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APPEARED IN BULLETIN OF THE

arXiv:math/9204237v1 [math.CV] 1 Apr 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 280-287A PERIOD MAPPING IN UNIVERSAL TEICHM ¨ULLER SPACESubhashis NagAbstract. In previous work it had been shown that the remarkable homogeneousspace M = Diff(S1)/ PSL(2, R) sits as a complex analytic and K¨ahler submanifoldof the Universal Teichm¨uller Space.There is a natural immersion Π of M intothe infinite-dimensional version (due to Segal) of the Siegel space of period matri-ces.

That map Π is proved to be injective, equivariant, holomorphic, and K¨ahler-isometric (with respect to the canonical metrics). Regarding a period mapping as amap describing the variation of complex structure, we explain why Π is an infinite-dimensional period mapping.IntroductionLet Diff(S1) denote, as usual, the infinite-dimensional Lie group of orientationpreserving C∞diffeomorphisms of the unit circle.In a previous paper [12] weshowed that the canonical complex analytic structure and K¨ahler metric, g, thatexist on the homogeneous space M = Diff(S1)/ M¨ob(S1) coincide exactly withthe canonical Ahlfors-Bers complex structure and the (generalised) Weil-PeterssonK¨ahler structure of the Universal Teichm¨uller Space T (1).

This identification wasvia the natural injection (I) of M into T (1), where T (1) is thought of as the homo-geneous space of all quasi-symmetric homeomorphisms of S1 modulo the M¨obiussubgroup.In this paper we study another natural embedding (Π) of M into the infinite-dimensional version, D∞, of the Siegel disc (or generalised upper halfspace) ofperiod matrices. These matrices (suitable symmetric Hilbert-Schmidt operators)comprising D∞are symmetric with complex entries, so D∞is a complex manifold(infinite-dimensional bounded domain).

As in finite dimensions, so also here theSiegel symplectic (K¨ahler) metric, h, exists on this homogeneous manifold. Onemain purpose here is to announce that, like I, the map Π is completely natural(in that it respects all the structures); namely, Π is an equivariant, holomorphic,and K¨ahler isometric immersion of M into D∞.

Thus, combining this with theresults proved by this author in [12, Part II], one can assert that on the complexsubmanifold M of the Teichm¨uller space, the Weil-Petersson metric coincides withthe Siegel symplectic metric.Considering M as the submanifold of “smooth” complex structures in the Te-ichm¨uller space, we can interpret the mapping Π as a period mapping that associates1991 Mathematics Subject Classification. Primary 32G15, 32G20, 30F10, 30F60, 81T30.Received by the editors May 21, 1991 and, in revised form, August 20, 1991This paper was presented as an invited lecture on May 27, 1991, at the second InternationalSymposium on Topological and Geometrical Methods in Field Theory, Turku, Finland, (May27–June 1, 1991).

Sponsored by Academy of Finland et alc⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2SUBHASHIS NAGto each point of M its corresponding matrix of periods. The idea is to look at theperiod map in P. Griffiths’s way as a map describing the variation of Hodge andcomplex structure.

We will provide an exposition.As explained in [12], that work was intimately related to and motivated by stringphysics. Once again, physicists have been looking at the map Π and have madeseveral claims.The various infinite-dimensional spaces appearing in the set-upare all somehow universal moduli spaces and may turn out to be important in anonperturbative formulation of string theory.

See Nag [10].1. The map ΠThe method of defining Π stems from a faithful representation of Diff(S1) in theinfinite-dimensional symplectic group, say Sp, that was introduced by G. Segal [13,§5].

(Segal’s motivation is to obtain the metaplectic representation of Sp, but thatneed not concern us here.) Recall that if a (real) vector space V is equipped witha skew-symmetric, nondegenerate bilinear form, S, then the linear automorphismsT ∈GL(V ) that preserve the form S (viz., S(T v1, T v2) = S(v1, v2) for all v1, v2 inV ) constitute the symplectic subgroup Sp(V, S) ⊂GL(V ).Consider the vector space(1)V = C∞Maps(S1, R)/R (constant maps).The quotienting by the one-dimensional subspace of constant maps means that oneis normalizing the 0-th Fourier coefficient to vanish.

V is equipped with a naturalnondegenerate skew-symmetric bilinear form S : V × V →R (Segal [13]), given by(2)S(σ, τ) =ZS1 σ(θ)(dτ/dθ) dθ.Then Diff(S1) acts on V , by the obvious precomposition, preserving this bilinearform S. Hence the diffeomorphism group becomes a subgroup of the real symplecticautomorphisms of (V, S).Actually one deals with a restricted subgroup of thesymplectic automorphisms, denoted Sp0(V ), which already contains Diff(S1) (see[13]). (This is an analytic detail to simplify the infinite-dimensional considerations,since Sp0(V ) elements have a natural norm.

)To proceed further we recall the definition of a positive polarization of the space Vwith respect to S. Let VC denote the complexification of V . Then a decompositionVC = W ⊕W such that (the complexification of) S takes zero values on arbitrarypairs from W is called a polarization and W is called a (maximal) isotropic subspacefor S. The assignment(3)⟨w1, w2⟩= iS( ¯w1, w2)is a Hermitian inner product on W, and the decomposition is a positive polarizationif this is positive definite.In that case W, its conjugate, and hence VC itself,can be completed to Hilbert spaces with respect to this Hermitian inner product.We will henceforth identify a positive polarization with the isotropic subspace Wdetermining it.Here is the fiducial positive polarization for (V, S): take W =W+ to be the subspace of VC = C∞Maps(S1, C)/C consisting of those mappingshaving only positive index Fourier components.

Note that W + = W−, and also

A PERIOD MAPPING IN UNIVERSAL TEICHM¨ULLER SPACE3the fundamental fact that the image under (the C-linear extension of) a symplecticautomorphism of a positive isotropic subspace is again such a subspace. In fact,Sp(V, S) acts transitively on the space of all positive polarizations, and the stabilizersubgroup at W is evidently identifiable with the unitary group U(W, ⟨, ⟩).Itfollows that the homogeneous space Sp /U can be identified with the family ofpositive polarizations of V .

Passing to the restricted subgroup Sp0 one finds thatSp0 /U represents a class of (“bounded”) positive polarizations of V , say Pol0(V ),and either of these spaces can be easily identified with the Siegel disc of infinitedimension (Segal [13, p. 316]),(4)D∞= {All Hilbert-Schmidt complex linear operators Z : W+ →W−such that Z is symmetric (w.r.t. S) and I −ZZ is positive definite}.For instance, the identification between D∞and Pol0(V ) is by associating toZ ∈D∞the positive isotropic subspace W that is the graph of the operator Z.

(The origin in D∞corresponds to the fiducial polarization W+.) In finite dimen-sion (genus g Riemann surfaces) this identification corresponds to taking W to bethe row span of the “full period matrix” (Z, I); then W is a positive isotropic g-dimensional complex subspace of C2g (equipped with standard skew form).

Thecurrent set-up thus agrees well with the classical case, and we have in infinite di-mension the Siegel disc appearing in the various incarnations(5)D∞= Sp0 /U = Pol0(V ) ֒→Gr(W+, VC).Here the subspaces W ∈Pol0(V ) comprise a complex analytic submanifold ofthe complex Grassmannian of subspaces of VC that are graphs of (not necessarilysymmetric) linear operators of Hilbert-Schmidt type from W+ to W−. This complexstructure on Pol0(V ) (identified as explained with D∞) is easily checked to be thesame as the complex structure of D∞as a family of (symmetric with respect to S)complex operators.We are finally ready to define Π.Lemma.

In Segal’s representation Diff(S1) →Sp0(V ), the diffeomorphisms thatmap into the unitary subgroup U of Sp0 comprise precisely the 3-parameter subgroupof M¨obius transformations on S1. Consequently one obtains an injective equivariantmapping(6)Π: Diff(S1)/M¨ob(S1) →Sp0 /U = D∞.The proof boils down to showing that a holomorphic self-map of the disc thatextends continuously to the boundary as a diffeomorphism of S1 must be a M¨obiustransformation.

This follows, for example, by the argument principle. That Π isequivariant (with respect to corresponding right translations on domain and target)is easily verified.2.

Holomorphy of ΠTheorem. Π is a holomorphic immersion.The method is to compute the derivative of Π at the origin of M and show thatit is a complex linear map of the tangent space of M to the tangent space of theGrassmannian above.

By equivariance it is enough to verify this at the origin.

4SUBHASHIS NAGLet me first explain the canonical complex structure on the homogeneous spaceM. The Lie algebra of the Lie group Diff(S1) is the algebra of smooth (C∞) realvector fields on S1.

The tangent space at the origin (i.e., coset of the identity) ofM corresponds to those smooth vector fields v = u(θ)∂/∂θ such that in the Fourierexpansion(7)u(θ) =∞Xm=−∞umeimθ,one has u−1 = u0 = u1 = 0. Note that reality of v implies um = u−m.Oneeasily sees that the three middle Fourier coefficients vanish because an infinitesimalM¨obius transformation allows one to normalize precisely these three coefficients.The canonical homogeneous almost-complex structure J on M is now definedsimply by “conjugation of Fourier Series,” namely,(8)Ju(θ) ∂∂θ= u∗(θ) ∂∂θ ,where(9)u∗(θ) = −∞Xm=−∞i sgn(m)umeimθ.This defines J at the origin on the homogeneous Fr´echet manifold M; elsewhere J istransported by right translations.

It is easy to check (at least at the formal level),that this almost complex structure on M is integrable.Thus M is canonicallya homogeneous complex manifold. The complex structure above arises naturallyfrom various points of view, for instance, from the Kirillov-Kostant coadjoint orbittheory and also in work of Pressley et al.

See references [1–4, 16, 17].Now, the tangent space to the Grassmannian at a point H can be canonicallyidentified with the linear space Hom(H, VC/H) (only those homomorphisms satis-fying a Hilbert-Schmidt boundedness condition are included). We now exhibit thederivative of Π from the tangent space of M to the tangent space of the Grassman-nian.Proposition.

The derivative of Π at the origin of M carries the vector field v =u(θ)∂/∂θ to the following homomorphism ηv ∈Hom(W+, W−):(10)ηv(σ) = π−(u(θ)σ′(θ))for any σ ∈W+,where π−denotes the projection of VC onto W−and prime denotes differentiationwith respect to θ.The proof of (10) is straightforward after one has properly identified the variousspaces involved.To establish the holomorphy of Π one needs to prove that the homomorphismηJv is i times ηv. Recall that the negative index Fourier coefficients of Jv are justthe corresponding coefficients of v multiplied by i.

On the other hand, any σ fromW+ (therefore any σ′) has only positive index Fourier components. Application offormula (10), therefore, gives what we wish.

A PERIOD MAPPING IN UNIVERSAL TEICHM¨ULLER SPACE53. Π is an isometryThere is a unique homogeneous K¨ahler metric, g, available on M. In Part IIof [12] I had proved that this metric is precisely the Weil-Petersson metric on theTeichm¨uller spaces.Here is one way to explain how g arises.

Since M is a coadjoint orbit in thecoadjoint representation of the Virasoro group, (see Witten [16]), one has, as always,the natural Kirillov-Kostant symplectic form on these homogeneous manifolds. Thissymplectic form is compatible with the complex structure seen above, and we thusget the K¨ahler structure g on M. In a less sophisticated fashion, it is actually easyto see that g is the unique homogeneous K¨ahler metric (up to a numerical scalingparameter) possible on M.The number 26, namely, the dimension of bosonicspacetime, appears entrenched in the curvature of g. See calculations of Bowick,Rajeev, Lahiri, and Zumino in the references.

Explicitly, this canonical g, at theorigin of M has the Hermitian pairing (see [12, Part II])(11)g(v, w) =" ∞Xm=2vmwm(m3 −m)#.Here v and w are two smooth real vector fields on S1 representing tangent vectorsto M (see equation (7)), and the vm, wm are the respective Fourier coefficients.Elsewhere on M, g is of course transported by right translations isometries.The Siegel symplectic metric, h, (note Siegel [15]) exists on each finite-dimensionalSiegel disc Dg as the unique K¨ahler metric on that bounded domain for which thefull holomorphic automorphism group Sp(2g, R) acts as isometries. It generaliseswithout trouble to Segal’s infinite-dimensional version D∞explained above.

Thepairing given by h on the symmetric elements of Hom(W+, W−), i.e., on the tangentspace at the origin of D∞, turns out, after some work, to be(12)h(φ, ψ) = trace of φ ◦¯ψ,where the conjugate of ψ maps W−to W+, so taking trace makes sense (and givesa finite answer).Theorem. Π is an isometric immersion of (M, g) into (D∞, h).Again utilising equivariance and the fact that g and h are homogeneous metricswe only need check at the origin.

One calculates the trace of ηv composed withconjugate of ηw and finds interestingly that the answer is as predicted by formula(11). The computation is finally a finite-series summation.

See [10] for details.4. Dependence of Π on Beltrami coefficientsHere is the explicit form of the period matrix in D∞associated to a givenBeltrami coefficient µ (representing a Teichm¨uller point [µ]).

The symplectic auto-morphism Tµ of V arising from µ is h 7→h◦wµ, h ∈V . Here wµ is the µ-conformalself-homeomorphism of the unit disc ∆, and we are using its boundary homeo-morphism.

The matrix for Tµ (complexified) in the standard orthonormal bases{ek = eikθ/p|k|} for VC = W+ ⊕W−has the form(13)Tµ =ABBA.

6SUBHASHIS NAGHere A: W+ →W+, B : W−→W+, A = ((apq)), B = ((bpq)).We obtain the fundamental formulasapq = 12π√p√qZ 2π0(wµ(eiθ))qe−ipθ dθ,p, q > 0;(14)brs = 12π√r√sZ 2π0(wµ(eiθ))−se−irθ dθ,r, s > 0. (15)Recall that the symplectic group acts by holomorphic automorphisms on D∞: Z 7→(A ◦Z + B) ◦(B ◦Z + A)−1 where the symplectic matrix “T ” is in the block form(13).

The zero matrix in D∞corresponds to µ = 0, and we see that(16)Π([µ]) = B ◦A−1with A and B given as above.First variation of the period matrix. That the matrix (16) varies holomorphi-cally with µ is already nontrivial (remember that wµ varies only real analyticallywith µ).

Since we know the perturbation formula for wµ, we can actually writedown the infinitesimal variation of the matrix entries of Π(µ) as µ varies.Proposition. For µ ∈L∞(∆) one has, as t →0,(17)Π([tµ])rs = t · π−1 · √rsZZ∆µ(z)zr+s−2 dx dy + O(t2).Remark.

The proposition corresponds precisely to the well-known Rauch varia-tional formulas for finite genus period mappings. In fact, the holomorphic 1-formson ∆are generated by φr = zr−1, r = 1, 2, .

. .

; so (17) is in harmony with theRauch formula “R Rµφrφs” for the variation of the rs entry. (See [11, Chapter 4].)5.

Π as a period mappingThe map Π links up with the variation of Hodge structure following P. Griffiths[5, 6]. The idea is to interpret the vector space V as the “first cohomology (with realcoefficients) of a surface of infinite genus.” The skew form S can then be recognisedas the cup product map on this cohomology vector space.Now, a “complex structure”, that is a point from the universal Teichm¨uller space,will produce a decomposition of the complexification of V into its H1,0 and H0,1subspaces.

The H1,0 subspace is a positive isotropic subspace for the cup productskew form because the condition of being positive isotropic is an implementationof the bilinear relations of Riemann. The decomposition so obtained is indeed apositive polarization of V .

For the base complex structure “0”, we take H1,0 tobe W+. Thus Π becomes realised as an association between the smooth complexstructures and their corresponding H1,0 subspaces, as is expected of a period map-ping in Griffiths’s set-up.

That Π is injective (as we saw) is then Torelli’s Theoremin this context.One can also prove that Π keeps track of the varying space of holomorphic 1-forms as the complex structure varies. That is precisely the characteristic of theclassical period mappings.

See [10] for this point of view.

A PERIOD MAPPING IN UNIVERSAL TEICHM¨ULLER SPACE7The infinitesimal Schottky locus. It is clear that the image Π(M) in D∞isthe analog of the Schottky locus of Jacobians in the Siegel disc Dg.

Using theformula (10) for dΠ we get a pretty description of the tangent to the “Schottkylocus” Π(M):Theorem. The tangent space to D∞at the origin is canonically identifiable withthe complex symmetric matrices ((λpq)) (p, q = 1, 2, 3, .

. .) satisfying the Hilbert-Schmidt condition P∞p=1P∞q=1 |λpq|2 < ∞.

The image of dΠ at the origin consistsof those that are of the form(18)λpq = i√pqa(p+q),where the sequence (a2, a3, a4, . .

. ) is an arbitrary complex sequence whose nth termgoes to zero faster than |n|−k for any k > 0.Remark.

In finite genus g, the (3g −3)-dimensional Teichm¨uller space sits via πg inthe g(g + 1)/2-dimensional Dg. Here also we see that the number of independententries in a D∞tangent vector is growing quadratically with the block size, whereasthe ones arising from Teichm¨uller points are determined just by their first column,i.e., grows linearly with block size.

In this aspect also Π generalises the πg.Remark. Since D∞sits within the infinite-dimensional Grassmannian, the theory(of the tau function) developed in [14] by Segal-Wilson suggests that it may bepossible to describe the image of Π by solutions of K −P equations; namely, aninfinite-dimensional form of the Novikov conjecture could be valid.Remark.

In Part II of [12] the present author had shown a form of Mostow rigiditythat implied that the Teichm¨uller spaces Tg of compact Riemann surfaces of genusg sit within the universal T (1) cutting transversely the submanifold M. That resultdisallows the possibility of relating the finite-dimensional period maps πg : Tg →Dgto Π by simple restriction of domains. Nevertheless, it was possible in Part II of[12] to relate the Weil-Petersson metric on Tg to the canonical metric g of M by aregulation of improper integrals.

Some similar analysis may relate πg to Π.In any case the result of §3 raises the following interesting question: How isπ∗g(Siegel symplectic metric) related to the Weil-Petersson metric on Tg? Certainlythey do not coincide in low genus!

A connection between the volume form inducedon Tg by the Siegel metric and volume forms arising from the complex geometry ofTg was shown in Nag [9].Hong-Rajeev [7] have considered the mapping Π as a map from the space ofunivalent functions (Riemann maps) to D∞; they have made several claims buthave offered no proofs. They appear to be thinking of points of Sp0 /U as complexstructures on V rather than as polarizations.

It seems to us that this does not quitefit requirements. Also, if one wishes to look at Π as a map on univalent functions,namely, as a map on domains bounded by smooth Jordan curves, then one is forcedto use the conformal welding correspondence to relate to Diff(S1)/ M¨ob(S1).

(See[12, Part II] and Katznelson-Nag-Sullivan [8]). Using the welding it is possible forus to write formulas for Π as a map on coefficients of Riemann maps (which areessentially nothing but Bers embedding holomorphic coordinates); however, we areunable to see how the formulas presented by Hong and Rajeev can be valid.More details on this work will appear in [10] and elsewhere.

8SUBHASHIS NAGReferences1. M. Bowick, The geometry of string theory, 8th Workshop on Grand Unification, Syracuse,NY 1987.2.

M. Bowick and A. Lahiri, The Ricci curvature of Diff(S1)/ SL(2, R), Syracuse Univ. Prepr.SU-4238-377 (Feb. 1988).3.

M. Bowick and S. Rajeev, String theory as the K¨ahler geometry of loop space, Phys. Rev.Lett.

58 (1987), 535–538.4., The holomorphic geometry of closed bosonic string, Nuclear Phys. B 293 (1987),348–384.5.

P. Griffiths, Periods of integrals on algebraic manifolds, Bull. Amer.

Math. Soc.

75 (1970),228–296.6., Periods of integrals on algebraic manifolds III, Publ. Math.

IHES, vol. 38, 1970.7.

D. Hong and S. Rajeev, Universal Teichm¨uller space and Diff(S1)/S1, Comm. Math.

Phys.135 (1991), 401–411.8. Y. Katznelson, S. Nag, and D. Sullivan, On conformal welding homeomorphisms associatedto Jordan curves, Ann.

Acad. Sci.

Fenn. Ser.

A I Math. 15 (1990), 293–306.9.

S. Nag, Canonical measures on the moduli spaces of compact Riemann surfaces, Proc.Indian Acad. Sci.

Math. Sci.

99 (1989), 103–111.10., Non-perturbative string theory and the diffeomorphism group of the circle, Proc.Internat. Sympos.

Topological and Geometrical Methods in Field Theory, Turku, Finland1991 (J. Mickelsson, ed. ), World Scientific (to appear).11., The complex analytic theory of Teichm¨uller spaces, John Wiley, Interscience, NewYork, 1988.12.

S. Nag and A. Verjovsky, Diff(S1) and the Teichm¨uller spaces, Comm. Math.

Phys. 130(1990), 123–138.

In two Parts. Part I by S. N. and A. V.; Part II by S. N.13.

G. Segal, Unitary representations of some infinite dimensional groups, Comm. Math.

Phys.80 (1981), 301–342.14. G. Segal and G. Wilson, Loop groups and equations of KdV type, Publ.

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61, 1985.15. C. L. Siegel, Symplectic geometry, Academic Press, New York, 1964.16.

E. Witten, Coadjoint orbits of the Virasoro group, Comm. Math.

Phys. 114 (1981), 1–53.17.

B. Zumino, The geometry of the Virosoro group for physicists, Cargese 1987 (R. Gastmans,ed.

), Plenum Press, New York.The Institute of Mathematical Sciences, C.I.T. Campus, Madras 600 113, INDIAE-mail address: nag@imsc.ernet.in

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