---

Central extensions of current groups in two dimensions

arXiv:hep-th/9303047v1 9 Mar 1993Central extensions of current groups in two dimensionsPavel I. Etingof and Igor B. FrenkelYale UniversityDepartment of Mathematics2155 Yale StationNew Haven, CT 06520 USAe-mail etingof@pascal.math.yale.eduApril 1992Submitted to Communications in Mathematical Physics in February 1993IntroductionThe theory of loop groups and their representations [13] has recently developed inan extensive field with deep connections to many areas of mathematics and theoret-ical physics. On the other hand, the theory of current groups in higher dimensionscontains rather isolated results which have not revealed so far any deep structurecomparable to the one-dimensional case.

In the present paper we investigate thegeometry of current groups in two dimensions and point out several remarkablesimilarities with loop groups. We believe that these observations give a few morehints about the existence of a new vast structure in dimension two.One of important problems in the theory of loop groups is integration of centralextensions of loop algebras.

It is known [13] that the non-trivial one dimensionalcentral extension of the loop algebra gS1 of a compact Lie algebra g integrates toa Lie group, which is a one dimensional central extension of the loop group GS1for the corresponding compact group G. Topologically this group turns out to bea nontrivial circle bundle over GS1, which plays a crucial role in the geometricrealization of representations of affine Lie algebras. The study of the coadjointaction of this group yields a very simple description of orbits by means of thetheory of ordinary differential equations [5,16], and it turns out that there is aperfect correspondence between orbits and representations [5].The Lie algebra of vector fields on the circle Vect(S1) arises as the Lie algebra ofouter derivations of a loop algebra and has many similarities with loop algebras.

Ithas a unique nontrivial one-dimensional central extension – the Virasoro algebra.The structure of coadjoint orbits of the Virasoro algebra can be obtained from thestudy of Hill’s operator[7,16]. The invariants of the coadjoint action of the complexVirasoro algebra are essentially the same as those of the extended algebra of loopsin sl2.In this paper we generalize some of these results for loop algebras and groups aswell as for the Virasoro algebra to the two-dimensional case.We define and study a class of infinite dimensional complex Lie groups which arecentral extensions of the group of smooth maps from a two dimensional orientablesurface without boundary to a simple complex Lie group G.These extensionsnaturally correspond to complex curves.

The kernel of such an extension is theJacobian of the curve. The study of the coadjoint action shows that its orbits arelabelled by moduli of holomorphic principal G-bundles over the curve and can bedescribed in the language of partial differential equations.

In genus one it is alsopossible to describe the orbits as conjugacy classes of the twisted loop group, which1

2leads to consideration of difference equations for holomorphic functions. This givesrise to a hope that the described groups should possess a counterpart of the richrepresentation theory that has been developed for loop groups.We also define a two-dimensional analogue of the Virasoro algebra associatedwith a complex curve.

In genus one, a study of a complex analogue of Hill’s operatoryields a description of invariants of the coadjoint action of this Lie algebra. Theanswer turns out to be the same as in dimension one: the invariants coincide withthose for the extended algebra of currents in sl2.Note that our main constructions are purely two-dimensional.

In particular, inthree and more dimensions the space of orbits tends to be infinite-dimensional andtopologically unsatisfactory.1. Lie algebra extensions.Let G be a simply connected simple complex Lie group, and let g be its Liealgebra.Denote by <, > a nonzero invariant bilinear form on g.Let Σ be anonsingular two-dimensional surface of genus g.Define GΣ as the group of allsmooth maps from Σ to G. Its Lie algebra consists of all smooth maps from Σ tog and will be denoted by gΣ.

These are called the current group and the currentalgebra .Now fix a complex structure on the surface Σ. Let HΣ be the space of holomor-phic differentials on Σ.

Its dimension is equal to g. Let ω be the identity element inHΣ ⊗H∗Σ. The element ω can be regarded as a holomorphic differential on Σ withvalues in H∗Σ.

Define a 2-cocycle on gΣ with values in H∗Σ (regarded as a trivialgΣ-module) by(1.1)Ω(X, Y ) =ZΣω∧< X, dY >,X, Y ∈gΣ.This cocycle defines a g-dimensional central extension of gΣ. This extension is non-trivial for surfaces of positive genus, and it has no nonzero trivial subextensions orquotient extensions.

Denote this new Lie algebra by ˆgΣ.If Σ is a complex torus, we can separate a subalgebra ˆgΣpol in ˆgΣ that consistsof all currents realized by trigonometric polynomials. The restriction of the abovecentral extension to this subalgebra in its natural basis looks very simple.

Namely,let L be a lattice in the complex plane C such that Σ = C/L. The Lie algebra ˆgΣpolis spanned by elements x(n), x ∈g, n ∈L, where x(n) depends linearly on x forany fixed n. These elements satisfy the following commutation relations:(1.2)[x(n), y(m)] = [x, y](n + m) + nδn,−m < x, y > kwhere k is the central element.

This extension is a natural two-dimensional coun-terpart of affine algebras. Similar extensions were considered in [10].Each of the extensions defined above can be obtained as a suitable quotient ofthe universal central extension UgΣ of the Lie algebra gΣ.Proposition 1.1.

( [13, Section 4.2]) The universal central extension UgΣ is anextension of gΣ by means of the infinite-dimensional space a = Ω1(Σ)/dΩ0(Σ) ofcomplex-valued 1-forms on Σ modulo exact forms. This extension is defined by thea-valued cocycle(1.3)u(ξ, η) =< ˜ξ, d˜η >,ξ, η ∈a,

3where ˜ξ, ˜η ∈Ω1(Σ) are any liftings of ξ, η.Note that this construction does not involve the complex structure on the surface.In order to obtain ˆgΣ for a specific complex structure on the surface Σ as aquotient of UgΣ, one needs to factorize it by the subgroup of all ξ ∈a such that(1.4)ZΣ˜ξ ∧ω = 0for any lifting ˜ξ ∈Ω1(Σ) of the element ξ.Proposition 1.2. (i) Let Σ1 and Σ2 be two Riemann surfaces.

The Lie algebrasˆgΣ1 and ˆgΣ2 are isomorphic if and only if Σ1 and Σ2 are conformally equivalent. (ii) Any automorphism f of ˆgΣ can be uniquely represented as a composition:f = h ◦φ∗where h is a conjugation by an element of Aut(g)Σ and φ∗is the directimage map induced by a conformal diffeomorphism φ : Σ →Σ.Proof.

Let f : ˆgΣ1 →ˆgΣ2 be any isomorphism. Then the dimensions of the centersof ˆgΣ1 and ˆgΣ2 must coincide, so the genera of Σ1 and Σ2 are the same.

Considerthe induced map f0 : gΣ1 →gΣ1 of current algebras. Since Σ1 and Σ2 are diffeomor-phic, we may actually regard this map as an automorphism of the current algebra.According to [13, Section 3.4], any such map uniquely decomposes as f0 = h ◦φ∗where h is a conjugation by an element of Aut(g)Σ1 and φ∗is the direct image mapinduced by a conformal diffeomorphism φ : Σ1 →Σ2.

In order for f0 to extend toan isomorphism of extensions, φ must take holomorphic differentials to holomorphicdifferentials which forces it to be a conformal equivalence. This proves both (i) and(ii).■One can also classify embeddings ˆgΣ1 ֒→ˆgΣ2.Indeed, any embedding f :gΣ1 ֒→gΣ2 is induced by a surjective map ϕ : Σ2 →Σ1, and conversely, anysuch map defines an embedding.

In order for this embedding to continue to cen-tral extensions, the map between surfaces must be holomorphic. However, in thiscase the continuation is not unique.

Continuations correspond to monomorphismsχ : HΣ1 ֒→HΣ2 with the property ϕ∗(χ(v)) = v for all v ∈HΣ1. Here ϕ∗is themap HΣ2 →HΣ1 induced by ϕ.We have shown that the group of outer automorphisms of the Lie algebra ˆgΣ isisomorphic to the group of holomorphic automorphisms of the Riemann surface Σ.If g > 1, this group is finite, and it is trivial for almost every surface.

In spite of it,ˆgΣ has plenty of outer derivations.Proposition 1.3. If g > 1, the Lie algebra of outer derivations of ˆgΣ coincides withthe Lie algebra Vect0,1(Σ) of all complex-valued vector fields on Σ of type (0,1), i.e.of the form u(z, ¯z) ∂∂¯z for any local complex coordinate z, u being a smooth function.If g = 1, the Lie algebra of outer derivations is <∂∂z > ⋉Vect0,1(Σ)Proof.

It is known that outer derivations of gΣ are in one-to-one correspondencewith complex vector fields on the surface Σ.In order for such a derivation tocontinue to the central extension ˆgΣ, the vector field must annihilate holomorphicdifferentials. If g > 1, such a field must have type (0,1).

If g = 1, any such field isa linear combination of a (0,1)-field and the constant field∂∂z ■

4The Lie algebra Vect0,1(Σ) should be thought of as an analogue of the Wittalgebra. It has a g-dimensional central extension which is a natural analogue of theVirasoro algebra.

This is the extension by the space HΣ defined by the cocycle(1.5)F(X, Y ) =ZΣω ∧¯∂X ¯∂2Y, X, Y ∈Vect0,1(Σ).In this expression, ¯∂X is a function and ¯∂2Y is a differential 1-form, since theoperator ¯∂maps vector fields to functions and functions to 1-forms. We will denotethis extension by Vir(Σ).

For Σ being the torus, the polynomial part Vect0,1(Σ)polof Vect0,1(Σ) is a graded Lie algebra of a very simple structure. Namely, let L bethe lattice in C such that Σ = C/L.

The basis of Vect0,1(Σ)pol consists of elementsen, n ∈L, which satisfy Witt’s relations(1.6)[en, em] = (m −n)en+m.The one dimensional extension we have described has an additional basis vector c– the central element, and the relations are analogous to Virasoro relations:(1.7)[en, em] = (m −n)en+m + m3δm,−nc.One can easily show that this is the universal central extension of Vect0,1(Σ)pol (i.e.it is the only possible nontrivial one-dimensional extension up to an isomorphism).The proof is similar to that for the Witt algebra and is given in [15].2. Group extensions.The following theorem describes the topology of the current group and can bededuced from the results of [13, Chapter 4].Theorem 2.1.

(i) GΣ is connected. (ii) π1(GΣ) = Z.

(iii) π2(GΣ) ⊗R = H2(GΣ, R) = R2g.It turns out that the Lie algebra ˆgΣ can be integrated to a complex Lie group.Theorem 2.2. There exists a central extension ˆGΣ of the current group GΣ bymeans of the Jacobian variety of Σ whose Lie algebra is ˆgΣ.Proof.

This group can be constructed by the procedure described in [13, Chapter4]. Namely, consider the left invariant holomorphic 2-form on GΣ equal to Ωon thetangent space at the identity.

It shall be denoted by the same letter. Integrals ofΩover integer 2-cycles in GΣ fill the lattice L = H1(Σ, Z) in H∗Σ.

Therefore, thereexists a holomorphic principal bundle over GΣ with fiber J = H∗Σ/L and with aholomorphic connection θ whose curvature form is 2πΩ. Note that J is the Jacobianof the surface Σ. Define the group ˆGΣ as the group of all transformations of theconstructed bundle that preserve the connection θ and project to left translationson the group GΣ.

It is a central extension of GΣ by J. One has an exact sequence(2.1)1 →J →ˆGΣ →GΣ →1,and it follows that the Lie algebra of ˆGΣ is isomorphic to gΣ.

■Obviously, this theorem will remain valid if we replace the current group GΣwith its universal covering ˜GΣ. Denote the extension obtained in this way by eˆGΣ.The following theorem characterizes the universal central extension of the currentgroup.

5Theorem 2.3. [13, Section 4.10] (i) The universal central extension UGΣ of thegroup GΣ is an extension of ˜GΣ by means of the infinite-dimensional abelian groupA of complex-valued 1-forms on Σ modulo closed 1-forms with integer periods.

(ii) πn(UGΣ) ⊗R = 0 for n < 3. (iii) The group A is homotopy equivalent to a 2g-dimensional torus.

The naturalmap π2(GΣ) →π1(A) associated to the fiber bundle UGΣ →GΣ is an isomorphismup to torsion.Proposition 2.4. The universal central extension UGΣ is homotopy equivalent toeˆGΣ.Proof.

Consider a homomorphism ε : A →J defined by(2.2)ε(a) =ZΣ˜a ∧ωmod L.where ˜a is any lifting of a ∈A into the space of 1-forms on Σ. It is easy to checkthat this map is well defined, i.e.

independent of the choice of ˜a. Obviously, ε isa homotopy equivalence.

This implies that the corresponding map of extensions˜ε : UGΣ →eˆGΣof the group ˜GΣ is also a homotopy equivalence.This theorem implies that eˆGΣis a nontrivial J-bundle on ˜GΣ and represents thehomotopically nontrivial part of the universal central extension.In fact, the universal extension can be constructed rather explicitly as a centralextension of ˜GΣ by the additive group of a vector space, as follows.Let K be the maximal compact subgroup of G, and let ˜KΣ be the subgroup ofall elements in ˜GΣ whose projections to GΣ are K-valued currents. Obviously,˜KΣis a central extension of KΣ by T 2g ×Z, where T 2g is the real 2g-dimensional torus.Choose a metric on Σ compatible to the complex structure.

A metric inducesan inner product on the space of differential forms. Define d∗to be the conjugateoperator to the de Rham differential d. This operator maps 2-forms to 1-forms.Let a be the space of 1-forms on Σ modulo exact 1-forms, and let W be the imageof the projection p : Imd∗→a.Lemma 2.5.

The subspace W has codimension 2g in a and is complementary tothe subspace H1(Σ, R) ⊂a.Proof. Let ∆: Ω1(Σ) →Ω1(Σ) be the Laplacian associated with the metric: ∆=dd∗+ d∗d.

Since ∆is self-adjoint, any form α ∈Ω1(Σ) can be represented in theform α = α1 + α2, where α1 ∈Ker∆, α2 ∈Im∆.Then dα1 = 0 by Hodge’stheorem, and α2 = ∆β = d∗dβ + dd∗β = d∗β1 + dβ2 by construction.Thus,p(α) = p(α1) + p(d∗β1). Since p(α1) ∈H1(Σ, R) and p(d∗β1) ∈W, we have shownthat any vector in a can be written as a sum of a vector in W and a vector in H1.Now let us show that W ∩H1(Σ, R) = 0.

Let w be an element of this intersection.Let ˜w be an inverse image of w in Imd∗. Then d ˜w = 0 and d∗˜w = 0, so w isharmonic.

By Hodge’s theorem, harmonic forms are in one-to-one correspondencewith cohomology classes, so any harmonic form in Imd∗has to be zero.Thus,w = 0.■Define a 2-cocycle on˜KΣ with values in W as follows:(2.3)C(˜a,˜b) = p(d∗< b−1db, da · a−1 >),˜a,˜b ∈˜KΣ

6where a, b ∈KΣ are images of ˜a,˜b.This cocycle defines a central extension of˜KΣ by W. Let us show that this newgroup is isomorphic to the universal central extension of KΣ.Since both groups are simply connected, it is enough to establish an isomorphismbetween their Lie algebras. Both Lie algebras are central extensions of kΣ, where kis the Lie algebra of K. The kernels of these extensions are ˜a = W ⊕HΣR and a,respectively, and the corresponding cocycles are ˜u = C ⊕ΩR and u (if V is a vectorspace then VR denotes the space V regarded as a real vector space).

All we need isto construct an isomorphism ρ : a →˜a such that ρ(u) = ˜u. It is easy to check thatsuch an isomorphism is provided by Lemma 2.5.The universal extension of ˜GΣ can now be obtained from the constructed exten-sion of KΣ by complexification.Remark 2.1.

(0,1)- vector fields on the surface are realized as J-invariant holo-morphic vector fields on ˆGΣ (=first order differential operators on holomorphicfunctions). Any such field v satisfies the condition v(gh) = v(g)h + gv(h) for allg, h ∈ˆGΣ.Remark 2.2.

A very challenging problem is to obtain a more explicit constructionof eˆGΣthan we have given, by presenting a 2-dimensional counterpart of the famousWess-Zumino-Witten construction for loop groups [19,9].It would also be interesting to obtain a direct construction of the universal centralextension of GΣ from eˆGΣby an analogue of formula (2.3) without referring to themaximal compact subgroup.3. Orbits of the coadjoint action.Let η be a holomorphic differential on Σ. Denote by Eη the one-dimensionalcentral extension of gΣ by means of the cocycle(3.1)Ωη(X, Y ) =ZΣη∧< X, dY > .Obviously, Eη is the quotient of ˆgΣ by the kernel of η as a linear function on HΣ.Denote by E∗the space of all operators D = λ¯∂+ ξ, where λ ∈C and ξ isa g-valued (0,1) form on Σ, i.e.

a 1-form which can be written as u(z, ¯z)d¯z forany local complex coordinate z, u being a g-valued smooth function.For anyrepresentation V of g, these operators take V -valued functions on Σ to V -valued(0,1)-forms according toDψ = λ¯∂ψ + ξψ,which allows to define an action of GΣ on E∗:(3.2)h ◦(λ¯∂+ ξ) = λ¯∂+ h−1 ¯∂h + Adh ◦ξ,h ∈GΣ.Consider a pairing between E∗and Eη given by(3.3)(λ¯∂+ ξ, µk + X) = λµ +ZΣη∧< ξ, X >,where k is the basis central element in Eη and µ ∈C.

7Proposition 3.1. Pairing (3.3) is GΣ-invariant and nondegenerate.The proof is straightforward.The constructed pairing allows us to interpret E∗as the smooth part of the dualE′η to the Lie algebra Eη, i.e.

as the proper coadjoint representation.Our goal now is to study the orbits of the action of GΣ in E∗. Hyperplanesλ = const are invariant under this action.

Let us fix a nonzero value of λ andexamine the orbits contained in the corresponding hyperplane Hλ.We will use the following classical construction [4].To an operator D = λ¯∂+ ξ one can naturally associate a holomorphic principalG-bundle on Σ as follows. Let Ui, i ∈I, be a cover of Σ by proper open subsets.For every i fix a solution ψi : Ui →G of the partial differential equation(3.4)λ¯∂ψi + ξψi = 0.

(the left hand side of (3.4) being a g-valued (0,1) form on Ui). Such solutions canalways be found.

Now for i, j ∈I define transition functions φij : Ui ∩Uj →Gby φij = ψ−1iψj.It is easy to see that these functions are holomorphic. Thusthey define a holomorphic principal G-bundle on Σ, which we denote by B(D).Moreover,it follows that the holomorphic bundles B(D1) and B(D2) associated to twooperators D1 = λ¯∂+ ξ1 and D2 = λ¯∂+ ξ2 are equivalent if and only if there existsan element h ∈GΣ such that h ◦D1 = D2.This h will be exactly the gaugetransformation establishing the equivalence between B(D1) and B(D2).Conversely, for any holomorphic principal G-bundle B on Σ there exists anoperator D = λ¯∂+ ξ such that B = B(D).

Indeed, any principal G-bundle on Σ istopologically trivial, since G is simply connected. If we choose a global trivializationthen the local holomorphic trivializations over open sets Ui will be expressed bysmooth functions ψi : Ui →G.

Since the transition functions φij = ψ−1iψj areholomorphic on Ui ∩Uj, we have ¯∂ψi · ψ−1i= ¯∂ψj · ψ−1jon Ui ∩Uj. Therefore thereexists a g-valued 1-form ξ on Σ such that ξ = −λ¯∂ψi · ψ−1ion Ui for all i ∈I.

LetD = λ¯∂+ ξ. Then B = B(D).This reasoning proves the followingProposition 3.2.

Orbits of the action of GΣ in Hλ are in one-to-one correspon-dence with equivalence classes of holomorphic G-bundles on Σ. The correspondenceis D ↔B(D).Remark 3.1.

The relation between differential operators and holomorphic prin-cipal bundles on Σ was used in computations of partition functions of the gaugedWess-Zumino-Witten model in [6] and [20].Remark 3.2. Holomorphic sections of the bundle B(D), i.e.

solutions of (3.4),are known as a special case of generalized analytic functions which were introducedin the fifties by L.Bers and I.Vekua [2,18]. If ξ = 0, they become usual analyticfunctions.Holomorphic principal G-bundles for G = SLn(C) were classified by Atiyah [1]for g = 1 and by Narasimhan and Sheshadri [11,12], for g > 1.

Their results weregeneralized to the case of any simple group G by Ramanathan [14]. We summarizehere some of them.

8Let Π be the fundamental group of Σ and K be a maximal compact subgroupof G. Let ρ : Π →K be any homomorphism. This homomorphism defines a flatG-bundle Bρ over Σ with a canonical holomorphic structure.

Bundles coming fromthis construction are called unitary. It is known [14] that the conjugacy class ofthe representation ρ is completely determined by the equivalence class of Bρ as aholomorphic principal bundle.The following theorem shows that almost all holomorphic principal bundles areflat and unitary.Theorem 3.3.

[14] Let {Bt}t∈T be a holomorphic family of holomorphic principalG-bundles parametrized by a complex space T . Then the subset T0 of such t ∈Tthat the bundle Bt is flat and unitary is Zariski open in T .Let us now define an appropriate notion of “almost everywhere”.Let V be any topological vector space.

We say that a set Z ⊂V is Zariski open iffor any finite dimensional complex manifold T and any holomorphic map f : T →Vthe set of points t ∈T such that f(t) ∈Z is Zariski open in T . Obviously, thisdefines a topology in V .

We will say that some property holds almost everywherein V if it holds on a nonempty Zariski open subset of V . Note that every nonemptyZariski open subset in V is dense, open, and connected in the usual topology.It is clear that a flat bundle B can be obtained from an operator D = λ¯∂+ ξwith ξ being a g-valued antiholomorphic differential on Σ: B = B(D).

Therefore,the above theorem in fact states that almost every (in Zariski sense) differentialoperator of the form λ¯∂+ξ can be reduced to an operator with an antiholomorphicξ by means of a gauge transformation from GΣ. This implies that the union S ofGΣ-orbits of all operators λ¯∂+ ξ with ξ being antiholomorphic is a Zariski opensubset of Hλ.Remark 3.3.

It is easy to construct an example of a holomorphic bundle whichis not flat and unitary. For instance, take any holomorphic line bundle β on Σof degree 1 and form a rank 2 bundle β ⊕β∗, where β∗is the dual bundle toβ.

This bundle has degree 0. Let B = Aut0(β ⊕β∗) be the SL2(C)-bundle ofautomorphisms of β ⊕β∗having determinant one.

It is easy to see that this bundleis not flat: the associated bundle β ⊕β∗has a nonzero holomorphic section s whichvanishes at a point z ∈Σ; this section cannot satisfy any equation λ¯∂s + ξs = 0with ξ being antiholomorphic. Another example would be Aut0(A) where A is theAtiyah’s bundle of rank 2 over a complex torus which is a semidirect sum of twotrivial line bundles.

This bundle is flat, but not unitary. However, according to theabove theorem, these bundles are exceptional and can be made flat and unitary byan arbitrarily small perturbation of transition functions.The space of equivalence classes of flat and unitary G-bundles is a complexmanifold with singularities.

Denote this manifold by M. This manifold can bethought of as the space of coadjoint orbits of generic position.We can now classify all holomorphic invariants of the action of gΣ in Hλ.Let b be a complex Lie algebra and V be a topological representation of b. Definea holomorphic invariant of the action of b in V as a holomorphic map from a Zariskiopen subset in V to a fixed finite-dimensional complex manifold which is invariantunder the action of b. The coefficient λ is an example of a holomorphic invariantof the coadjoint action of gΣ.

Another example would be the M-valued function b

9defined on a Zariski open subset in E∗which takes an operator D to the equivalenceclass of B(D). The orbit classification result impliesProposition 3.4.

Any holomorphic invariant of the action of gΣ in E∗is a func-tion of b and λ.4. Orbital structure in genus 1.In this section we assume the surface Σ to be a complex torus C/L, L = {p +qτ|p, q ∈Z}, Imτ > 0.

In this case it is possible to give a more explicit descriptionof the space of orbits.The coadjoint representation E∗of the group ˆGΣ can be identified with the spaceof differential operators λ ∂∂¯z + ξ where ξ is a g-valued function on the torus. Theseoperators act on V -valued functions on the torus for any representation V of G.According to the previous section, orbits of the coadjoint action of GΣ restrictedto a hyperplane Hλ correspond to equivalence classes of holomorphic principal G-bundles over the complex torus Σ.

Almost every such bundle is flat and unitary,i.e.comes from a homomorphism from the fundamental group to the maximalcompact subgroup K in G. The fundamental group of the torus is Π = Z2, so wemay assume that this homomorphism lands in a maximal torus T ⊂K. Since allmaximal tori are conjugate, we can assume T to be a fixed maximal torus in K.The images of the two generators of Π can be any two elements in T .

Thus, we geta covering of the set of non-equivalent unitary representations of Π by the productT × T . Two elements of this product correspond to isomorphic representations ofΠ if and only if one of them can be obtained from the other by the action of anelement of the Weyl group W of K. Therefore, the set of equivalence classes ofunitary representations of Π is (T × T )/W.Thus, we have found that topologically the space M of orbits of generic positionis (T ×T )/W.

However, this realization does not tell us anything about the complexstructure on M. Therefore let us describe a different realization of M.Let D = λ ∂∂¯z + ξ. If the bundle B(D) is flat (which happens, as we know, foralmost every ξ) then the equation Dψ = 0 on a G-valued function ψ will have asolution ψ(z) with the properties ψ(z + 1) = ψ(z), ψ(z + τ) = ψ(z)A, where A isa fixed element of G. If the bundle B(D) is also unitary, the element A will besemisimple.

Then we may assume that it belongs to a fixed complex maximal torusTC ⊂G.The element A completely determines the bundle B(D). Indeed, let a ∈g satisfyA = exp(a), and set F(z) = ψ(z) exp(−a z−¯zτ−¯τ ).

Obviously F is doubly periodic, i.e.F ∈GΣ, and we have(4.1)F ◦(λ ∂∂¯z + ξ) = λ ∂∂¯z −aτ −¯τ .Moreover, different elements of TC may correspond to equivalent bundles. Letr be the rank of G and t be the Lie algebra of T .

If hj ∈it, 1 ≤j ≤r denotethe standard basis of the Cartan subalgebra of g then exp(2πihj) = 1, and thesolution ψ(z) of the equation Dψ = 0 can be replaced by another solution ψn(z) =ψ(z) exp(2πiz(Prj=1 njhj)) where nj ∈Z.This solution satisfies ψn(z + 1) =ψn(z), ψn(z + τ) = ψ(z)An, where An = A exp(2πiτ(Prj=1 njhj)).Obviously,An corresponds to the same bundle as A. Furthermore, if we conjugate A by any

10element of the Weyl group W, we will obtain an element A′ of TC that correspondsto the same bundle as A.Let Q∨be the lattice generated by hj (the dual weight lattice of G). We haveshown that the bundle B(D) is completely determined by the projection of A intothe complex space TC/(W ⋉exp(2πiτQ∨)) = tC/(W ⋉(Q∨⊕τQ∨)).

It is easyto see that different points of this space correspond to non-isomorphic bundles.Therefore, we haveProposition 4.1. The space M of equivalence classes of flat and unitary holo-morphic G-bundles over the complex torus is isomorphic to tC/(W ⋉(Q∨⊕τQ∨)).An alternative way to obtain the classification of generic coadjoint orbits of ˆGΣis based on a study of conjugacy classes of the twisted loop group.

Let us give abrief description of this method.By a loop group GS1hwe mean the group of holomorphic maps from the cylinderC/Z to G. The abelian group C/Z acts by automorphisms on GS1hthrough trans-lations of the argument. Form the semidirect product ˇGS1h = C/Z⋉GS1h associatedto this action.

Let us analyze the conjugacy classes of ˇGS1h . For (z, f), (τ, g) ∈ˇGS1hwe have(4.2)(z, f)−1(τ, g)(z, f) = (τ, h),h(u) = f(u −τ)−1g(u −z)f(u),u ∈C/Z.This implies that τ is preserved under conjugations, so we may assume that it isfixed, and study conjugacy classes inside the coset Cτ = {(τ, g)|g ∈GS1h }.

We willtreat the case τ /∈R/Z. In this case we may assume, without loss of generality,that Imτ > 0.Let ξ be a smooth function on the torus Στ.

Consider the differential equation(4.3)λ∂ψ∂¯z + ξψ = 0with respect to a G-valued function ψ on the cylinder C/Z.Let ψ0(z) be a solution of equation (4.3). Then ψ0(z +τ) is also a solution, sincethe equation is invariant under a translation by τ.

Therefore, ν0(z) = ψ0(z)−1ψ0(z+τ) is a holomorphic G-valued function on the cylinder. If we choose another solutionof (4.3), say, ψ1, it will have the form ψ1(z) = ψ0(z)µ(z) where µ is holomorphic.Therefore, the function ν1(z) = ψ1(z)−1ψ1(z + τ) can be expressed as follows:ν1(z) = µ(z)−1ν0(z)µ(z + τ).

This implies that the conjugacy class of the element(τ, ν0) is independent on the choice of the solution ψ0 of (4.3).Thus we havecanonically associated a conjugacy class of ˇGS1hto any equation of form (4.3). Itis clear that every conjugacy class in Cτ comes from a certain equation.

Thus wehave established a one-to-one correspondence between orbits of the action of GΣ inHλ and conjugacy classes of ˇGS1hin Cτ.Now the result that generic orbits of GΣ in Hλ are parametrized by points ofthe space M can be deduced from the classical theory of difference equations. Thistheory was developed in the beginnning of the twentieth century, by G.D.Birkhoff,R.D.Carmichael, C.R.Adams, W.J.Trjitinsky and others [3,17].Consider the equation(4.4)F(z + τ) = A(z)F(z)where A(z) is an entire periodic G-valued function with period 1.

11Proposition 4.2. For almost every A(z) (i.e.

for A(z) belonging to a Zariski opensubset in GS1h ) there exists a solution of (4.4) of the form(4.5)F(z) = Φ(z) exp(Pz),where P ∈g and Φ is an entire G-valued function with period 1.This proposition, for G = SLn, can be deduced from the Fundamental ExistenceTheorem in the theory of linear difference equations [17].Clearly, Proposition 4.2 implies the classification result for generic orbits.Despite we did not use the notion of a vector bundle in this argument, it wasimplicitly present in our considerations.Indeed, E.Looijenga [8] observed thatthe conjugacy classes of GS1hwhich lie inside Cτ are in one-to-one correspondencewith equivalence classes of holomorphic principal G-bundles over the complex torusΣτ = C/(Z ⊕τZ). This correspondence is constructed as follows.The complex torus Στ can be obtained from the annulus {z ∈C/Z|0 ≤Imz ≤Imτ} by gluing together the boundary components according to the rule z ↔z +τ.In order to define a holomorphic bundle on the torus, it is enough to present a G-valued holomorphic transition function A(z) in a neighborhood of the seam Imz = 0.Therefore, we can naturally associate a holomorphic G-bundle to every element(τ, g) ∈Cw by setting A(z) = g(z).

Now observe that equation (4.2) expressesexactly the fact that the equivalence class of this bundle does not depend on thechoice of the element inside the conjugacy class, and that different conjugacy classesgive rise to non-equivalent bundles. It remains to make sure that every holomorphicG-bundle over Στ comes from a certain conjugacy class in Cτ.

To see this, let uspick a bundle B over Στ, and pull it back to the cylinder C/Z. Of course, theobtained bundle ˜B will be trivial.

Let us pick a global holomorphic section χ(z) of˜B. Then χ(z+τ) is another holomorphic section.

Therefore, g(z) = χ(z)−1χ(z+τ)is a holomorphic function. Evidently, the bundle B is associated to the conjugacyclass of (τ, g).To conclude this section, let us discuss the coadjoint action of Vir(Σ) in the casewhen Σ is a complex torus.

Obviously, this Lie algebra cannot be integrated to aLie group – this is impossible even in the one-dimensional case as long as we workover C. Therefore, we cannot define orbits of the coadjoint action in the usual way.What we can do, though, is define and fully describe holomorphic invariants of thecoadjoint action.Coadjoint orbits of the real Virasoro algebra have been described in [7,16]. Thesmooth part of the coadjoint representation can be interpreted as the space of Hill’soperators λ d2dx2 + q(x) taking densities of weight −1/2 to densities of weight 3/2on the circle.

The coefficient λ is invariant under the coadjoint action, so we mayrestrict this action to the hyperplane Hλ of operators with a fixed value of thiscoefficient. Orbits of the group Diff+(S1) lying in this hyperplane for λ ̸= 0 arelabelled by conjugacy classes of the universal covering of the group SL2(R).For the complex Virasoro algebra, orbits are not defined since the algebra doesnot integrate to a Lie group.

However, one can study holomorphic invariants of thecoadjoint action which carry the same information as orbits. Repeating the aboveargument with obvious modifications, one finds that besides λ there is essentiallya unique holomorphic invariant of the coadjoint action – the monodromy of thecorresponding Hill’s operator.

This invariant takes values in the space of conjugacy

12classes of SL2(C).Since almost every element in SL2(C) is diagonalizable, wemay assume that the monodromy invariant takes values in C∗/ ∼where ∼is theequivalence relation that identifies z with z−1. Any holomorphic invariant of thecoadjoint action will then be a function of the coefficient λ and the monodromyinvariant.It is surprising that the same procedure works in two dimensions.Define adensity of weight µ ∈C on the torus Σ as a formal expression u(d¯z)µ where u is asmooth function on Σ.

The action of (0,1)-vector fields on densities is given by(4.6)v ∂∂¯z ◦u(d¯z)µ =v ∂u∂¯z + µu∂v∂¯z(d¯z)µ.The smooth part of the coadjoint representation of Vir(Σ) can now be identifiedwith the space of two-dimensional Hill’s operators H = λ ∂2∂¯z2 + q(z, ¯z) taking den-sities of weight −1/2 to densities of weight 3/2 on the torus.The monodromy of such an operator can be defined as follows. Assume λ ̸= 0.Consider the differential equation(4.7)Hu = 0.This equation is equivalent to the system(4.8)∂∂¯z U + QU = 0,Q =01q0,U =uu′,u′ = ∂u∂¯z .This system, as we know, defines a holomorphic SL2(C)-bundle on the torus, whichis almost always flat and unitary.

Therefore, for almost every H defined is a pointon M corresponding to this bundle. Let us call this point the monodromy of H.Proposition 4.3.

The monodromy is a holomorphic invariant of the coadjointaction of Vir(Σ).Proof. The proof is by giving another definition of the monodromy.

Let S(H) bethe set of all s ∈C such that there exist two linearly independent solutions u1, u2of (4.7) which are functions on the cylinder C/Z and satisfy the conditions(4.9)u1(z + τ) = su1(z),u2(z + τ) = s−1u2(z).S(H) is non-empty whenever the bundle associated to H is flat and unitary, i.e. foralmost every H. Also, if s ∈S(H) then for any n ∈Z s(n) = s exp(2πinτ) ∈S(H).Indeed, if we consider new solutions u(n)1= u1 exp(2πinz), u(n)2= u2 exp(−2πinz)of (4.8), they will satisfy (4.9) with s replaced by s(n).

Finally, if s ∈S(H) thens−1 ∈S(H). Moreover, it is easy to check that conversely, if s1, s2 ∈S(H) theneither s2 = s(n)1for some integer n or s−12= s(n)1 .Define ˆs as the projection of s into the space C∗/ ∼where ∼is the equivalencerelation that identifies s with s(n) for any n, and s with s−1.

If s ∈S(H) thenˆs depends only on H. It is obvious from the definition that ˆs is a holomorphicinvariant. It remains to observe that C∗/ ∼is isomorphic to M, and that ˆs isnothing else but the monodromy of H.

13Remark 4.1. 1.

A little more extra work is needed to prove the following:Any holomorphic invariant of the coadjoint action of Vir(Σ) is a function of λand the monodromy.This statement motivates a definition of coadjoint orbits for Vir(Σ) as level setsof the monodromy inside hyperplanes λ = const in the space of two-dimensionalHill’s operators.Remark 4.2. Observe that the space of “orbits” for Vir(Σ) is the same as forfor SL2(C)Σ.

A similar coincidence takes place also in the one-dimensional theory,where it reflects the deep relationship between the representation theories of VirCand csl2(C).5. Geometry of orbits.Define an analogue of Kirillov-Kostant structure on orbits of the coadjoint actionof the group GΣ.

We will assume that Σ is a complex torus. The construction isthe same as for classical Lie groups.Let X1, X2 be tangent vectors to an orbit O in E∗at a point f. Then there existelements ˜X1, ˜X2 ∈gΣ such that(5.1)ddt(et ˜Xj ◦f) = Xj.Set(5.2)K(X1, X2) = f([ ˜X1, ˜X2])Despite the liftings ˜X1, ˜X2 are not unique, equation (5.2) presents a well definedantisymmetric bilinear form on the tangent space to O at f for every f ∈O.

Thus,K is a holomorphic differential 2-form on O.The following properties of K are proved similarly to the classical orbit method.Proposition 5.1. (i) K is closed.

(ii) K is nondegenerate. (iii) K is invariant under the action of GΣ.Remark 5.1.

Unfortunately, we are unable to define a proper counterpart ofKahler structure on the orbits. This appears to be the main obstacle in attemptsto construct projective representations of GΣ via orbit method.

Attempts to con-struct representations algebraically encounter quite similar difficulties arising fromthe inability to generalize the notion of polarization for loop algebras to gΣ.Let us describe the topological structure of orbits. For this purpose we need todescribe the isotropy subgroup of a vector in the coadjoint representation.Recall the realization of M as (T × T )/W.

Let m ∈M, and let ˜m ∈T × Tbe an inverse image of m. Let W0 ⊂W be the stabilizer of ˜m. We choose ˜mso that W0 be generated by a set of Weyl reflections corresponding to the nodesof the Dynkin diagram of G.Also, pick an inverse image ˆm ∈tC of the point˜m in the complex Cartan subalgebra.

Denote by Stab( ˆm) the stabilizer of ˆm inthe Weyl group. Obviously, we have Stab( ˆm) ⊂W0.

Also, it is easy to show thatamong the groups Stab( ˆm) for various inverse images ˆm there is a maximal one, i.e.

14one containing all the others. We denote this group by W ′ and the correspondinginverse image by m′.

The group W ′0 coincides with the group of all w ∈W0 suchthat w ˆm −ˆm = wq −q, q ∈Q∨, for an inverse image ˆm. It is also generated byWeyl reflections.It follows from this definition that W ′0 is a normal subgroup in W0.

Denote thequotient W0/W ′0 by F. Let R be the centralizer of m′ in G. It is known that Ris a connected reductive subgroup of G (Levi subgroup). The Weyl group of R isW ′0.

Obviously, W0 normalizes R, since it normalizes TC and W ′0. Therefore, wecan form a semidirect product ˜R = F ⋉R.Proposition 5.2.

The stabilizer of the element D = λ ∂∂¯z + λm′ ∈Hλ in GΣ isisomorphic to ˜R. Its intersection with the subgroup of constant currents is equal toR.Remark 5.2.

A quite similar statement is true in the case of loop groups.Remark 5.3. If the quotient F is nontrivial, the stabilizer of D is not conjugatein GΣ to any subgroup of constant currents.Idea of proof.

For the sake of brevity let us assume that G = SLn.Then theoperator D naturally acts in the space of Cn-valued functions on the torus. If welet µk, 1 ≤k ≤n be the eigenvalues of m′ as an operator in Cn, and vk, 1 ≤k ≤nform the corresponding eigenbasis, then the spectrum of D is the union of n latticesµk +L, and the corresponding eigenbasis of D would be vk exp(2πi(px+qy)) wherep, q ∈Z and x, y are real coordinates such that z = x + τy.

In the generic case,when W0 is trivial and all the eigenvalues of D are distinct, any element h fromthe stabilizer of D in GΣ has to be diagonal in this basis, which shows that thestabilizer of D is just TC. When D has multiple eigenvalues, it is still true that itseigenspaces are invariant under h. It allows one to classify all possible h and obtainthe result of the theorem.

A similar argument works for an arbitrary G.Corollary 5.3. The orbit OD of the vector D is isomorphic to GΣ/ ˜R.Corollary 5.4.

Let s=dim(R/[R,R]). Then H2(O) = Zs+2 ⊕torsion.Proof.

Pick a point z ∈Σ and decompose GΣ as a smooth manifold as follows:GΣ = GΣz ×G, GΣz being the group of all currents equal to the identity at z. Since Gis 3-connected, we have H2(GΣz ) = H2(GΣ) = Z2⊕torsion.

Therefore, H2(GΣ/R) =H2(GΣz × G/R) = H2(GΣz ) ⊕H2(G/R) (since G/R is simply connected).Thehomology of the flag space G/R is known; in particular, H2(G/R) = Zs. Thus,H2(GΣ/R) = Zs+2 ⊕torsion.

According to the above proposition, the space GΣ/Ris a finite covering of the orbit OD with the fiber F = ˜R/R. Since F acts triviallyon R/[R, R], it also acts trivially on H1(R) = H2(G/R).

Therefore, H2(OD) =H2(GΣ). ■Let us construct a fundamental system of 2-cycles on OD.

First of all, we havetwo independent cycles Cx and Cy that descend from the group GΣ. They areobtained by pushing forward the generator of H2(GS1) = Z by the embeddingsfx, fy : GS1 →GΣ induced by the projections of Σ = S1 × S1 onto the first andthe second circle, respectively.Let us call a node ν of the Dynkin diagram of G regular if the correspondingWeyl reflection wν does not lie in W ′0.

There are exactly s regular nodes: ν1, ..., νs.

15To each regular node νj we can canonically associate a 2-cycle Cνj in G/R, andtransfer this cycle to OD. This transfer, however, is not canonical: it depends onthe choice of the element m′.

If we change this element to another one, the cycleCνj will be shifted by pjCx + qjCy where pj, qj are integers.The cycles Cx, Cy, Cν1, ..., Cνs constitute a basis in the free part of H2(OD).At this point we need to assume a standard normalization of the invariant form<, >. Namely, we pick the one in which the minimal length of a root of g equals√2.Proposition 5.5.

(5.3)12πZCxK = λ,12πZCyK = λτ,12πZCνjK = λ < hνj, m′ > .Idea of proof. Since the cycles Cx and Cy lie inside the images of GS1, and thecycles Cνj lie inside the subgroup G of constant currents, these identities followfrom the corresponding results for GS1.Let us now define and classify integral orbits, by analogy with the theory of loopgroups [13],[5].

For loop groups, an integral orbit is defined as an orbit on whichthe form K/2π has integral periods. A theorem in [13, Section 4.5] states that anorbit is integral if and only if there exists a circle bundle on it with curvature K.Definition 5.1.

Let us say that an orbit OD ⊂Hλ is integral if for any C ∈H2(OD)(5.4)12πZCK ∈LProposition 5.6. (i) An orbit OD is integral if and only if there exists a holomorphic principalΣ-bundle E over OD with a connection θ whose curvature is K.(ii) If OD is an integral orbit and E is the corresponding principal bundle thenthe action of GΣ on OD can be uniquely lifted to an action of ˆGΣ on E preservingthe connection θ.The proof is similar to that for loop groups.Identities (5.3) allow us to classify integral orbits.Denote by O(λ, m) the orbit in Hλ corresponding to the point m ∈M.

Weassume that λ ̸= 0.Let us call λ the level and m the weight of the orbit, byanalogy with loop groups.We also assume that τ is generic, i.e.that Σ has exactly two holomorphicautomorphisms modulo translations.Denote by Z the center of G. Let λ ∈C∗, m ∈M, and let ˜m ∈T × T be alifting of m.Proposition 5.7. An orbit O(λ, m) is integral if and only if λ is an integer, andλ ˜m ∈Z × Z ⊂T × T .

Thus, the number of integral orbits at each level is finite.This statement follows immediately from the two previous propositions.

16The number of integral orbits at each level can be easily calculated for anyparticular group. For instance, if G = SL2 then it is equal to 2λ2 + 2.Acknowledgements.We would like to thank R.Beals and H.Garland for interesting discussions.

Weare grateful to R.Beals for showing us an elementary analytic proof of Theorem 3.3for the case of a surface of genus one and G = SLn. The work of I.F.

was supportedby NSF grant DMS-8906772.

17References1. Atiyah, M., Vector bundles over an elliptic curve, Proc.

Lond. Math.

Soc. 7 (1957), 414-452.2.

Bers, L., Theory of pseudoanalytic functions, New York, 1953.3. Birkhoff, G.D., The generalized Riemann problem for linear differential equations and theallied problems for linear difference and q-difference equations, Proc.

Amer. Acad.

Arts 49(1913), 521-568.4. Donaldson, S.K., A new proof of a theorem of Narasimhan and Seshadri, J. Diff.

Geom. 18(1983), 269-277.5.

Frenkel, I.B., Orbital theory for affine Lie algebras, Inventiones Mathematicae 77 (1984),301-352.6. Gawedzki, K., and Kupianen, A., Coset construction from functional integrals, Nucl.

Phys.B 320(FS) (1989), 649.7. Kirillov, A.A., Infinite-dimensional groups, their orbits, invariants, and representations, inbook: Twistor geometry and nonlinear systems, Lecture notes in Math.

970 (1982), 101-123.8. Loojenga, E., private communication.9.

Mickelsson, J., Kac-Moody groups, topology of the Dirac determinant bundle, and fermion-ization, Comm. Math.

Phys. 110 (1986), 173-183.10.

Moody, R.V., Rao, S.E., and Yokonuma, T., Toroidal Lie algebras and vertex representations,Geometriae Dedicata 35 (1990), 283-307.11. Narasimhan, M.S., and Seshadri, C.S., Holomorphic vector bundles on a compact Riemannsurface, Math.Ann.

155 (1964), 69-80.12. Narasimhan, M.S., and Seshadri, C.S., Stable and unitary vector bundles on a compact Rie-mann surface, Ann.Math.

82 (1965), 540-567.13. Pressley, A., and Segal, G., Loop groups, Clarendon Press, Oxford, 1986.14.

Ramanathan, A., Stable principal bundles on a compact Riemann surface, Math.Ann. 213(1975), 129-152.15.

Ramos, E., Sah, C.H., and Shrock, R.E., Algebras of diffeomorphisms of the N-torus, J.Math.Phys.31(8) (1990).16. Segal, G., Unitary representations of some infinite dimensional groups, Commun.

Math. Phys.80 (1981), 301-342.17.

Trjitzinsky, W.J., Analytic theory of linear q-difference equations, Acta Math. 61 (1933),1-38.18.

Vekua, I.N., Generalized analytic functions, Pergamon Press, London, 1962.19. Witten, E., Non-abelian bozonization in two dimensions, Comm.

Math. Phys.

92 (1984),455-472.20. Witten, E., On holomorphic factorization of WZW and coset models, Comm.

Math. Phys.144 (1992), 189-212.

---

출처: arXiv:9303.047 • 원문 보기