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A K¨ahler Structure on the Space

arXiv:alg-geom/9305012v1 26 May 1993A K¨ahler Structure on the Spaceof String World-SheetsClaude LeBrun∗Department of MathematicsSUNY Stony BrookAbstractLet (M, g) be an oriented Lorentzian 4-manifold, and consider thespace S of oriented, unparameterized time-like 2-surfaces in M (stringworld-sheets) with fixed boundary conditions.Then the infinite-dimensional manifold S carries a natural complex structure and acompatible (positive-definite) K¨ahler metric h on S determined by theLorentz metric g. Similar results are proved for other dimensions andsignatures, thus generalizing results of Brylinski [2] regarding knots in3-manifolds. Generalizing the framework of Lempert [5], we also inves-tigate the precise sense in which S is an infinite-dimensional complexmanifold.Running title: The Space of String World-Sheets∗Supported in part by NSF grant DMS-92-04093.1

1IntroductionGiven a collection of circles in a 4-dimensional oriented Lorentzian space-time, one may consider the space S of unparameterized oriented time-likecompact 2-surfaces with the given circles as boundary. The main purpose ofthe present note is to endow S with the structure of an infinite-dimensionalK¨ahler manifold— i.e.

with both a complex structure and a Riemannianmetric for which this complex structure is covariantly constant. This wasmotivated by a construction of Brylinski [2], whereby a K¨ahler structureis given to the space of knots in a Riemannian 3-manifold.In fact, ourdiscussion will be structured so as to apply to codimension 2 submanifolds ofa space-time of arbitrary dimension and metrics of arbitrary signature, withthe proviso that we only consider those submanifolds for which the normalbundle is orientable and has (positive- or negative-)definite induced metric;thus Brylinski’s construction becomes subsumed as a special case.As the reader will therefore see, complex manifold theory thus comesnaturally into play when one studies codimension 2 submanifolds of a space-time.

On the other hand, complex manifold theory makes a quite differentkind of appearance when one attempts to study the intrinsic geometry of2-dimensional manifolds. If some interesting modification of string theorycould be found which invoked both of these observations simultaneously,one might hope to thereby explain the puzzling four-dimensionality of theobserved world.Many of the key technical ideas in the present note are straightforwardgeneralizations of arguments due to L´aszl´o Lempert [5], whose lucid study ofBrylinski’s complex structure is based on the theory of twistor CR manifolds[4].

One of the most striking features of the complex structures in question isthat, while they are formally integrable and may even admit legions of localholomorphic functions, they do not admit enough finite-dimensional com-plex submanifolds to be locally modeled on any complex topological vectorspace. This beautifully illustrates the fact, emphasized by Lempert, that theNewlander-Nirenberg Theorem [6] fails in infinite dimensions.2

2The Space of World-SheetsLet (M, g) be a smooth oriented pseudo-Riemannian n-manifold. We usethe term world-sheet to refer to a smooth compact oriented codimension-2submanifold-with-boundary Σn−2 ⊂Mn for which the inner product inducedby g on the conormal bundleν∗Σ := {φ ∈T ∗M|Σ| φ|TΣ ≡0}of Σ is definite at each point.

If g is Riemannian, this just means an orientedsubmanifold of codimension 2; on the other hand, if (M, g) is a Lorentzian4-manifold, a world-sheet is exactly an oriented time-like 2-surface.Definition 1 Let (M, g) be a smooth oriented pseudo-Riemannian n-manifold, and let Bn−3 ⊂Mn be a smooth codimension-3 submanifold whichis compact, without boundary. We will then let SM,B denote the space ofsmooth oriented world-sheets Σn−2 ⊂M such that ∂Σ = B.Of course, this space is sometimes empty— as happens, for example,if B is a single space-like circle in Minkowski 4-space.

This said, SM,B isautomatically a Fr´echet manifold, and its tangent space at Σ isTΣSM,B = {v ∈Γ(Σ, C∞(νΣ)) | v|∂Σ ≡0} .Indeed, if we choose a tubular neighborhood of Σ which is identified withthe normal bundle of an open extension Σε of Σ beyond its boundary, ev-ery section of νΣ →Σ which vanishes on ∂Σ is thereby identified with animbedded submanifold of M, and this submanifold is still a world-sheet pro-vided the C1 norm of the section is sufficiently small. This provides SM,Bwith charts which take values in Fr´echet spaces, thus giving it the desiredmanifold structure.Since the normal bundle νΣ = (ν∗Σ)∗= TM/TΣ = (TΣ)⊥of our world-sheet is of rank 2 and comes equipped with an orientation as well as a metricinduced by g, we may identify νΣ with a complex line bundle by takingJ : νΣ →νΣ, J2 = −1 to be rotation by +90◦.This then defines anendomorphism J of TS byJ : TΣSM,B →TΣSM,B : v →J ◦v .3

Clearly J 2 = −1, so that J gives S the structure of an almost-complexFr´echet manifold— i.e. every tangent space of S can now be thought of asa complex Fr´echet space by defining J to be multiplication by √−1.

In thenext sections, we shall investigate the integrability properties of this almost-complex structure.3Integrability of the Complex StructureLet (M, g) denote, as before, an oriented pseudo-Riemannian manifold. LetGr+2 (M) denote the bundle of oriented 2-planes in T ∗M on which the innerproduct induced by g is definite.

This smooth (3n−4)-dimensional manifoldthen has a natural CR structure [4, 7] of codimension n −2. Let us reviewhow this comes about.Let ˆN ⊂[(C ⊗T ∗M) −T ∗M] denote the set of non-real null covectors ofg, and let N ⊂P(C⊗T ∗M) be its image in the fiber-wise projectivization ofthe complexified cotangent bundle.

There is then a natural identification ofN with Gr+2 (M). Namely, using pairs u, v ∈T ∗xM of real covectors satisfying⟨u, v⟩= 0 and ⟨u, u⟩= ⟨v, v⟩, we define a bijection between these two spacesbyGr+2 (M) ∋oriented span(u, v) ↔[u + iv] ∈N ⊂P(C ⊗T ∗xM)which is independent of the representatives u and v. But, letting ϑ = P pjdxjdenote the canonical complex-valued 1-form on the total space of C⊗T ∗M →M, and letting ω be the restriction of dϑ to ˆN, the distributionˆD = ker(ω : C ⊗T ˆN →C ⊗T ∗ˆN)is involutive by virtue of the fact that ω is closed; since ˆD also containsno non-zero real vectors as a consequence of the fact that ˆN ∩T ∗M = ∅,ˆD is a CR structure on ˆN, the codimension of which can be checked tobe n −2.

This CR structure is invariant under the natural action of C×on ˆN by scalar multiplication, and thus descends to a CR structure D onN = Gr+2 (M), again of codimension n −2. Moreover, ϑ| ˆN descends to N asa CR line-bundle-valued 1-formθ ∈Γ(N, E1,0(L)) ,¯∂bθ = 0 ,4

where, letting T 1,0N := (C ⊗TN)/D, L⊗(n−1) = V(2n−3) T 1,0N, E1,0(L) :=C∞(L ⊗(T 1,0N)∗), and ¯∂b is naturally induced by d|D.The CR structure D of N may be expressed in the formD = {v −iJv | v ∈H}for a unique rank 2n −2 sub-bundle H of the real tangent bundle TN anda unique endomorphism J of H satisfying J2 = −1.In these terms thegeometric meaning of the CR structure of N is fairly easy to describe. Indeed,if ̟ : Gr+2 (M) →M is the tautological projection, then HP = (̟∗P)−1(P)for every oriented definite 2-plane P ⊂TM.

On vertical vectors, J acts bythe standard complex structure on the quadric fibers of N →M; whereas Jacts on horizontal vectors by 90◦rotation in the 2-plane P ⊂TM. This pointof view, however, obscures the fact that D is both involutive and unalteredby conformal changes g 7→efg.A compact (n −2)-dimensional submanifold-with-boundary S ⊂N, willbe called a transverse sheet if its tangent space is everywhere tansverse tothe CR tangent space of N:TN|S = TY ⊕H|Y .As before, let Bn−3 ⊂M denote a compact codimension-3 submanifold, andlet ̟ : N →M be the canonical projection.

We will then let ˆSN,B denotethe set of transverse sheets S ⊂N such that ̟ maps ∂S diffeomorphicallyonto B. Thus ˆSN,B is a Fr´echet manifold whose tangent space at S is givenbyT ˆSN,B|S = {v ∈Γ(S, C∞(H|S)) | ̟∗(v|∂S) ≡0} ,and hence J : H →H induces an almost-complex structure ˆJ on ˆSN,B byˆJ (v) := J ◦v.Proposition 3.1 The almost-complex structureˆJ on the space ˆSN,B oftransverse sheets is formally integrable— i.e.τ(v, w) := ˆJ [v, w] −[v, ˆJ w] −[ ˆJ v, w] −ˆJ [ ˆJ v, ˆJ w] = 0for all smooth vector fields v, w on ˆSN,B.5

Proof.The Fr¨ohlicher-Nijenhuis torsion τ(v, w) is tensorial in the sensethat its value at S only depends on the values of v and w at S.GivenvS, wS ∈{v ∈Γ(S, C∞(H|S)) | ̟∗(v|∂S) ≡0}, we will now define preferredextensions of them as vector fields near S ∈ˆSN,B in such a manner as tosimplify the computation of τ(v, w) = τ(vS, wS). To do this, we may firstuse a partition of unity to extend vS and wS as sections ˆv, ˆw ∈Γ(N, C∞(H))defined on all of of N in such a manner that ˆv and ˆw are tangent to thefibers of ̟ along all of ̟−1(B).

Now let U ⊂N be a tubular neighborhoodof S which is identified with the normal bundle H of some open extensionSǫ of S, and let ˆU ⊂ˆSN,B be the set of transverse sheets S′ ⊂U. We maynow define our preferred extensions of v and w of vS and wS on the domainˆU by letting the values of v and w at S′ ⊂U be the restrictions of ˆv and ˆwto S′.

Notice that [v, w] is then precisely the vector field on ˆU induced by[ˆv, ˆw], whereas ˆJ v is the vector field induced by Jˆv. Since the integrabilitycondition for (N, D) says thatJ([ˆv, ˆw] −[Jˆv, J ˆw]) = [ˆv, J ˆw] + [Jˆv, ˆw] ,it therefore follows thatˆJ [v, w] −ˆJ [ ˆJ v, ˆJ w] = [v, ˆJ w] + [ ˆJ v, w] ,so that τ(v, w) = 0, as claimed.We now observe that there is a canonical imbeddingSM,BΨ֒→ˆSN,BΣ7→νΣobtained by sending a world-sheet to its normal-bundle, thought of as theimage of a section of Gr+2 (M)|Σ = N|Σ; thought of in this way, it is easy tosee that νΣ ⊂N is a transverse submanifold.Theorem 3.2 The imbedding Ψ realizes (SM,B, J ) as a complex submani-fold of ( ˆSN,B, ˆJ ).

In particular, the almost-complex structure J of SM,B isformally integrable.Proof. The projection ̟ : N →M induces a map ˆ̟ : ˆSN,B →SM,B whichis a left inverse of Ψ and satisfies ˆ̟∗ˆJ = J ˆ̟∗.

It therefore suffices to show6

that the tangent space of the image of Ψ is ˆJ -invariant. Now the conditionfor a transverse sheet S ⊂N to be the Ψ-lifting of the world-sheet ̟(S) ⊂Mis exactly that θ|S ≡0.

When S satisfies this condition, a connecting fieldv ∈Γ(S, E(H)) then represents a vector ˆv ∈T ˆS which is tangent to theimage of Ψ iffvdθ)|TS + d(vθ)|TS ≡0 ;(1)the exterior derivative of θ may here be calculated in any local trivializa-tion for the line bundle L, since the left-hand side rescales properly underchanges of trivialization so as define an L-valued 1-form on S. But sinceθ ∈Γ(N, E1,0(L)) satisfies ¯∂bθ = 0, it follows that(Jvdθ)|TS + d(Jvθ)|TS = i(Jvdθ)|TS + id(Jvθ)|TSbecause θ and dθ are of types (1,0) and (2,0), respectively. The tangent spaceof the image of Ψ is therefore ˆJ -invariant, and the claim follows.Definition 2 Let (X, J) be an almost-complex Fr´echet manifold, and let f :U →C be a differentiable function defined on an open subset of X.

We willsay that f is J-holomorphic if(Jv)f = ivf∀v ∈TU .Definition 3 An almost-complex Fr´echet manifold (X, J) is called weaklyintegrable if for each real tangent vector w ∈TX there is a J-holomorphicfunction f defined on a neighborhood of the base-point of w such that wf ̸= 0.Theorem 3.3 Suppose that (M, g) is real-analytic. Then ( ˆSN,B, ˆJ ) is weaklyintegrable.Proof.

If (M, g) is real-analytic, so is the CR manifold (N, D), and we cantherefore realize (N, D) as a real submanifold of a complex manifold (2n−3)-manifold N . This can even be done explicitly by taking N to be a space ofcomplex null geodesics for a suitable complexification of (M, g).Now let S ⊂N ⊂N be any transverse sheet.

Then there is a neigh-borhood V ⊂N of S which can be holomorphically imbedded in some Cℓ.7

Indeed, let Y ⊂N be a totally real (2n −3)-manifold containing S, letf : Y →Rℓbe a smooth imbedding, and let Y0 be a precompact neighbor-hood of S ⊂Y with smooth boundary. By [3],the component functions f j|Y0are limits in the C1 topology of the restrictions of holomorphic functions.Using such an approximation of f, we may therefore imbed Y0 as a totallyreal submanifold of Cℓby a map which extends holomorphically to a neigh-borhood of Y0, and this holomorphic extension then automatically yields aholomorphic imbedding of some open neighborhood V ⊃S in Cℓ.Now suppose that v is a smooth section of H along S. We may expressv uniquely as u + Jw, where u and w are tangent to the Y .

By changingY if necessary, we can furthermore assume that u ̸≡0. Let F be a smoothfunction on Y which vanishes on S and such that the derivative vF is non-negative and supported near some interior point of S; and let ϕ be a real-valued smooth (n −2)-form on Y whose restriction to S is positive on thesupport of vF.

Set ψ = Fϕ. Using our imbedding of Y in Cℓ, we can expressψ as a family of component functions— e.g.

by arbitrarily declaring that allcontractions of ψ with elements of the normal bundle (TY )⊥⊂TCℓshallvanish. But, again by [3], these component functions are C1-limits on Y0 ofrestrictions of holomorphic functions from a neighborhood of Y0 ⊂Cℓ.

Thus,by perhaps replacing V with a smaller neighborhood, there is a holomorphic(n −2)-form β on V which approximates ψ well enough thatℜeZS vdβ > 12ZS udψ > 0andℜeZ∂S vβ > −12ZS udψ .Let ˆV := {S′ ∈SN,B | S′ ⊂V }, and define fβ : ˆV →C by fβ(S′) =RS′ β.Then fβ is a holomorphic function on the open set ˆV ⊂SN,B. Indeed, if γ isany smooth (n−2)-form on V , and if we set fγ(S′) =RS′ γ, then, for S′ ⊂V ,the derivative of fγ in the direction of w ∈Γ(S′, C∞(H)), ̟∗(w)|∂S′ ≡0, isgiven bywfγ|S =ZS wdγ +Z∂S′ wγ ;for if w is extended to V as a smooth vector field ˆw tangent to the fibers of8

̟ and St is obtained by pushing S′ along the flow of the vector field ˆw, thenwfγ|S′=ddtZSt γt=0=ZS′ £ˆwγ=ZS′[ ˆwdγ + d( ˆwγ)]=ZS′ wdγ +Z∂S′ wγ .But since β is the restriction of a holomorphic (n −2)-form from a region ofN , it therefore follows that( ˆJ w)f|S′=ZS′(Jw)dβ +Z∂S′(Jw)β=iZS′ wdβ + iZ∂S′ wβ=iwf|S′ ,showing that the function fβ induced by β is ˆJ -holomorphic, as claimed.However, we have also carefully chosen β so that the real part of the ex-pressionRS vdβ +R∂S vβ = vfβ is positive. For every real tangent vectorv on SN,B, one can thus find a locally-defined J -holomorphic function whosederivative is non-trivial in the direction v. Hence ˆJ is weakly integrable, asclaimed.Corollary 3.4 If (M, g) is real-analytic, then (SM,B, J ) is weakly integrable.Proof.

By Theorem 3.2 and 3.3, (SM,B, J ) can be imbedded in the weaklyintegrable almost-complex manifold ( ˆSN,B, ˆJ ).Since the restriction of aholomorphic function to an almost-complex submanifold is holomorphic, itfollows that (SM,B, J ) is weakly integrable.One might instead ask whether (SM,B, J ) is strongly integrable— i.e. lo-cally biholomorphic to a ball in some complex vector space.

The answer is no;in contrast to any strongly integrable almost-complex manifold, (SM,B, J )contains very few finite-dimensional complex submanifolds:9

Proposition 3.5 Suppose that (M, g) is real-analytic. At a generic pointS ∈SM,B, a generic (n −1)-plane is not tangent to any (n −1)-dimensionalJ -complex submanifold.Proof.

Let S ⊂M be a world-sheet which is not real-analytic near p ∈S.Let q ∈N = Gr+2 (M) be given by q = TpS⊥, and let v1, . .

. , vn−1 ∈Hq be aset of real vectors such that the vj + iJvj form a basis for Dq.

Extend thesevectors as sections ˆvj of H|Ψ(S) which satisfy equation (1) along the sheet;this may be done, for example, by first extending each vj to just a 1-jet at psatisfying (1) at p, projecting this to a 1-jet via ̟ to yield a 1-jet of a normalvector field on S ⊂M, extending this 1-jet as a section of the normal bundleof S, and finally lifting this section using Ψ∗. Let u1, .

. .

, un be the elementsof TSSM,B represented by ˆv1, . .

. , ˆvn−1, and let P ⊂T 1,0S SM,B be spanned byu1 −iJ u1, .

. .

, un −iJ un.Now suppose there were an J -holomorphic submanifold X ⊂SM,Bthrough S with (1,0)-tangent plane equal to P. Then X represents a fam-ily of transverse sheets in N which foliates a neighborhood U ⊂N of q;moreover, because X represents a holomorphic family, the leaf-space projec-tion ℓ: U →X is CR in the sense that ℓ∗(D) ⊂T 0,1X. Since we haveassumed that (M, g) is real-analytic, we may also assume that U has areal-analytic CR imbedding U ֒→C2n−3.

Moreover the twistor CR mani-fold N is automatically “anticlastic,” by which I mean that the Levi formL : D →TN/H : v 7→[v, ¯v] mod H is surjective at each point of N. Thisgives rise to a Bochner-Hartogs extension phenomenon: every CR functionon U extends to a holomorphic function on some neighborhood of U ⊂C2n−3.In particular, every CR map defined on U must be real-analytic, and this ap-plies in particular to the leaf-space projection ℓ. Thus Ψ(S) is real-analyticnear q, and S is therefore real-analytic near p. This proves the result bycontradiction.4K¨ahler StructureThe complex structure J on S depends only on the conformal class [g] ={efg} of our metric, but we will now specialize by fixing a specific pseudo-Riemannian metric g. Our reason for doing so is that we thereby induce an10

L2-metric on S. Indeed, each tangent spaceTΣSM,B = {v ∈Γ(Σ, C∞(νΣ)) | v|∂Σ ≡0}may be equipped with a positive-definite inner product by settingh(v, w) :=ZΣ g(v, w) dvolg|Σ.We shall now see that this metric has some quite remarkable properties.Theorem 4.1 The Riemannian metric h on S is K¨ahler with respect to thepreviously-defined complex structure J .Proof. Let Ωdenote the volume n-form of g, and define a 2-form ω on S byω(v, w) :=ZΣ(v ∧w)Ω.Obviously, ω is J -invariant andh(v, w) = ω(J v, w).We therefore just need to show that ω is closed.1To check this, let usintroduce the universal familyFSMpπ❏❏❏❏❏❫✡✡✡✡✡✢1 The reader may ask whether it is actually legitimate to call a Riemannian mani-fold K¨ahler when the almost-complex structure in question is at best weakly integrable.However, formal integrability and the closure of the K¨ahler form are the only conditionsnecessary to insure that the almost-complex structure tensor is parallel, even in infinitedimensions.11

where the fiber of π over Σ ∈S is defined to be Σ ⊂M. We can then pull Ωback to F to obtain a closed n-form α = p∗Ωwhich vanishes on the boundaryB = ∂Σ of every fiber of π.

But ω is just obtained from α by integrating onthe fibers of π:ω = π∗α .Since π∗commutes with d on forms which vanish along the fiber-wise bound-ary (cf. [1], Prop.

6.14.1), it follows thatdω = d(π∗α) = π∗dα = π∗d(p∗Ω) = π∗p∗dΩ= π∗p∗0 = 0.Thus h is a K¨ahler metric, with K¨ahler form ω.To conclude this note, we now observe that (S, h) is formally of Hodgetype— but non-compact, of course!Proposition 4.2 Modulo a multiplicative constant, the K¨ahler form ω of hrepresents an integer class in cohomology. If, moreover, M is non-compact,ω is actualy an exact form, and its cohomology class thus vanishes.Proof.

If M is compact, we may assume that g has total volume 1, so thatits volume form Ωthen represents an element of integer cohomology. Sinceω = π∗p∗Ω, its cohomology class [ω] = π∗p∗[Ω] is therefore integral.

If, onthe other hand, M is non-compact, Ω= dΥ for some (n −1)-form Υ, andhence ω = d(π∗p∗Υ).Acknowledgements. The author would like to thank L´aszl´o Lempert andEdward Witten for their suggestions and encouragement.References[1] R. Bott and L.T.

Tu, Differential Forms in Algebraic Topology,Springer-Verlag, 1982. [2] J.-L. Brylinski, The K¨ahler Geometry of the Space of Knots in a SmoothThreefold, preprint, Penn State, 1990.12

[3] R. Harvey and R.O. Wells, Jr. Holomorphic Approximation and Hyper-function Theory on a C1 Totally Real Submanifold of a Complex Mani-fold, Math.

Ann. 197 (1972) 287–318.

[4] C. LeBrun, Twistor CR Manifolds and Three-Dimensional ConformalGeometry, Trans. Am.

Math. Soc.

284 (1984) 601–616. [5] L. Lempert, Loop Spaces as Complex Manifolds, J. Diff.

Geom, toappear. [6] A. Newlander and L. Nirenberg, Complex Analytic Coordinates in Al-most Complex Manifolds, Ann.

Math. 65 (1957) 391–404.

[7] H. Rossi, LeBrun’s Non-Imbeddibility Theorem in Higher Dimensions,Duke Math. J.

52 (1985) 457–474.13

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