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DEFORMATIONS, SYMMETRIES AND TOPOLOGICAL

arXiv:hep-th/9109055v1 27 Sep 1991RU91-14DOE/ER40325-2-TASKBDEFORMATIONS, SYMMETRIES AND TOPOLOGICALDEGREES OF FREEDOM OF THE STRING*Mark Evans and Ioannis Giannakis†The Rockefeller University1230 York AvenueNew York, NY 10021-6399AbstractWe discuss three closely related questions; i) Given a conformal field theory,how may we deform it? ii) What are the symmetries of string theory?

and iii) Doesstring theory have free parameters? We show that there is a distinct deformationof the stress tensor for every solution to the linearised covariant equations of mo-tion for the massless modes of the Bosonic string, and use this result to discuss thesymmetries of the string.

We also find an additional finite dimensional space ofdeformations which may correspond to free parameters of string theory, or alterna-tively may be interpreted as topological degrees of freedom, perhaps analogous tothe isolated states found in two dimensions. * Talk presented at the XX Meeting on Differential Geometric Methods in Theoretical Physics,June 3–7, 1991, at Baruch College, New York City.† Work supported in part by the Department of Energy Contract Number DOE-ACO2-87ER40325, TASK B

Deformations, Symmetries and Topological . .

.1. IntroductionConformal field theories (with appropriate central charge) are solutions tothe classical equations of motion of string theory [1], so that by studying infinites-imal deformations which preserve this conformal structure, we are examining thelinearised classical equation of motion about the corresponding solution.

This is aninteresting problem in its own right, but it also gives us insight into the symmetrystructure of string theory. We normally think of finding symmetries by lookingfor transformations on the fields which leave invariant the action of a theory.

Thisrequires that we solve the daunting problems of string field theory before we can ad-dress the problem of symmetry. However, there is an alternative approach; we maysimply work with the equations of motion of the theory and find transformationswhich take one solution into another without changing the physics.

A symmetry istherefore a particular case of a deformation.Finally, we may shed some light on the question of whether string theoryhas free parameters. For a given conformal field theory, the interactions of all thephysical states appear to be uniquely prescribed, if we want both Lorentz invarianceand unitarity in space-time.Different conformal field theories are supposed tocorrespond simply to different background solutions of the equations of motion.This supposition has a testable consequence, that every deformation of a conformalfield theory correspond to a deformation of the background value of some physicalfield of the string.

We shall find that there are in fact deformations of the usualcritical bosonic string which do not correspond to physical fields, and so may beinterpreted as free parameters, or, equivalently, as isolated states or topologicaldegrees of freedom. As such they are perhaps higher dimensional analogues of theisolated states of two-dimensional string theory which have been the focus of somuch discussion at this conference [2].2.

Conformal Field Theory.There are several equivalent definitions of a conformal field theory, of varyingdegrees of sophistication, but for our purposes we shall take the simplest which hasthe added advantage of making the algebraic properties manifest. In the parlanceof physicists we shall consider a Hamiltonian formulation; that is we shall takeour world-sheet to be a cylinder (twice-punctured sphere if you prefer) with anarbitrarily chosen cycle around it, parameterised by a single real coordinate σ,running from 0 to 2π.

A Conformal Field Theory consists of the following, definedon this cycle:1) an algebra A of operator valued distributions, usually called fields2) a representation of this algebra, and3) two distinguished fields, T(σ) and T(σ) which satisfy two mutually commut-ing copies of the Virasoro algebra:2

Deformations, Symmetries and Topological . .

. [T(σ), T(σ′)] = −ic24πδ′′′(σ −σ′) + 2iT(σ′)δ′(σ −σ′) −iT ′(σ′)δ(σ −σ′)(1a)[T(σ), T(σ′)] = ic24πδ′′′(σ −σ′) −2iT(σ′)δ′(σ −σ′) + iT′(σ′)δ(σ −σ′)(1b)[T(σ), T(σ′)] = 0(1c)and which generate motion around the cycle:L0 −L0, φ(σ′)=Zdσ[T(σ) −T(σ), φ(σ′)] = −iφ′(σ′)∀φ(σ′) ∈A(2)(A prime may denote differentiation with respect to σ).

These special fields are thetwo non-vanishing components of the energy momentum tensor of the field theory,and so include the Hamiltonian, H,H = L0 + L0 =Zdσ(T(σ) + T(σ))(3)which may be used to define the evolution of fields offthe cycle, although we shallnot use this fact.Since V ir × V ir ⊂A, and any subalgebra acts on its parent through com-mutation (the adjoint action), the elements of A will themselves be grouped intorepresentations of V ir × V ir. It is therefore natural to define Primary Fields ofdimension (d, d), which transform simply under the adjoint action, by[T(σ), Φ(d,d)(σ′)] = idΦ(d,d)(σ′)δ′(σ −σ′) −(i/√2)∂Φ(d,d)(σ′)δ(σ −σ′)[T(σ), Φ(d,d)(σ′)] = −idΦ(d,d)(σ′)δ′(σ −σ′) −(i/√2)∂Φ(d,d)(σ′)δ(σ −σ′)(4)The symbols ∂and ∂indicate differentiation with respect to the light-cone coordi-nates x± = (σ ± τ)/√2, and so take us offthe space-like cycle.

From our algebraicpoint of view, we should think of the symbol ∂φ(σ′) as meaning i√2[L0, φ(σ′)] forany field φ(σ′) ∈A, with a similar meaning for ∂. The definition of primary field,Eq.

4, is thus an empty tautology for the zero modes of the energy momentumtensor, but is non-trivial for the others.To conclude this section, we will try to make clearer what is meant by thebelief that conformal field theories are solutions of the classical equations of motionof the string. Consider a string moving in some space-time with metric G, then anatural choice for the two-dimensional field theory to describe this situation isT(σ) = 12Gµν(X)∂Xµ∂Xν(σ)T(σ) = 12Gµν(X)∂Xµ∂Xν(σ)(5)where the X are scalar fields which can be thought of as coordinates for the stringin space-time, and∂Xµ(σ) = 1√2(πµ(σ) + Gµν(X)Xν′(σ))∂Xµ(σ) = 1√2(πµ(σ) −Gµν(X)Xν′(σ))(6)3

Deformations, Symmetries and Topological . .

.and π(σ) is the momentum conjugate to X; the only non-vanishing bracket amongthe X and π is[πµ(σ), Xν(σ′)] = −iδνµδ(σ −σ′)(7)(This definition of ∂X is consistent with the one given above). If our bracket isthe Poisson bracket, then the T and T defined in Eq.

5 satisfy V ir × V ir for allchoices of G. However, if, as we want, they are defined as normal ordered (withrespect to the Fourier modes of X and π) products of fields, and the bracket is acommutator, then they satisfy V ir × V ir only when G satisfies certain conditionswhich look something like the Einstein equations of motion. Since the spectrumof the bosonic string includes a state which has all the properties of a disturbanceof the space-time metric (a, “graviton,”), this condition of conformal invariance isnaturally interpreted as an equation of motion for this physical field.

It is one of thegoals of the work described in this talk to clarify this relationship between equationsof motion for space-time fields and conformal field theories, and so to generalise itto include the full, infinite set of space-time fields.3. Deformations of Conformal Field Theories.3.1 The Deformation Equations.Having defined a conformal field theory in the previous section, we may nowconsider making an infinitesimal deformation which preserves the axioms listedabove.

We may in principle deform a conformal field theory through any of itselements, viz. the algebra A, the distinguished fields T(σ) and T(σ) or even therepresentation (deforming the cycle should make no difference).

However, physicistsusually have a canonical choice for all elements of a theory except its Hamiltonian,and so we shall consider only changes in the fields T(σ) and T(σ). We are thusinterested in deforming the embedding V ir×V ir ⊂A.

The more general problem ofdeforming a morphism of algebras has been discussed in the mathematical literature[3].We must preserve V ir × V ir, including the value of the central charge, c,and the fact that L0 −L0 generates translations. This last fact means that L0 −L0may deform at most by a central element, and will generally be invariant.

To firstorder, then, δT(σ) and δT(σ) must satisfy[δT(σ), T(σ′)] + [T(σ), δT(σ′)] = 2iδT(σ′)δ′(σ −σ′) −iδT ′(σ′)δ(σ −σ′)(8a)[δT(σ), T(σ′)] + [T(σ), δT(σ′)] = −2iδT(σ′)δ′(σ −σ′) + iδT′(σ′)δ(σ −σ′) (8b)[δT(σ), T(σ′)] + [T(σ), δT(σ′)] = 0(8c)We shall refer to Eq. 8 as the deformation equations.3.2 Canonical Deformations.Let Φ(1,1)(σ) be a primary field of dimension (1,1), then the deformationequations 8 are satisfied byδT(σ) = δT(σ) = Φ(1,1)(σ)(9)4

Deformations, Symmetries and Topological . .

.This result [4] follows from the definition of a primary field, Eq. 4, and weshall call such deformations canonical.

An alternative, and world-sheet covariant,discussion of canonical deformations has been given by Campbell, Nelson and Wong[5]. Note that such a deformation mixes left and right moving sectors, so that itis not sufficient to consider just one sector.

Also, since δT(σ) = δT(σ), T(σ) −T(σ) is an invariant of the canonical deformation class, and its zero mode thereforecontinues to generate translations, satisfying Eq. 2.Canonical deformations have a number of features which indicate that theyare indeed the way we should, “turn on,” a space-time field:1) They agree with our preconceptions of how massless fields appear in theenergy momentum tensor.

Varying the space-time metric G in Eq. 5, in-cluding the implicit dependence made explicit in Eq.

6, yields a canonicaldeformation.2) (1,1) primary fields are in natural correspondence with the physical states ofstring theory, being the vertex operators which create asymptotic physicalstates and describe their scattering. This means that canonical deformationshave a straightforward interpretation as changes in space-time fields.3) Canonical deformations work for massive states just as well as they do formassless, and avoid certain ambiguities and pathologies which may be im-plicit in other approaches.Note that the third of these virtues seems to involve some small revision of thestandard lore on the relationship between sigma models and strings: in particular,the, “standard,” sigma model, containing only terms of naive dimension two, notonly puts the massive fields on shell, but also puts them equal to zero.Appealing though they are, canonical deformations have a significant draw-back; they correspond to turning on space-time fields in a particular gauge.

This ismost easily seen in an example. For simplicity, consider a conformal field theory offree scalars, defined by the energy momentum tensor of Eq.

5 with G the standardflat Minkowski metric. A short calculation soon shows that primary (1,1) fields ofnaive dimension two are of the formΦ(1,1) = Hµν(X)∂Xµ∂Xν(10)where the coefficient functions H must satisfy certain conditions, if we are workingin the quantum case where all fields are understood to be normal ordered and thebracket is a commutatorHµν(X) = 0(11)∂µHµν(X) = ∂νHµν(X) = 0(12)The first of these is an equation of motion, something that we would expect to arisein making a conformal deformation, but Eq.

12 is a gauge condition (of course, (11)is only the correct equation of motion when this gauge condition holds). As weshall explain in the next section, this gauge condition is a serious nuisance whenwe come to the problem of symmetry, and motivated us to explore [6] more generalsolutions of the deformation equations (see an alternative approach in [7]).5

Deformations, Symmetries and Topological . .

.4. Symmetries.A symmetry is a change in the space-time fields which does not change thephysics (i.e.

all masses and S-matrix elements are unchanged). Since the physics isdetermined by the conformal field theory corresponding to the field configurationin question, a transformation on the fields will be a symmetry if the correspondingconformal field theories are isomorphic.

From our definition of a conformal fieldtheory in section 2, it is clear what we mean by such an isomorphism; there mustexist an isomorphism of the two operator algebras, ρ: A1 →A2 which maps energymomentum tensors on to one another. In particular, if ρ is an automorphism suchthat ρ(TΦ(σ)) = TΦ+δΦ(σ), then Φ →Φ + δΦ is a symmetry transformation.

HereΦ is a space-time field configuration, and as such indexes the energy momentumtensors of conformal field theories; for example, in Eq. 5, all space-time fields Φ arezero except for the metric G.Even inner automorphisms are interesting in this context, and appear to giverise to the gauge symmetries of string theory.

In this talk we shall restrict ourselvesto inner automorphisms, and so we are interested in infinitesimal operators h ∈Asuch thati[h, TΦ(σ)] = TΦ+δΦ(σ) −TΦ(σ)(13)(Changing all operators by their commutator with a fixed infinitesimal operator, h,is an algebra isomorphism by virtue of the Jacobi identity; this is the infinitesimalversion of a similarity transformation). The right hand side of Eq.

13 is a deforma-tion, which is one reason for being interested in the subject. There is a large classof operators h which make the right hand side of Eq.

13 a canonical deformation:Let h be the sum of zero modes of (1,0) and (0,1) primary fields, thenh generates a canonical deformation.The proof of this statement [4] is a straightforward application of the definition ofa primary field, Eq. 4, and the Jacobi identity.Despite its simplicity, this is a very interesting class of symmetries.

It in-cludes the familiar general coordinate and two-form gauge invariances, generatedby the, “obvious,” such currents ξµ(X)∂Xµ and ζµ(X)∂Xµ, as well as an infiniteclass of higher symmetries generated by currents which classically would have higherdimension, such as ψµνλ(X)∂Xµ∂Xν∂Xλ. In each case, these currents are primaryfields of dimension (1, 0) or (0, 1) only if the parameters of the transformations, ξ,ζ, ψ .

. ., satisfy certain differential constraints.Differential constraints on parameters are quite familiar from field theory.They are nothing to do with the equations of motion for the fields, but arise fromthe demand that a transformation preserve a gauge condition.

Since canonical de-formations are associated with with gauge conditions, as in Eq. 12, it is inevitablethat gauge transformations corresponding to canonical deformations have differ-ential constraints on their parameters.Correspondingly, if we wish to lift theseconstraints we must find a more general set of deformations, unaccompanied bysuch gauge conditions.6

Deformations, Symmetries and Topological . .

.Exhibiting the explicit transformations on the space-time fields is possibleonly for those conformal field theories over which we have good computationalcontrol, such as free bosons.Nevertheless, with such examples as guides, it ispossible to draw certain conclusions about this class of symmetries [4]. They are allgauge symmetries, and most or all of the states of the string are the correspondinggauge fields.

The higher symmetries are spontaneously broken (so that the massivestates are so by virtue of the Higgs mechanism), and mix states at different masslevels. It is also possible to argue that the symmetric solutions of string theoryshould correspond to topological world-sheet field theories, as suggested by Witten.However, to exhibit the claims of the previous paragraph more concretely,and to use these insights about symmetry to do physics, we need more.

We mustunderstand precisely what the set of generators is for a wider class of conformalfield theories, and we must relax the differential constraints on the parameters ofthe transformations, alluded to above (for the higher symmetries, these constraintsexclude the global transformations which encode so much of the physics). We shalladdress the second of these problems and, in so doing, learn something about thefirst [6].

As we argued above, if we are to relax the differential constraints, we mustfirst understand more than canonical deformations, and that will be the subject ofthe next section.5. Beyond Canonical Deformations.Can we find fully gauge covariant deformations, or does conformal invariancenecessarily come accompanied with a gauge condition?

We shall consider the sim-plest possible case [6], that of turning on massless fields of the bosonic string aboutflat twenty-six dimensional space-time. Thus our starting conformal field theory istwenty-six free bosons, defined by Eq.

5, with G the canonical Minkowski metric.We shall attempt to solve the deformation equations, Eq. 8, but will take a moregeneral ans¨atz than usual, the most general operator of naive dimension two:δT = Hνλ(X)∂Xν∂Xλ + Aνλ(X)∂Xν∂Xλ+ Bνλ(X)∂Xν∂Xλ + Cν(X)∂2Xν + Dλ(X)∂2XλδT = Hνλ(X)∂Xν∂Xλ + Aνλ(X)∂Xν∂Xλ+ Bνλ(X)∂Xν∂Xλ + Cν(X)∂2Xν + Dλ(X)∂2Xλ(14)The tensors Hνλ .

. .

Dλ are initially taken to be completely independent.Thisans¨atz is then substituted into the deformation equations, Eq. 8.

Some tediousbut straightforward manipulation reduces these conditions, after an appropriateredefinition, to the following:δT(σ) = Kνλ∂Xν∂Xλ +∂−∂[Cν∂Xν −Dλ∂Xλ]δT(σ) = Kνλ∂Xν∂Xλ −∂−∂[Cν∂Xν −Dλ∂Xλ](15)7

Deformations, Symmetries and Topological . .

.Note that δT and δT differ only by a derivative with respect to σ, so that L0 −L0is invariant, as argued in section 3.1. The quantities C, C, D, D are give in terms ofK, which we interpret as the sum of the graviton and two-form, by∂νCν = 0(16)Dλ = −12∂µKµλ(17)Dλ = −12∂µKλµ(18)∂λCν = 12Kνλ −12∂ν∂µKµλ(19)∂λCν = 12Kλν −12∂ν∂µKλµ(20)At first sight there is no equation of motion for the physical field K, and thedilaton is nowhere to be seen.

However Eqs. 19 and 20 cannot be solved for C andC for arbitrary K. There is an integrability condition which K must satisfy whichturns out to yield both an equation of motion and the dilaton:Kνλ −∂ν∂µKµλ −∂λ∂µKνµ + ∂ν∂λKµµ = ∂ν∂λφ + ανλ(21)for some scalar function φ, which we identify as the dilaton.Eq.

16 yields thedilaton equation of motion,φ = 0.We have thus achieved our goal of finding deformations which are associatedwith a covariant equation of motion and no gauge condition. We get exactly theexpected linearised equations, except for the last term on the right hand side ofEq.

21. α is antisymmetric in its indices and constant. This term therefore indicatesthat there is a finite dimensional space of additional deformations, over and abovethose associated with turning on the physical fields.What are we to make of these additional deformations?

Some caution isappropriate [6], since our calculations are only to first order in the deformation, butthere is nonetheless a natural interpretation of these parameters α. With loweredindices, α is a harmonic two-form, and so may be integrated over compact two-dimensional submanifolds of space-time.

This integration may be pulled back tothe world-sheet to yieldSα =ZαµνdXµdXν(22)which, since α is closed, is invariant under deformations of the map X. In thelanguage of non-linear sigma models, Sα is proportional to an instanton number, andwhen added to the usual action (a “θ-term”) will usually have physical consequences.In particular, if Sα is added to the usual action with a coefficient proportionalto ln ǫ (where ǫ is, say, the parameter of dimensional regularisation), then it willclearly amend the condition for conformal invariance [8].

The equations of motionwill then be affected in just the way we see in Eq. 21.

It is worthy of note that, atleast in some cases, the topological charge density does have just such an ultravioletdivergence at one loop [9].8

Deformations, Symmetries and Topological . .

.Given this interpretation of these additional deformations, we may quitereasonably refer to α either as free parameters or topological degrees of freedom ofstring theory. Their description as θ-terms is very similar to that given by otherspeakers at this conference of the isolated states found in two-dimensional stringtheory [2].6.

Symmetries Redux.Finally, let us set α = 0 and return to the question of symmetries. With theapparently covariant set of deformations found in the previous section, we can nowfind the full set of unbroken gauge transformations as inner automorphisms.

It isstraightforward to see thath =Zdσξµ(X)∂Xµ + ζµ(X)∂Xµ,(23)generates deformations of the type given in Eq. 15, and that the fields transformin the conventional way.

Note that we have achieved our goal of eliminating thedifferential constraints on the parameters ξ and η.These explicit calculations need to be extended in two ways; we would like tounderstand the higher symmetries, and we would like to know how the generatorsof symmetry deform, if at all, with the conformal field theory. To do this let us firstunderstand why the operators of Eq.

23 had to generate a symmetry. The argumenthas two parts: i) turning on the space-time fields without restricting their gaugecorresponded to the most general conformal deformation by world-sheet fields, andii) commuting the free energy-momentum tensor with the h of Eq.

23 produces adeformation of this form (by conservation of naive dimension). Hence it has tobe possible to pull back the inner automorphism generated by h to a symmetrytransformation on the space-time fields.It is very tempting to generalise this argument to the massive fields and thehigher symmetries.

That is, we might conjecture that arbitrary space-time fields areturned on, without imposing any restriction as to gauge, by considering an ans¨atzwhich is the obvious generalisation of Eq. 14 to the appropriate naive dimension, anddemanding that it be a conformal deformation, satisfying Eq.

8. Then a deforma-tion of this form would necessarily be generated by the corresponding dimensionalgeneralisation of the generator in Eq.

23. The problem is that this symmetry ap-pears to be larger than we need; working out some examples [6] it is easy to seethat h contains many more degrees of freedom than there are gauge conditions torelax.It therefore seems likely that the most general conformal deformation ofhigher naive dimension contains not just the physical space-time fields, but alsounphysical auxiliary fields, which are pure gauge artifacts.

This is very much akinto the situation which arises in the superspace formulation of supersymmetric gaugetheories, where there exist auxiliary gauge artifacts over and above those neededto account for the halving of fermionic degrees of freedom in going on shell. If weare willing to live with these auxiliary fields, then we have answered the question9

Deformations, Symmetries and Topological . .

.of which operators generate the higher symmetries. On the other hand, if we wishto restrict ourselves to the physical, propagating fields with non-trivial dynamics(“Wess-Zumino gauge”), then it remains to be determined what the appropriaterestrictions on the generators are.What happens as when we consider a theory other than that of free bosons?In the discussion of section four, the generators were associated with primary fields,which deform with the conformal field theory and so are not known explicitly overthe whole deformation class.

However the two preceeding paragraphs contain nomention of primary fields. The only place where a specific property of the free theorywas used was an occasional appeal to conservation of naive dimension, but this wasonly a convenience which allowed us to discuss one mass-level at a time, and maybe dispensed with.

It would seem, then, that deforming the set of operators whichgenerate canonical deformations is complicated because of the need to preserve agauge condition, rather than to ensure an interpretation as a transformation onspace-time fields. Space-time fields (including auxiliaries) are turned on with themost general solution of the deformation equations in terms of world-sheet fields,and so symmetries are generated by all operators which commute with the generatorof translations, L0 −L0.

Since this operator is an invariant of the deformation class,so are the symmetry generators. In summary:Symmetries are generated by the centraliser of L0 −L0, and this setis invariant over the whole deformation class.Acknowledgements.M.E.

would like to thank Burt Ovrut for collaboration on some of the earlywork described here, and S. Catto and A. Rocha for the invitation to talk at a verystimulating conference.References. [1] C. Lovelace, Phys.

Lett. 135B (1984), 75; C. Callan, D. Friedan, E. Martinecand M. Perry, Nucl.

Phys. B262 (1985), 593; A. Sen, Phys.

Rev. D32 (1985),2102.

[2] A. Polyakov, Mod. Phys.

Lett. A6 (1991), 635; B. Lian and G. Zuckerman,Phys.

Lett. 254B (1991), 417 and Yale preprint YCTP-P18-91; D. Gross, I.Klebanov and M. Newman, Nucl.

Phys. B350 (1990), 621.

[3] M. Gerstenhaber and S. Schack in Deformation Theory of Algebras andStructures, ed. M. Hazewinkel and M. Gerstenhaber (Kluwer, Dordrecht,1988).

[4] M. Evans and B. Ovrut, Phys. Rev.

D41 (1990), 3149; Phys. Lett.

231B(1989), 80. [5] M. Campbell, P. Nelson and E. Wong, University of Pennsylvania ReportUPR-0439T (1990)[6] M. Evans and I. Giannakis, Rockefeller Preprint RU91-2 (1991)10

Deformations, Symmetries and Topological . .

. [7] P. Nelson, Phys.

Rev. Lett.

62 (1989), 993; H-S. La and P. Nelson, Nucl.Phys. B332 (1990), 83.

[8] A. Tseytlin, private communication[9] M. Evans, Nucl. Phys.

B208 (1982), 122; Ph.D. Thesis, Princeton University(1983).11

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