in principle, solve (2.3) for φ. Theslight problem with this approach is that (2.3) is only the leading order in α′ approximationto the true anomaly and the best we can hope to do is to construct solutions as a powerseries in α′. Since our goal is to find exact solutions, we adopt the different strategy oflooking for special backgrounds where the R ∧R −F ∧F anomaly on the r.h.s of (2.3)cancels.

If that is possible, the equation for d ∧H becomese−2φ = 0, an equation whichcan be solved once and for all with no expansion in powers of α′. The cancellation ofthe anomaly of course means that the underlying sigma model has been made effectivelyleft-right symmetric, a property which will play a key role in the proof that our solutionsare exact in α′.

It remains to show that desired cancellation can, in fact, be achieved.What is required, according to (2.3) , is that the curvature R(Ω−) should cancelagainst the instanton Yang-Mills field F. We will take the instanton to be embedded in anSU(2) subgroup of the gauge group (this is always the lowest-action instanton), so whatis needed is that Ω−be a self-dual SU(2) connection. The SU(2) condition we alreadyknow to be met, so the only issue is self-duality.

Given the special ansatz and coordinatesystem of (2.7) , it is easy to calculate the curvature of Ω±:R(Ω±)[mn]µν=δnν∇m∇νφ −δmµ∇m∇µφ −δmν∇n∇µφ + δmµ∇n∇νφ± ǫµmnα∇α∇µφ ∓ǫνmnα∇α∇νφ ,(2.12)where∇µ∇νφ =∂µ∂νφ −δµν∂m∂mφ + 2∂µ∂νφ ,∇2φ =2e2φe−2φ . (2.13)It is then a trivial arithmetical exercise to show that, in four dimensions and under thecondition that δmn∇m∇nφ = 0, Ωis self-dual:R(Ω±)[mn]µν= ∓12ǫµνλσR(Ω±)[mn]λσ.

(2.14)Since a self-dual SU(2) connection is an instanton connection, it will be possible to choosea gauge instanton which exactly matches the ”metric” instanton Ω−and makes the r.h.sof (2.3) vanish, thus making the whole solution self-consistent. In the next section, we willexplore the qualitative properties of the solutions which we have constructed in the aboverather roundabout manner.A feature of the above development which could cause confusion is the intricate in-terplay of the two non-Riemannian connections Ω(±).

To refresh the reader’s memory,5

we will summarize the essentials of this phenomenon (we denote the (4+, 2+) spinor byǫ+): The gravitino supersymmetry variation equation boils down to Ω[ab]+µ γ[ab]ǫ+ = 0 whichin turn implies that Rµν(Ω+)abγabǫ+ = 0. The index-pair interchange symmetry for anon-Riemannian connection, which reads R(Ω+)ab,cd = R(Ω−)cd,ab, allows us to convertthe previous condition for ǫ+ to Rµν(Ω−)abγµνǫ+ = 0.

If we then make the identifica-tion Fµν[ab] ∼Rµν(Ω−)[ab], we see that we have reproduced the gaugino supersymmetryvariation equation. This is simply to emphasize that, because of the crucial role of theantisymmetric tensor in these solutions, the precise way in which the Ω+’s and Ω−’s appearin the various equations we deal with is tightly constrained and quite critical.3.

Development and Interpretation of the SolutionsNow we will work out the geometry and physical interpretation of the solution de-scribed in the previous section. To recapitulate, we have found the following solution ofthe low-energy spacetime effective action of the heterotic string:ds2 =e−2φ(x)δµνdxµdxν + ηαβdyαdyβHµmn =ǫµmnp∂pφHµνλ = −12ǫµνλσ∂σe−2φFµν[mn] = eFµν[mn] = Rµν(Ω−)[mn] ,(3.1)where µν = 6 .

. .9 , αβ = 0 .

. .5 and ηαβ is the Minkowski metric.

The last equationexpresses the fact that the gauge field is a self-dual instanton with moduli chosen so thatit coincides (up to gauge transformations of course) with the curvature of the generalizedconnection of the theory. The consistency condition for all this is juste−2φ = 0.The solution of the consistency condition on φ is just a constant plus a sum of poles:e−2φ = e−2φ0 +NXi=1Qi(x −xi)2(3.2)The constant term is fixed by the (arbitrary) asymptotic value of the dilaton field, φ0 .

Instring theory, e−φ is identified with the local value of the string loop coupling constant,gstr. For the solution described by (3.2) , gstr goes to a constant at spatial infinity and goesto infinity at the locations of the poles!

We shall worry about the physical interpretation ofthis fact in due course. Now, the metric of our solution is conformally flat with conformalfactor given by (3.2) .

Since φ goes to a constant at infinity, the geometry is asymptoticallyflat, which is precisely what we want for a soliton interpretation. In the neighborhood ofa singularity, we can replace e−2φ by a simple pole Q/r2 and obtain the approximate lineelementds2 ∼Qr2 (dr2 + r2dΩ23)=dt2 + QdΩ23(3.3)6

where dΩ23 is the line element on the unit three-sphere and we have introduced a newradial coordinate t = √Qlog(r/√Q). This expression becomes more and more accurate ast →−∞.

In this same limit, the other fields are given byφ = t/pQH = Qǫ3 ,(3.4)where ǫ3 is the volume form on the three sphere. In Sect.

5 we will see that the linearbehavior of the dilaton field plays a crucial role in the underlying exact conformal fieldtheory. The geometry described by (3.3) is a cylinder whose cross-section is a three-sphereof constant area 2π2Q.

The global geometry is that of a collection of semi-infinite cylinders,or wormholes (one for each pole in e−2φ), glued into asymptotically flat four-dimensionalspace. The wormholes are semi-infinite since the approximation of (3.3) becomes betterand better as t →−∞and breaks down as t →+∞.

It is these wormholes which wepropose to interpret as solitons.A further crucial fact is that the residues, Q, are quantized. Consider an S3 whichsurrounds a single pole, of residue Q, in e−2φ.

The net flux of H through this S3 is entirelydue to the enclosed pole and can easily be calculated:Hijk = −12ǫijkl∂le−2φ = Qǫijkl∂lxl/x4ZS3H =2π2Q(3.5)The flux of H through the S3 at in finity is thus proportional to the sum of all the residues.By a familiar cohomology argument, however, the flux of H through any S3 must be aninteger multiple of some basic unit [12]. The point is that, if the flux of H is non-zero,then there cannot be a unique two-form potential B covering the whole sphere.

The bestone can do is to have two sections, B±, covering the upper and lower halves of the S3 andrelated to each other by a gauge transformation in an overlap region which is topologicallyan S2. Since the sigma model action involves B, not H, the non-uniqueness of B could leadto an ill-defined sigma model path integral.

It is possible to show that, with our definitionsof the sigma model action, this danger is avoided if and only if the flux of H is an integralmultiple of α′. The details of this argument can be found in [12] .

The consequence for usis that the residues of the poles in e−2φ are discretely quantized: Qi = niα′. As a result,the cross-sectional areas of the individual wormholes are quantized in units of 2π2α′ andthere is thus a minimal transverse scale size of the fivebrane.

(This fact may be useful infuture attempts to quantize the transverse fluctuations of the fivebrane. )Finally, we want to characterize the instanton component of this solution.

The keypoint is that, when the dilaton field satisfiese−2φ = 0, we can construct a self-dual SU(2)connection out of the scalar field φ and we want to identify this connection with the gaugeinstanton. But there is a well-known ansatz [13] for constructing multi-instanton solutionswhich has precisely this character: if we define an SU(2) connection byAµ(x) = ¯Σµν∂νlogf¯Σµν =12 ¯ηµνiσi(3.6)where ¯ηµνi is ’tHooft’s anti-self-dual symbol, then the condition of self-duality of the con-nection reduces tof = 0, an equation which has the general solutionf = 1 +NXi=1ρ2i(x −xi)2(3.7)7

(we normalize the behavior at infinity to unity since Aµ is invariant to rescaling f by anoverall constant). An important point is that the total instanton number of the solutionbuilt on f is N, the number of poles.

The gauge potential which follows from taking f tohave a single pole can be shown to beAµ = −2ρ2 ¯Σµνxνx2(x2 + ρ2) ,(3.8)an expression which one immediately recognizes as the singular gauge instanton of scalesize ρ centered at x = 0. The only way this can match our construction of a self-dualgeneralized connection is if we make the identificationf(x) = e2φ0e−2φ = 1 +NXi=1e2φ0Qi(x −xi)2 .

(3.9)Thus, given the solution (3.2) for the dilaton field, we can assert that the associated in-stanton has instanton number N, with instantons of scale size e2φ0Qi localized at positionsxi. Since the Qi are quantized, so are the instanton scale sizes.

The only free parameters(moduli) are the 4N center locations of the instantons. In ten dimensions, the multipleinstanton solution corresponds to multiple fivebranes with the locations in the transversefour-dimensional space of the individual fivebranes given by the center coordinates of theindividual instantons.An important fact about the solution we have just constructed is that it is not per-turbative in α′.

As we saw in the discussion following (3.1), the wormhole associated witha pole of residue Q = nα′ has a cross-section which is a sphere of area 2π2Q and thereforehas curvature R ∼1/Q. In the perturbative sigma model approach to strings in back-ground fields, one finds that the sigma model expansion parameter is α′R.

In the case athand, this becomes α′/Q = 1/n, which is obviously not small for the elementary fivebrane,which has n = 1. Since our solution has been constructed by solving the leading-order-in-α′beta function equations, ignoring all higher-order corrections, one can legitimately worrywhether it makes any sense.

In the next two sections we will present evidence that allhigher-order corrections to this particular solution actually vanish, and the leading-ordersolution is exact. We will briefly discuss other solutions of interest for which higher-ordercorrections don’t vanish.There is another perturbation theory issue to bring up here.

String theory has twoexpansion parameters: the string tension α′ and the string loop coupling constant gstr ∼e−2φ0. The latter is the quantum expansion parameter of string theory and, in this paper,we are working to zeroth-order in an expansion in gstr.

In effect, we are producing anexact solution, to all orders in α′, of classical string field theory. However, as we havealready pointed out, our solution has the unusual feature that gstr grows without limitdown the throat of a wormhole so that there is, strictly speaking, no reliable classicallimit!

Since virtually nothing is known about non-perturbative-in-gstr physics, we don’tknow what this means for the ultimate validity of this sort of solution. Similar issues arisein the matrix model/Liouville theory approach to two-dimensional quantum gravity, and8

we hope eventually to gain some insight from that source (for a review, see the contributionof David Gross to this School).Before proceeding to show that our solution is exact (in the sense just described),we want to briefly describe some inexact, but instructive, solutions of the basic heteroticfield equations. A particularly interesting possibility (and this was Strominger’s originalapproach to this problem [7]) is to proceed along the lines of Sect.

2, but to determinethe dilaton by solving the curl equation for H perturbatively in apm. To do this, combine(2.11) and (2.3)to givee−2φ ∼α′(trR ∧R −130TrF ∧F) .

(3.10)In a perturbative solution, this equation implies that ∂φ ∼O(α′) which, via (2.12) , impliesthat R ∼O(α′). Therefore, to leading order in α′, one is entitled to drop the R ∧R termin (3.10) .

Substituting the explicit gauge field strength for an instanton of scale size ρ,one obtains the following dilaton solution:e−2φ = e−2φ0 + 8α′ (x2 + 2ρ2)(x2 + ρ2)2 + O(α′2) . (3.11)The metric and antisymetric tensor fields are built out of this dilaton field according to thespacetime-supersymmetric ansatz of (2.7) and (2.8).

This solution is very different from theprevious one, obtained by setting d∧H = 0: The dilaton field and the metric are everywherefinite and the topology of the solution is R4 rather than R4 with semi-wormholes gluedin. One can examine higher-order in α′ corrections to the beta functions and verify thatthe solution must receive corrections.

At the same time, one can examine higher-ordercorrections to the supersymmetry transformations [14] and verify that it is possible tomaintain spacetime supersymmetry in the α′-corrected solution. These solutions are veryinteresting in their own right and certainly have a soliton interpretation.

On the otherhand, since we do not know how to deal with the higher-order in α′ corrections in anygeneral manner, we will not pursue this line of development here.Another interesting point concerns what happens when we lift the requirement ofspacetime supersymmetry and look for solutions of the beta function equations ratherthan the condition that some supersymmetry charges annihilate the solution. Our solu-tions have the property that the mass (the ADM mass, to be precise) per unit fivebranearea is proportional to the axion charge: M5 = 2πα′−3Q.

This equality can be understoodvia a Bogomolny bound: any solution of the leading-order field equations with the fivebranetopology must satisfy the inequality M5 ≥2πα′−3Q and our solution saturates the inequal-ity. One can easily imagine a process in which mass, but not axion charge, is increased bysending a dilaton wave down one of the wormhole throats.

Since the wormhole throat issemi-infinite, this wave need not be reflected back: It can continue to propagate down thethroat forever, leaving an exterior solution for which M5 > 2πα′−3Q. Such solutions of theleading-order beta function equations have indeed been found [15] and they resemble thefamiliar Reissner-Nordstrom sequence of charged black holes: They have an event horizonand a singularity, but the singularity retreats to infinity as the mass is decreased to the9

extremal value that saturates the Bogomolny bound. Perhaps not surprisingly, the non-extremal solutions are not annihilated by any spacetime supersymmetries, and we do notexpect to be able to find the corresponding exact conformal field theories.

Nonetheless,the fact that the extremal black hole is under exact string theory control should eventuallyallow us to make progress on understanding the string physics of black holes, Hawkingradiation and the like.In the rest of these lectures, we will pursue the much more limited goal of showingthat our special solution is an exact solution of string theory.4. Worldsheet Sigma Model ApproachTo show conclusively that a given spacetime configuration is a solution of string theory,we must show that it derives from an appropriate superconformal worldsheet sigma model.In this section we will show that the worldsheet sigma models corresponding to the five-branes constructed in section 2 possess extended worldsheet supersymmetry of type (4,4)The notation derives from the fact that in a conformal field theory, the left-moving fields(functions of z) and the right-moving fields (functions of ¯z) are dynamically independent.It is therefore possible to have different numbers of right- and left-moving superchargesQI±.

The general case, referred to as (p, q) supersymmetry, is described by the algebra{QI+, QJ+} =δIJ∂zI, J = 1 . .

.p{QI−, QJ−} =δIJ∂¯zI, J = 1 . .

.q{QI+, QJ−} =0 . (4.1)The minimal possibility, corresponding to a generic solution of the heterotic string, has(1, 0) supersymmetry.

Any left-right-symmetric, and therefore non-anomalous theory, willhave (p, p) supersymmetry (this is sometimes referred to as N=p supersymmetry). Themaximal possibility is (4, 4) which, it turns out, is what is realized in our fivebrane solution.We will argue that, in the (4,4) case, there is a nonrenormalization theorem which makesthe lowest-order in α′ solution for the spacetime fields exact.

The latter issue is closelyrelated to the question of finiteness of sigma models with torsion and with extended su-persymmetry [16,17] and the results we find are slightly at variance with the conventionalwisdom, at least as we understand it. We will comment upon this at the appropriate point.First we digress to explain why we expect four-fold extended supersymmetry in thisproblem.

The models of interest to us are structurally equivalent to a compactification often-dimensional spacetime down to six dimensions: there are six flat dimensions (along thefivebrane) described by a free field theory and four ‘compactified’ dimensions (transverseto the fivebrane) described by a nontrivial field theory. The fact that the ‘compactified’space is not really compact has no bearing on the supersymmetry issue.The definingproperty of all the fivebranes of section 2 is that they are annihilated by the generators ofa six-dimensional N=1 spacetime supersymmetry.

That is, they provide a compactificationto six dimensions which maintains N=1 spacetime supersymmetry. Now, it is well-knownthat in compactifications to four dimensions, the sigma model describing the six compact-ified dimensions must possess (2,0) worldsheet supersymmetry in order for the theory to10

possess N=1 four-dimensional spacetime supersymmetry [18]. Roughly speaking, the con-served U(1) current of the (2,0) superconformal algebra defines a free boson which is usedto construct the spacetime supersymmetry charges.

It is also known that, if one wantsto impose N=2 four-dimensional spacetime supersymmetry, the compactification sigmamodel must have (4,0) supersymmetry [19]. The conserved SU(2) currents of the (4,0)superconformal algebra are precisely what are needed to construct the larger set of N=2spacetime supersymmetry charges.

Since, by dimensional reduction, N=1 supersymmetryin six dimensions is equivalent to N=2 in four dimensions, the above line of argumentimplies that spacetime supersymmetric compactifications to six dimensions (including ourfivebrane) require a compactification sigma model with at least (4,0) worldsheet super-symmetry. Since our solution is constructed to cancel the anomaly, it will be left-rightsymmetric and therefore automatically of type (4, 4).Now we turn to a study of string sigma models.

The generic sigma model underlyingthe heterotic string describes the dynamics of D worldsheet bosons XM and D right-moving worldsheet fermions ψMR (where D, typically ten, is the dimension of spacetime)plus left-moving worldsheet fermions λaL which lie in a representation of the gauge groupG (typically SO(32) or E8 ⊗E8). The generic Lagrangian for this sigma model is writtenin terms of coupling functions GMN, BMN and AM which eventually get interpreted asspacetime metric, antisymmetric tensor and Yang-Mills gauge fields.

This Lagrangian hasthe explicit form [20]14πα′Zd2σ{GMN(X)∂+XM∂−XN + 2BMN(X)∂+XM∂−XN+iGMNψMR D−ψNR + iδabλaLD+λbL + 12(FMN)abψMR ψNR λaLλbL}(4.2)where H = dB. In this expression, the covariant derivatives on the left-moving fermionsare defined in terms of the Yang-Mills connection, while the covariant derivatives on theright-moving fermions are defined in terms of a non-riemannian connection involving thetorsion (which already appeared in section 2):D−ψAR =∂−ψAR + Ω−NAB∂−XNψBR,D+λaL =∂+λaL + ANab∂+XNλbL.

(4.3)As in section 2 we use indices of type M for coordinate space indices, type A for the tangentspace and type a for the gauge group. An absolutely crucial feature of this action is thatthe connection appearing in the covariant derivative of the right-moving fermions is thegeneralized connection Ω−, not the Christoffel connection.

This action has a naive (1, 0)worldsheet supersymmetry and can be written in terms of (1,0) superfields. Superconfor-mal invariance is broken by anomalies of various kinds unless the coupling functions satisfycertain ‘beta function’ conditions [1] which are equivalent to the spacetime field equationsdiscussed in section 2.

The dilaton enters these equations in a rather roundabout, but bynow well-understood, way [20].To proceed further, we must construct the specific sigma models corresponding tothe fivebrane solutions. For the generic fivebrane, (4.2) undergoes a split into a nontrivial11

four-dimensional theory and a free six-dimensional theory: the sigma model metric (asopposed to the canonical general relativity metric) then describes a flat six-dimensionalspacetime times four curved dimensions. The right-moving fermions couple via the kineticterm to the generalized connection Ω−, which acts only on the four right-movers lying inthe tangent space orthogonal to the fivebrane.

The other six right-movers are free (wemomentarily ignore the four-fermi coupling) so there is a six-four split of the right-moversas well. The left-moving fermions couple to an instanton gauge field which may or maynot be identified with the other generalized connection, Ω+.

In all the cases of interest tous, the gauge connection is an instanton connection and acts only in some SU(2) subgroupof the full gauge group, so that four of the left-movers couple nontrivially, while the other28 are free. Finally, the four-fermion interaction term couples together precisely those left-and right-movers which couple to the nontrivial gauge and Ω−connections and is thereforeconsistent with the six-four split defined by the kinetic terms.

The remaining variables canbe regarded as defining a heterotic, but free, theory (6 X, 6 ψR and 28 λL) living in the six‘uncompactified’ dimensions along the fivebrane. From now on, we focus our attention onthe nontrivial piece of (4.2) referring to the four-dimensional part of the split.

For stringtheory consistency, it must have a central charge of 6, which would be trivially true if theconnections were all flat, but is far from obvious for a fivebrane.Now let us further specialize to the sigma model underlying the left-right symmetric(and therefore non-anomalous) fivebrane solution of section 2. It is constructed by iden-tifying the gauge connection with the ‘other’ generalized connection Ω+ and making thatconnection self-dual by imposing the conditione−2φ = 0 on the metric conformal factor.The result of this is that the four bosonic coordinates transverse to the fivebrane and thefour nontrivially-coupled left- and right-moving fermions are governed by the worldsheetaction14πα′Zd2σ{Gµν(X)∂+Xµ∂−Xν + 2Bµν(X)∂+Xµ∂−Xν+iGµνψµRD−ψνR + iGµνλµLD+λνL + 12R(Ω+)µνλρψµRψνRλλLλρL}(4.4)where D± are the covariant derivatives built out of the generalized connections Ω± .

In fact,as long as the H appearing in Ω± is given by d ∧B, (4.4) is identical to the basic left-rightsymmetric, (1, 1) supersymmetric nonlinear sigma model with torsion [21] . Despite theapparent asymmetry of the coupling of λL to Ω+ and ψR to Ω−, the theory nonethelesshas an overall left-right symmetry (under which B →−B) and is non-anomalous.

Toexchange the roles of ψR and λL one has to replace the curvature of Ω−by that of Ω+.This exchange symmetry property relies on the non-riemannian relationR(Ω+)µνλρ = R(Ω−)λρµν(4.5)which indeed holds for the generalized connection (2.4) when d ∧H = 0. To summarize,we have shown that the heterotic sigma model describing the nontrivial four-dimensionalgeometry of the fivebrane is actually an example of a left-right symmetric sigma modelwith at least (1, 1) supersymmetry.

As we will now show, it actually has (4, 4) worldsheetsupersymmetry.12

We now turn to the question of extended supersymmetry. The basic worldsheet su-persymmetry of a (1, 1) model like (4.4) isδXM = ǫLψMR +ǫRψMLδψAL + [Ω+M]ABδXMψBL =∂XAǫR + .

. .δψAR + [Ω−M]ABδXMψBR =∂XAǫL + .

. ..(4.6)The worldsheet supersymmetry of the (1, 0) model is obtained by dropping the contribu-tions of ǫR and ψL.

The general structure of a possible second supersymmetry transfor-mation isˆδXM = ǫLfR(X)MNψNR +ǫRfL(X)MNψNLˆδψAL + [Ω+M]ABδXMψBL = −fL(X)AB∂XAǫR + . .

.ˆδψAR + [Ω−M]ABδXMψBR = −fR(X)AB∂XAǫL + . .

. .

(4.7)The function f is normalized and fully defined by the requirements that {ˆδ, δ} = 0 andthat ˆδ anticommute with itself to give ordinary translations as in (4.1) . The question is,what conditions must f satisfy in order for ˆδ to be a symmetry and how many of themcan there be?This question was first addressed in [22] for the case of left-right symmetric theorieswithout torsion (i.e.without an antisymmetric tensor coupling term).

The more complexcase of left-right symmetry with torsion was subsequently dealt with in [23,16,17]. Thebasic result is that the pair of tensors fR,L must be complex structures, covariantly constantwith respect to the appropriate connection:f 2± = −1D±Af±BC =∂Af±BC + Ω(±)ADBf±DC −Ω(±)ACDf±BD = 0 ,(4.8)where the ± notation is equivalent to the L, R notation.

The tensors in (4.8) are writtenin tangent space indices which is why the generalized spin connections Ω(±) appear in thecovariant derivative. The equation could, of course, also have been written in coordinateindices.

In general, it is not obvious that such a pair of complex structures can be found,but, if one can, we know that the sigma model actually possesses (2, 2) worldsheet super-symmetry. A further question is whether multiple pairs f (r)±of such complex structures canbe found.

If we can find p −1 of them, then the sigma model has (p, p) supersymmetry. Itturns out that the only consistent possibility for multiple complex structures is that therebe three of them [22] and that they satisfy the Clifford algebraf (r)± f (s)±= −δrs + ǫrstf (t)± .

(4.9)This corresponds to the case of (4, 4) supersymmetry. It is worth noting that each complexstructure leads to a conserved (chiral) current:J(r)±= ψA±(f (r)± )ABψB± .

(4.10)13

This yields a U(1) symmetry in the (2, 2) case and an SU(2) symmetry in the (4, 4) case.The question of left-right asymmetric theories, such as those which underlie the ‘non-exact’ fivebranes discussed in section 3, is more delicate. According to [16] , a heteroticsigma model will have (p, 0) supersymmetry if there are p−1 complex structures f (r)+whichare covariantly constant under the connection which couples to the right-moving fermions(those which do not couple to the gauge field) and if the gauge field (which affects the left-moving fermions) satisfies a condition which reduces, for a four-dimensional base space,to self-duality.

The latter condition is met for all of the fivebranes of interest to us sincethey are all built on instanton gauge fields. Thus, in all cases, the essential issue is theexistence of complex structures.To count complex structures, we will use the theorem that a complex structure canbe constructed from any covariantly constant spinor [24].

We start with a spinor η (in ourcase four-dimensional) of definite chirality (γ5η = ±η, say) and unit normalized (η†η = 1).Then we define a tensorJAB = −iη†γABη(4.11)which we will identify as a complex structure tensor (in tangent space indices and withindices raised and lowered by the identity metric). It is then automatic that if the spinoris covariantly constant with respect to some connection, so is JAB.

A simple Fierz identityargument, quite similar to that found on p.52 of [24] , then shows that J squares to −1(JABJBC = −δAC) and is indeed a complex structure.We are now ready to construct the explicit complex structures. As was explained inthe discussion after (2.10) , on the fivebrane, constant spinors of definite four-dimensionalchirality are covariantly constant.

Using the Weyl representation for the four-dimensionalgamma matrices, one has the following solutions of the two covariant constancy conditions:Dµ(Ω+)ǫ+ = 0 ⇒ǫ+ =χ0Dµ(Ω−)ǫ−= 0 ⇒ǫ−=0χ,(4.12)where χ is any constant two-spinor (which we might as well unit normalize). Since thereare three parameters needed to specify the general normalized two-spinor, there should bethree independent choices for the two-spinor χ and therefore three choices for both ǫ+ andǫ−.

We will define the independent χr (r = 1, 2, 3) as those which give expectation valuesof the spin operator along the three coordinate axes:χ†rσiχr = δir. (4.13)This finally leads, with the help of (4.11) , to the following set of three right- and left-handed complex structures:J+1 =iσ200iσ2J−1 =−iσ200−iσ2J+2 =01−10J−2 =0−σ3σ30J+3 =0iσ2iσ20J−3 =0−σ1σ10.

(4.14)14

It is trivial to show that the J+ commute with all the J−and that they satisfy the Cliffordalgebra (4.9). These are precisely the conditions needed to generate (4, 4) supersymmetryin a left-right symmetric theory (or (4, 0) supersymmetry in a heterotic theory).Thecomplex structures are thus extremely simple indeed.Finally, we come to the questions of finiteness and need for higher-order in α′ correc-tions to our solutions.

It is rather firmly established that two-dimensional nonlinear sigmamodels with (4, 4) supersymmetry are in fact finite. The general proof was given quitesome time ago by Alvarez-Gaume and Freedman [22] and it consists in showing that no(4, 4)-invariant counterterms of the needed dimension can be constructed.

If the theoryis finite, the beta-functions get no higher-order corrections and the choice of backgroundfields which made the beta functions vanish at leading order must continue to make themvanish at all orders in α′. Confirmation of this comes from a construction by Gates et.al.

[23], using (2, 2) superfields, of the most general (4, 4)-invariant action. The functionalform of the action must satisfy certain conditions in order to have (4, 4) supersymme-try and, with hindsight, one can see that the most general solution of these conditionscorresponds precisely to our special multi-fivebrane solution.As an aside, we mention that it has been argued that one really only needs (4, 0)supersymmetry to achieve finiteness [16].

This would apply to variations on the solutiondescribed in Sect. 2 in which, for example, the gauge instanton scale size did not matchthe wormhole throat transverse scale size.

In the discussion given earlier in this section,we recall that the existence and properties of the right-moving complex structures f (+)ihave nothing to do with the properties of the gauge field (which governs the left-movingcomplex structures). So, if we keep the same metric then we should have the same f (+)iand thus at least a (4, 0) supersymmetry.

While the solution may well exist, the anomaliesprobably mean that there will be corrections to the beta functions so that the theory isnot finite, but constructible order by order. This subject has yet to be explored in anydetail.5.

Algebraic CFT ApproachIt is one thing to show that a sigma model is a superconformal field theory, as wehave done in the previous section, and quite another to be able to classify its primaryfield content and calculate n-point functions of its vertex operators. Indeed, in order toanswer all the interesting questions about string solitons, it would be desirable to have asdetailed an algebraic understanding of the underlying conformal field theory as we alreadyhave for, say, the minimal models.

We are far from having such an understanding, but inthis section we will see that useful insight can be gained by studying a special limit whichemphasizes the semi-wormhole throat.Recall from section 2 that the (four-dimensional part of the) metric of the symmetricsolution has the formds2 = e−2φdx2(5.1)where dx2 is the flat metric on R4 ande−2φ(x) = e−2φ0 +nX1Qi(x −xi)2 . (5.2)15

The singularities in e−2φ are associated with the semi-wormholes. Taking n = 1 and thelimit e−2φ0 →0 givese−2φ = Qx2 ,(5.3)which is the solution corresponding to the wormhole throat itself.Using spherical co-ordinates centered on the singularity, and defining a logarithmic radial coordinate byt = √Qlnpx2/Q, the metric, dilaton and axion field strength of the throat may be writtenin the formds2 = dt2 + QdΩ23,φ = −t/pQ,H = −Qǫ,(5.4)where dΩ23 is the line element and ǫ the volume form of the unit 3-sphere obeyingRǫ = 2π2.The geometry of the wormhole is thus a 3-sphere of radius √Q times the open line R1and the dilaton is linear in the coordinate of the R1.Remarkably, these metric andantisymmetric tensor fields are such that the curvatures constructed from the generalizedconnections, defined in (2.4) , are identically zero, reflecting the parallelizability of S3.The axion charge Q is integrally quantized.

So, since Q appears in the metric, the radiusof the S3 is quantized as well.The sigma model defined by these background fields is an interesting variant of theWess-Zumino-Witten model and the underlying conformal field theory can, it turns out,be analyzed in complete detail. The basic observation along these lines was made in [25] inthe lorentzian context and euclideanized in [26,27]: the S3 and the antisymmetric tensorfield are equivalent to the O(3) Wess-Zumino-Witten model of levelk = Qα′ ,(5.5)while the R1 and the linear dilaton define a Feigin-Fuks-like free field theory with a back-ground charge induced by the linear dilaton.

Both systems are conformal field theories ofknown central charges:cwzw =3kk + 2cff = 1 + 6k. (5.6)The shift of the R1 central charge away from unity is a familiar background charge effectwhich has been exploited in constructions of the minimal models [28] and in cosmologicalsolutions [25].For the combined theory to make sense, the net central charge must be four.

Let usfor the the moment consider the bosonic string. If we expand cwzw in powers of k−1 (thiscorresponds to the usual perturbative expansion in powers of α′), we see instead thatctot = cwzw + cff = 4 + O(k−2 ∼α′2) .

(5.7)But, we should not have expected to do any better: the field equations we solved in section2 to get this solution are only the leading order in α′ approximation to the full bosonic16

string theory field equations and we must expect higher-order corrections to the fields andcentral charges. In fact, this issue can be studied in detail and it can be shown [29] thatthe metric and antisymmetric tensor fields are not modified and that the only modificationof the dilaton is to adjust the background charge of the R1 (i.e.

the coefficient of the linearterm in φ) so as to maintain ctot exactly equal to four.While this is quite interesting, we are really interested in the superstring case. Theleading-order-in-α′ metric, dilaton etc.

fields are the same as in the bosonic case (and,because of the non-renormalization theorems, we expect no corrections to them) but variousfermionic terms are added to the previous purely bosonic sigma model. The structure isthat of the (1,1) worldsheet supersymmetric sigma model (4.4) discussed in section 3.

Thereis still an S3×R1 split, but the component theories are supersymmetrized versions of Wess-Zumino-Witten and Feigin-Fuks. The Feigin-Fuks theory is still essentially free.

In thesupersymmetric WZW theory, the four-fermi terms vanish identically because, as pointedout above, the generalized curvature vanishes for this background. As a consequence, thegeneralized connections are locally pure gauge and can be eliminated from the fermionkinetic terms by a gauge rotation of the frame field.

Since the fermions are effectively free,they make a trivial addition to the central charges of both the S3 and the R1 models:cwzw =3kk + 2 + 32cff = 1 + 6k + 12 . (5.8)There is, however, a small subtlety: the gauge rotation which decouples the fermions ischiral, and therefore anomalous, because the left- and right-moving fermions couple to twodifferent pure gauge generalized connections, Ω+ and Ω−.

The entire effect of this anomalyon the central charge turns out to be the replacement in cwzw of k by k −2 (the detailscan be found in [30]) with the result thatcwzw = 3(k −2)k+ 32ctot = cwzw + cff = 6 . (5.9)Six is, of course, exactly the value we want for the central charge.

The remarkable factis that, in the supersymmetric theory, the expansion of cwzw in powers of k−1 terminatesat first non-trivial order and no modification of the dilaton field is needed to maintain thedesired central charge of six. These results are consistent with the non-renormalizationtheorems discussed in section 4, but are not tied to perturbation theory, since they de-rive from exactly-solved conformal field theories.

On the other hand, since the presentdiscussion makes no reference to the (4,4) supersymmetry which was crucial in provingthe perturbative non-renormalization theorems of section 4, an important element is stillmissing.This is a good point to remind the reader of the hierarchy of superconformal algebras.Much of what we know about conformal field theory comes from studying the represen-tation theory of these algebras. The basic N=1 superconformal algebra is contains anenergy-momentum tensor T(z) and its superpartner G(z).

The essential information is17

contained in the singular terms in their operator product expansion:T(z)T(w) =c/2(z −w)4 +2T(w)(z −w)2 + ∂wT(w)(z −w) + . .

.T(z)G(w) =32G(w)(z −w)2 + ∂wG(w)(z −w) + . .

.G(z)G(w) =c/6(z −w)3 +12T(w)(z −w) + . .

. .

(5.10)The central charge c is unconstrained. All superstring theories have at least this muchworldsheet supersymmetry.

The N=2 superconformal algebras differ from this by havinga conserved current J(z) and two supercharges G±(z) distinguished by the value (±1) oftheir charge with respect to the current J(z). This charge also plays a key role in the GSOprojection which rids the theory of tachyons.

The important new algebraic relations arecontained in the operator productsJ(z)G±(w) = ± G±(w)(z −w) + . .

.G±(z)G±(w) ∼1G+(z)G−(w) =c/6(z −w)3 +12J(w)(z −w)2 + . .

. .

(5.11)There is an N=1 subalgebra generated by T(z) and G(z) =1√2(G+(z) + G−(z)). The‘practical’ utility of the N=2 algebra is that the conserved current defines a free field Hby the relation J(z) = ip c3∂zH(z) and this free field can be used to construct the N=1spacetime supersymmetry charge in a compactification to four dimensions.

Once again,the central charge, c, is unconstrained. One further extension, to four supercharges, turnsout to be possible (and it can be shown [22] that this is the maximal extension).

Thereare now three conserved currents Ji which generate an SU(2) Kac-Moody algebra and thesupercharges Gα(z), ¯Gα(z) are in I = 1/2 representations of the conserved SU(2). Therelevant operator product expansions areT(z)T(w) =3k(z −w)4 +2T(w)(z −w)2 + ∂wT(w)(z −w) + .

. .Gα(z) ¯Gβ(w) =kδαβ(z −w)3 + 12Ji(w)σiαβ(z −w)2 + 12T(w)δαβ(z −w) + .

. .Ji(z)Jj(w) = −12kδij(z −w)2 + ǫijkJk(w)(z −w) + .

. .Ji(z)Gα(w) = −12σiαβGβ(w)(z −w) + .

. .

. (5.12)The triplet of conserved charges is what is needed to construct the larger spacetime su-persymmetry algebra associated with a compactification down to six, rather that four,18

dimensions.The SU(2) Kac-Moody algebra is of arbitrary level k, but we can see bycomparison with (5.10) that the central charge is constrained to be 6k. Since the level isconstrained by unitarity to be integer, the only allowed values of the central charge are6, 12, .

. .

. Fortunately, c = 6 is just what we need, and this suggests that the N=4 algebrawill be important role for us.We will now show that a closer examination of the algebraic structure of the wormholeconformal field theory reveals the existence of just the right extended supersymmetry.

Animportant clue to understanding the structure of the (4, 4) superconformal symmetry comesfrom the fact that there must be two SU(2) Kac-Moody symmetries: The first is part ofthe standard N=4 superalgebra. This algebra contains the energy-momentum tensor T(z),four supercurrents Ga(z) and three currents Ji(z) of conformal weight 1, which generatean SU(2) Kac-Moody algebra of a level tied to the conformal anomaly (in our case, levelone).

The second is the SU(2) Kac-Moody algebra of the Wess-Zumino-Witten part ofthe wormhole conformal field theory. It has a general level n, related to the area of thewormhole cross-section (or, equivalently, its axion charge) and is clearly distinct from theN=4 SU(2) Kac-Moody.

Since the superconformal algebra is quite tightly constrained, itis not a priori obvious that such an SU(2) ⊗SU(2) Kac-Moody is compatible with N=4supersymmetry and useful information, such as restrictions on allowed values of the centralcharge, might be obtained by explicitly constructing the algebra (assuming a consistent oneto exist). Quite remarkably, precisely the algebra we need has already been constructedby Sevrin et.al.

[31] , who discovered an alternate N=4 superalgebra, containing anSU(2) ⊗SU(2) ⊗U(1) Kac-Moody algebra, which had been missed in previous attemptsat a general classification of extended superalgebras. In what follows* we briefly summarizeenough of their work to explain its significance for the wormhole problem and, in particular,to verify the assertions made in section 4 about the rˆole of (4,4) supersymmetry.

In additionto establishing the presence of a (4, 4) superconformal symmetry, this construction is auseful starting point for studying the correspondence between the instanton moduli spaceand perturbations of the superconformal field theory.The construction discovered by Sevrin et.al. goes as follows: Start with the bosonicWZW model for an SU(2) ⊗U(1) group manifold (this is the geometry of the wormhole ifwe let the radius of the U(1) be infinite).

The conformal model contains four dimension-oneKac-Moody currents Ja satisfying the usual KM algebra:J0(z)J0(w) = −12(z −w)−2J0(z)Ji(w) =O((z −w)0)Ji(z)Jj(w) = −12nδij(z −w)−2 + ǫijk(z −w)−1Jk(w)(5.13)where i = 1, 2, 3 indexes the currents of an SU(2) algebra of level n and J0 is the currentof the U(1) algebra. This is supersymmetrized by adding a set of four dimension-1/2 fieldsψa satisfying the free fermion algebraψa(z)ψb(w) = −12δab(z −w)−2(5.14)* This discussion was developed in collaboration with E. Martinec19

(this is motivated by the arguments given earlier in this section that the fermions in asupersymmetric wzw model are, modulo anomalies, free).As usual, the Sugawara construction provides an energy-momentum tensorT(z) = −J0J0 −1n + 2JiJi −∂ψaψa(5.15)with respect to which the fields Ja (ψa) are primaries of weight 1 (1/2) and which has theexpected Swzw conformal anomalycswzw =3nn + 2 + 3 = 6(n + 1)/(n + 2) . (5.16)There is also a Sugawara-like construction of four real supersymmetry charges Ga, witha = 0, .., 3 :G0 =2[J0ψ0 + (1/√n + 2)Jiψi + (2/√n + 2)ψ1ψ2ψ3]G1 =2[J0ψ1 + (1/√n + 2)(−J1ψ0 + J2ψ3 −J3ψ2) −(2/√n + 2)ψ0ψ2ψ3](5.17)(plus cyclic expressions for G2 and G3).

These supercharges could have been packagedas a complex I = 1/2 multiplet, as in (5.12) . The operator product expansion of thesesupercharges with themselves readsGa(z)Gb(w) =4(n + 1)(n + 2)δab(z −w)−3 + 2δabT(w)(z −w)−1−8[1n + 2α+iab A+i (w) + n + 1n + 2α−iab A−i (w)](z −w)−2−4[1n + 2α+iab ∂A+i (w) + n + 1n + 2α−iab ∂A−i (w)](z −w)−1(5.18)whereα±iab = ±δi[aδjb] + 12ǫijk(5.19)andA−i = ψ0ψi + ǫijkψjψkA+i = −ψ0ψi + ǫijkψjψk + Ji(5.20)are commuting SU(2) Kac-Moody algebras of levels 1 and n+1, respectively.The c-number term (the central charge) and the term involving T(z) are obligatory in any higher-N superalgebra, while the terms involving dimension 1 operators are what differentiatethe various possible extended superalgebras.With further effort, one shows that theG · A± OPE generates combinations of Ga and ψa while the G · ψ OPE yields A± and J0.No new operators appear in further iterations, so the complete algebra generated by thesupercharges contains just T (dimension 2), Ga (dimension 3/2), Ai± and J0 (dimension1) and ψa (dimension 1/2).

The Kac-Moody algebra defined by the dimension 1 operatorsis evidently SU(2) ×SU(2) ×U(1) , which accords with our expectations derived from thewormhole geometry.20

The superalgebra whose construction we have outlined above is a particular exampleof a one-parameter family of N=4 algebras dubbed the Aγ algebras. The only problemwith it is that the sigma model analysis of extended supersymmetry (see for example[17]) makes quite clear that the canonically defined supercharges and energy-momentumtensor must satisfy the standard N=4 algebra, which closes on T, Ga and a single level-oneSU(2) Kac-Moody algebra Ji.

The supercharges defined above obviously do not have thatproperty. However, if we ‘improve’ them as follows˜T = T −1n + 2∂J0˜Ga = Ga −2n + 2∂ψa ,(5.21)we can show that ˜T , ˜Ga and Ai−(the level-one Kac-Moody current) close on themselvesand enjoy precisely the standard N=4 superalgebra.

This says that the full algebra hasthe standard algebra as a subalgebra, perhaps no great surprise.This improvement has a simple physical interpretation: J0 generates a U(1) symmetrywhich can be regarded as a translation in a free coordinate ρ (that is, we can write J(z) ∼∂zρ(z) where ρ is a free scalar field).The original algebra (5.18) makes no referenceto the dilaton and corresponds physically to a constant dilaton field. It is well-knownthat, if one turns on a dilaton which is linear in a free coordinate ρ, this has the effectof adding a term proportional to ∂2zφ ∼∂zJ0(z) to T(z) and shifting the central chargeof the superconformal algebra by a constant.

With a little care we can show that thelinear dilaton implicit in (5.21) is precisely what we obtained earlier in this section inour discussion of the WZW-Feigin-Fuks conformal field theory of the wormhole. This is afurther piece of evidence that the improved energy-momentum tensor ˜T is the physicallyrelevant one.

Now comes the miracle: T is, in any event, not physically acceptable becauseit has a central charge of 6(n + 1)/(n + 2). The central charge of ˜T , however, can easilybe shown to be 6, precisely the required value!This shows that there is an exact conformal field theory of just the right central chargeassociated with the wormhole geometry and verifies the key rˆole of N=4 extended super-symmetry in establishing the physics of the model.

There are many fivebrane applicationsof this exact wormhole conformal field theory which are just beginning to be worked out.Perhaps the most interesting concern the vertex operators of excitations about the worm-hole, among which one must find the moduli of the exact solutions. In any event, thisline of argument shows that the dramatic consequences of (4,4) superconformal symmetry,which we first extracted from perturbative considerations, seem to have nonperturbativestatus.6.

ConclusionIn these lectures, we have constructed a special set of conformal field theories whichhave the interpretation of soliton solutions of heterotic string theory. We first constructedthem as solutions of the leading order in α′ beta function conditions and then showed that,owing to an extended worldsheet supersymmetry, the associated nonlinear sigma modelis an exact conformal field theory.

It is the existence of an explicit and exact conformalfield theory associated with the soliton solution which distinguishes the solution described21

here from previous attempts to construct string theory solitons. There are several lines ofinquiry which can be pursued now that ”exact” string solitons exist.

One issue concernsthe mass of the soliton. In all previous discussions of string solitons, the mass has beencomputed using the lowest-order spacetime effective action (2.1) and is therefore knownonly to lowest order in α′.

It would obviously be desirable to know the mass exactly,but for that one needs to develop a conformal field theory method of computing solitonmasses. Our exact soliton conformal field theory should provide a useful laboratory fordeveloping such methods.

A second issue is the question of stringy collective coordinatesand their semiclassical quantization. It should be an instructive challenge to translate thewell-known standard field theory physics of collective coordinates into the string theorycontext.

This is a nontrivial matter because motion in collective coordinate space becomesmotion in a space of conformal field theories and it is a nontrivial matter to find the actionassociated with such motions (and knowing the exact underlying conformal field theoriesshould help). Yet another question to pursue is that of stringy black hole physics.

Wenoted in Sect. 3 that our solitons were similar to the extreme Reissner-Nordstrom blackholes in the sense that, while they have no singularity or event horizon, if one increasestheir mass by any finite amount (while keeping the axion charge fixed), an event horizonand a singularity (lying at a finite geodesic distance from any finite point) will appear.Such black hole solitons can easily be created by scattering some external particle on anextremal soliton and, by studying stringy scattering theory about the extremal soliton,one should be able to explore, in a controlled way, how a stringy black hole Hawkingradiates and the nature of the final state it approaches.

These are quite difficult questions,but having precise control of the underlying conformal field theory may allow us to makeprogress on them. Perhaps it will be possible to report on progress along these lines at thenext Swieca Summer School.22

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