Meromorphic open-string vertex algebras and modules over two-dimensional orientable space forms
MMEROMORPHIC OPEN-STRING VERTEX ALGEBRAS AND MODULESOVER TWO-DIMENSIONAL ORIENTABLE SPACE FORMS
FEI QI
Abstract.
We study the meromorphic open-string vertex algebras and their modules over thetwo-dimensional Riemannian manifolds that are complete, connected, orientable, and of constantsectional curvature K (cid:54) = 0 . Using the parallel tensors, we explicitly determine a basis for themeromorphic open-string vertex algebra, its modules generated by eigenfunctions of the Laplace-Beltrami operator, and their irreducible quotients. We also study the modules generated bylowest weight subspace satisfying a geometrically interesting condition. It is showed that everyirreducible module of this type is generated by some (local) eigenfunction on the manifold. Aclassification is given for modules of this type admitting a composition series of finite length.In particular and remarkably, if every composition factor is generated by eigenfunctions ofeigenvalue p ( p − K for some p ∈ Z + , then the module is completely reducible. Introduction
Vertex algebras are algebraic structures formed by vertex operators satisfying commutativityand associativity. In mathematics, they arose naturally in the study of representations of infinite-dimensional Lie algebras and the Monster group (see [B] and [FLM]). In physics, they are used inthe study of two-dimensional conformal field theory (see [BPZ] and [MS]). The commutativityand associativity allow us to view vertex algebras as analogs to the commutative associativealgebras.Meromorphic open-string vertex algebras (MOSVAs hereafter) are algebraic structures formedby meromorphic vertex operators satisfying only associativity without commutativity. They wereintroduced by Huang in 2012 (see [H1]), as special cases of the open-string vertex algebras in-troduced by Huang and Kong in 2003 (see [HK]) where all correlation functions are rationalfunctions. Similar to the case of vertex algebras, MOSVAs can be viewed as analogues of asso-ciative algebras that are not necessarily commutative.As a nontrivial example, Huang also introduced the MOSVA associated with a Riemannianmanifold in 2012 (see [H2]), using the parallel sections of the tensor algebra of the affinized tangentbundle. Given a complex-valued smooth function f on an open subset U of the manifold, Huangalso constructed the module generated by f , and proved that association of U with the sum ofmodules generated by all smooth functions over U gives a presheaf of modules for the MOSVA.Of particular interest are the modules generated by eigenfunctions of the Laplace-Beltramioperator. As such functions can be understood as quantum states in quantum mechanics, themodules they generate can be understood as the string-theoretic excitement to the quantumstates. It is Huang’s idea that the modules for the MOSVA generated by the eigenfunctions and a r X i v : . [ m a t h . QA ] J u l OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 2 the yet-to-be-defined intertwining operators among these modules may lead to a mathematicalconstruction of the quantum two-dimensional nonlinear σ -model.This paper studies the example of MOSVAs and modules over two-dimensional orientable spaceforms, i.e., two-dimensional Riemannian manifolds that are complete, connected, orientable, andwith constant curvature K (cid:54) = 0 . The paper is organized as follows,In Section 2, we briefly review the previously known results in [H1] and [H2]. The axiomsfor the MOSVA and modules are slightly modified using the results in [Q1] to make it easier toverify.In Section 3, we determine a basis for the MOSVA. We study the holonomy groups of all thebundles involved in Huang’s construction. Then we write down the parallel sections, use themto determine a basis explicitly for the MOSVA, and compute its graded dimension. It turns outthat the graded dimension is related to the hypergeometric function F .In Section 4, we focus on the modules generated by the eigenfunctions for the Laplace-Beltramioperator. We first prove a lemma on higher-order covariant derivatives, which is then used toshow that the covariant derivative of an eigenfunction along every parallel tensor is a scalarmultiple of the eigenfunction. The scalar turns out to be a polynomial function in terms of theeigenvalue with roots sitting in { p ( p − K : p ∈ Z + } . Using these results, we determine a basisexplicitly for the modules generated by the eigenfunctions. We also show that two such modulesare isomorphic if and only if they are generated by eigenfunctions with the same eigenvalues overspace forms of the same sectional curvature.In Section 5, we study the irreducible modules generated by the eigenfunctions, which arequotients of the modules constructed in Section 4 by their unique maximal submodules. Toidentify these quotients, we prove a formula on the lowest weight projection of Y ( v, x ) w forhomogeneous v ∈ V and w ∈ W , which plays a crucial role in the subsequent discussions. Thenwe take two quotients consecutively to obtain the irreducible modules that are generated byeigenfunctions of generic eigenvalues. Remarkably, for modules generated by eigenfunctions ofspecific eigenvalues p ( p − K for some p ∈ Z + , an additional quotient is needed, making thestructure different to those with generic eigenvalues.In Section 6, we study the lowest weight modules in general, which are modules generated by itslowest weight subspace. We show that there exists a bijective correspondence between irreduciblelowest weight modules for the MOSVA and irreducible modules for the algebra of parallel tensors.We then focus on lowest weight modules whose lowest weight subspace is a module for the paralleltensors satisfying a geometrically interesting condition, called the covariant derivative condition.It turns out that every irreducible module of this type is isomorphic to some irreducible modulesconstructed in Section 5. In other words, every irreducible module of this type is generatedby a (local) eigenfunction on the manifold. We also give a classification of modules of thistype admitting a composition series of finite length. In particular, such a module is completely OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 3 reducible if every composition factor is generated by eigenfunctions with special eigenvalues p ( p − K for some p ∈ Z + . Acknowledgements.
The author would like to thank Yi-Zhi Huang for his long-term supportand patient guidance. The author would also like to thank Robert Bryant for his guidanceon holonomy groups over the tensor powers of the tangent bundle and his correction on thestatement of Lemma 3.5. The author would also like to thank Igor Frenkel, Eric Schippers, andNolan Wallach for helpful discussion.2.
Previously known results
In this section, we recall some background knowledge in [H1], [H2] and [Q1].2.1.
Axioms of the meromorphic open-string vertex algebra and its left module.Definition 2.1. A meromorphic open-string vertex algebra (hereafter MOSVA) is a Z -gradedvector space V = (cid:96) n ∈ Z V ( n ) (graded by weights ) equipped with a vertex operator map Y V : V ⊗ V → V [[ x, x − ]] u ⊗ v (cid:55)→ Y V ( u, x ) v, and a vacuum ∈ V , satisfying the following axioms:(1) Axioms for the grading:(a) Lower bound condition : When n is sufficiently negative, V ( n ) = 0 .(b) d -commutator formula : Let d V : V → V be defined by d V v = nv for v ∈ V ( n ) . Thenfor every v ∈ V [ d V , Y V ( v, x )] = x ddx Y V ( v, x ) + Y V ( d V v, x ) . (2) Axioms for the vacuum:(a) Identity property : Let V be the identity operator on V . Then Y V ( , x ) = 1 V .(b) Creation property : For u ∈ V , Y V ( u, x ) ∈ V [[ x ]] and lim x → Y V ( u, x ) = u .(3) D -derivative property and D -commutator formula : Let D V : V → V be the operatorgiven by D V v = lim x → ddx Y V ( v, x ) for v ∈ V . Then for v ∈ V , ddx Y V ( v, x ) = Y V ( D V v, x ) = [ D V , Y V ( v, x )] . (4) Weak associativity with pole-order condition : For every u , v ∈ V , there exists p ∈ N suchthat for every u ∈ V , ( x + x ) p Y V ( u , x + x ) Y V ( u , x ) v = ( x + x ) p Y V ( Y V ( u , x ) u , x ) v. OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 4
Remark 2.2.
This definition is slightly more special than the definition given by Huang in [H1],where p is not necessarily depending only on u and v . This dependence is called pole-ordercondition and can be used to simplify the verification of axioms. Please see [Q1] for a detaileddiscussion. Definition 2.3.
Let V be a meromorphic open-string vertex algebra. A left V -module is a C -graded vector space W = (cid:96) m ∈ C W [ m ] (graded by weights ), equipped with a vertex operatormap Y LW : V ⊗ W → W [[ x, x − ]] u ⊗ w (cid:55)→ Y LW ( u, x ) w, an operator d W of weight and an operator D W of weight , satisfying the following axioms:(1) Axioms for the grading:(a) Lower bound condition : When Re ( m ) is sufficiently negative, W [ m ] = 0 .(b) d -grading condition : for every w ∈ W [ m ] , d W w = mw .(c) d -commutator formula : For u ∈ V , [ d W , Y LW ( u, x )] = Y LW ( d V u, x ) + x ddx Y LW ( u, x ) . (2) The identity property : Y LW ( , x ) = 1 W .(3) The D -derivative property and the D -commutator formula : For u ∈ V , ddx Y LW ( u, x ) = Y LW ( D V u, x )= [ D W , Y LW ( u, x )] . (4) Weak associativity with pole-order condition : For every v ∈ V, w ∈ W , there exists p ∈ N such that for every v ∈ V , ( x + x ) p Y LW ( v , x + x ) Y LW ( v , x ) w = ( x + x ) p Y LW ( Y V ( v , x ) v , x ) w. Example: Noncommutative Heisenberg.
The first nontrivial example of MOSVA isconstructed by Huang in [H1]. We should recall the construction here.Let h be a finite-dimensional Euclidean space over R . We define a vector space ˆ h = h ⊗ R C [ t, t − ] ⊕ C k , which is the ambient vector space of the Heisenberg Lie algebra. Note that ˆ h = ˆ h − ⊕ ˆ h ⊕ ˆ h + , where ˆ h + = h ⊗ R t C [ t ] , ˆ h = h ⊗ R C ⊕ C k , ˆ h + = h ⊗ R t − C [ t − ] . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 5
Let N (ˆ h ) be the quotient of the tensor algebra T (ˆ h ) of ˆ h modulo the two-sided ideal I generatedby ( a ⊗ t m ) ⊗ ( b ⊗ t n ) − ( b ⊗ t n ) ⊗ ( a ⊗ t m ) − m ( a, b ) δ m + n, k , ( a ⊗ t k ) ⊗ ( b ⊗ t ) − ( b ⊗ t ) ⊗ ( a ⊗ t k ) , ( a ⊗ t k ) ⊗ k − k ⊗ ( a ⊗ t k ) , (1)for a, b ∈ h , m ∈ Z + , n ∈ − Z + , k ∈ Z . Note that in the quotient, there are no relations between X ⊗ t m and Y ⊗ t n for m, n ∈ Z + and for m, n ∈ Z − . This is the main difference to the usualconstruction of Bosonic Fock space, where ( a ⊗ t m ) ⊗ ( b ⊗ t n ) − ( b ⊗ t n ) ⊗ ( a ⊗ t m ) is also included in the generators of I , for each a, b ∈ h , m, n ∈ Z ± . Nevertheless, the PBWstructure still holds for this quotient: N (ˆ h ) (cid:39) T (ˆ h − ) ⊗ T (ˆ h + ) ⊗ T ( h ) ⊗ T ( C k ) as vector spaces(see [H1], Proposition 3.1)Let C = C be a one-dimensional vector space on which h acts by 0. Define the action of k by 1and ˆ h + by 0. One can prove that the induced module N (ˆ h ) ⊗ N (ˆ h + ⊕ ˆ h ) C is isomorphic to T (ˆ h − ) asa vector space. We regard T (ˆ h − ) now as an N (ˆ h ) -module and denote the action of h ⊗ t j by h ( j ) .Then T (ˆ h − ) is spanned by h ( − m ) · · · h k ( − m k ) for k ∈ N , h , ..., h k ∈ h , m , ..., m k ∈ Z + .Huang proved the following theorem in [H1]. Theorem 2.4 ([H1], Theorem 5.1) . The left N (ˆ h ) -module T (ˆ h − ) forms a grading-restrictedMOSVA with the following vertex operator action: Y ( h ( − m ) · · · h k ( − m k )1 , x )= ◦◦ m − d m − dx m − h ( x ) · · · m k − d m k − dx m k − h k ( x ) ◦◦ , where h i ( x ) = (cid:80) n ∈ Z h i ( n ) x − n − . The normal ordering is defined as follows: ◦◦ h ( m ) · · · h k ( m k ) ◦◦ = h σ (1) ( m σ (1) ) · · · h σ ( k ) ( m σ ( k ) ) , where σ ∈ S k is the unique permutation such that σ (1) < · · · < σ ( α ) , σ ( α + 1) < · · · < σ ( β ) , σ ( β ) < · · · < σ ( k ) ,m σ (1) , ..., m σ ( α ) < , m σ ( α +1) , ..., m σ ( β ) > , m σ ( β +1) , ..., m σ ( k ) = 0 . (2)2.3. MOSVA of a Riemannian manifold.
In [H2], Huang further constructed a MOSVAfor any Riemannian manifold M , using the parallel sections of a certain bundle. We recall theconstruction here.Let M be a Riemannian manifold. Let p ∈ M . We consider the affinization of tangent bundle (cid:100) T M = T M ⊗ R ( M × C [ t, t − ]) ⊕ ( M × C k ) , OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 6 where M × C [ t, t − ] and M × C k are trivial bundles over M . The fiber of this bundle at p isnothing but (cid:91) T p M = T p M. ⊗ R C [ t, t − ] ⊕ C k . Analogous to the constructions in the previous subsection, we have (cid:100)
T M = (cid:100) T M + ⊕ (cid:100) T M ⊕ (cid:100) T M − , with (cid:100) T M ± = T M ⊗ R ( M × t ± C [ t ± ]) , (cid:100) T M = T M ⊗ R ( M × C t ) ⊕ T M ⊗ R ( M × C k ) . Now we look into the tensor algebra bundle T ( (cid:100) T M ) . We similarly construct a bundle N ( (cid:100) T M ) ,whose fiber at each point is obtained by taking the quotient of T ( T p M ) versus the two-sided idealgenerated by elements in (1), for a, b ∈ T p M . Likewise, N ( (cid:100) T M ) is isomorphic to T ( (cid:100) T M − ) ⊗ T ( (cid:100) T M + ) ⊗ T ( (cid:100) T M ) ⊗ T ( C k ) .We now consider the space Π( T ( (cid:100) T M − )) consisting of parallel sections of the tensor algebrabundle T ( (cid:100) T M − ) of (cid:100) T M − . It is well known that Π( T ( (cid:100) T M − )) , as a vector space, is isomorphicto subspace of fixed points T ( (cid:91) T p M − ) Hol ( T ( (cid:100) T M − ) , where Hol means the holonomy group of thebundle. Recall that Theorem 2.4 endows the space T ( (cid:91) T p M − ) with a MOSVA structure. In [H2],Huang proved that the elements of the holonomy group could be realized as automorphismsMOSVA, and on the fixed point subspace of automorphism there is a MOSVA structure. ThusHuang proved the following theorem: Theorem 2.5.
On the space Π( T ( (cid:100) T M − )) of parallel sections there is a MOSVA structure, whichis a subalgebra of the MOSVA of T ( (cid:91) T p M − ) in Theorem 2.4.Moreover, by an analogous argument to the Segal-Sugawara construction, Huang also showedthat the Laplace-Beltrami operator on the manifold M is realized as a component of some vertexoperator.2.4. Modules generated by smooth functions.
In [H2], Huang considered that the action ofthe parallel sections Π( T ( T M C )) on the space of smooth functions via the covariant derivatives.More precisely, let U be an open subset of M ; let f be a smooth function on U . Define a parallelsection X ∈ Π((
T M C ) ⊗ k ) acts on f by ψ U ( X ) f := ( √− k ( ∇ k f )( X ) . Huang proved (Theorem 4.1, [H2]) that the action respects the associative algebra structuredefined by ⊗ , namely, for X, Y ∈ Π( T ( T M ) C ) , ψ U ( X ⊗ Y ) = ψ U ( X ) ψ U ( Y ) . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 7
Thus the space of complex-valued smooth functions can be viewed as a module for the associativealgebra Π( T ( T M ) C ) .Let p ∈ U . Identify Π( T ( T M ) C ) with T ( T p M C ) Hol ( T ( T p M C )) , which can be viewed as a subal-gebra of T ( T p M C ) . Thus the module C ∞ ( U ) can be induced, defining C p ( U ) = T ( T p M C ) ⊗ T ( T p M C ) Hol ( T ( TpM ) C C ∞ ( U ) . By Theorem 6.5 of [H1], on the vector space T ( (cid:91) T p M − ) ⊗ C p ( U ) there is a natural module structure for the MOSVA T ( (cid:91) T p M − ) . In particular, it is a module for theMOSVA T ( (cid:91) T p M − ) Hol ( (cid:91) T p M − ) = Π( T ( (cid:100) T M − )) . We can thus consider the Π( T ( (cid:100) T M − )) -submodulegenerated by ⊗ (1 ⊗ f ) for some f ∈ C ∞ ( U ) . Since the Laplace-Beltrami operator is a component of some vertexoperator, it would be natural to consider the submodule generated by its eigenfunction. Asthe Laplace-Beltrami operator plays the role of energy operator in quantum mechanics, thesubmodule can then be interpreted as string-theoretical excitement of the quantum states.3. Basis of the MOSVA and modules
Let M be a two-dimensional Riemannian manifold with constant sectional curvature K . Forconvenience, we assume M is orientable, connected and complete. We will also focus on thecase K (cid:54) = 0 . In this section, we will first determine the parallel sections of the tensor algebra ofthe affinized tangent bundle. Using these parallel sections, we explicitly identify a basis for theMOSVA and modules generated by the eigenfunctions constructed in [H2].3.1. Holonomy of the tensor powers of the complexified tangent bundle.
Recall thatthe holonomy group of a bundle E based at a point p ∈ M is the subgroup generated by all theparallel translations along piecewise smooth contractible loops based on p . We will simply denotethe holonomy group by Hol ( E ) since holonomy groups based at different points are isomorphic. Lemma 3.1. If K (cid:54) = 0 , then the holonomy group Hol ( T M ) of the tangent bundle T M is SO (2 , R ) . Proof.
Since M is orientable, we know that Hol ( T M ) ⊂ SO (2 , R ) (see [P]). Fix p, q, r ∈ M . Let γ , γ , γ be geodesics connecting pq, qr and rp . Let α p , α q , α r be the angles of geodesic triangle pqr . Let v ∈ T p M be a unit vector. One sees easily that that composition of parallel transportalong the concatenation of γ , γ and γ ends up with a unit vector w ∈ T p M that is obtainedfrom rotating v by the angle π − ( α p + α q + α r ) , as shown in the following two figures. OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 8
By Gauss-Bonnet theorem (geodesic triangle version, see [D]), α p + α q + α r = π + K · Area ( pqr ) . Now let q, r vary near p , so that the area of the geodesic triangle varies continuously within someinterval [0 , s ] . Then we see that rotations by angles between π − Ks and π are all included inthe holonomy group. These rotations generate all SO (2 , R ) . (cid:3) We focus on the complexified tangent bundle C ⊗ R T M , where C is regarded as a trivial bundleover M . Notation 3.2.
From now on, we shall denote the complexified tangent bundle C ⊗ R T M by E .We will also omit the ⊗ R symbol when writing the smooth sections in Γ( E ) . All ⊗ symbol willmean ⊗ C by default unless otherwise stated.The connection on E is related to that on T M by ∇ ( X + iY ) = ∇ ( X ) + √− ∇ ( Y ) , X, Y ∈ Γ( T M ) . Lemma 3.3.
Hol ( E ) = Hol ( T M ) = SO (2 , R ) . Proof.
This follows from the observation that the bundle E is essentially the direct sum T M ⊕√− · T M . (cid:3) We will also consider the tensor bundle E ⊗ k for each k ∈ Z + . Lemma 3.4.
There is a natural surjective homomorphism Hol ( E ) → Hol ( E ⊗ k ) of holonomygroups, where g ∈ Hol ( E ) acts on each fiber E ⊗ kp by g ( v ⊗ · · · ⊗ v n ) = gv ⊗ · · · ⊗ gv n . for any v , ..., v n ∈ E p . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 9
Proof.
For any piecewise smooth path γ : [0 , → M with γ (0) = p , let P γ ( t ) : E p → E γ (1) bethe parallel transport along γ on the bundle E ; let P kγ ( t ) : E ⊗ kp → E ⊗ kγ (1) be the parallel transportalong γ with respect to the bundle E ⊗ k . Then from the definition of the connection on E ⊗ k : ∇ ( X ⊗ · · · ⊗ X n ) = n (cid:88) i =1 X ⊗ · · · ⊗ ∇ ( X i ) ⊗ · · · ⊗ X n , it follows that P kγ ( t ) ( v ⊗ · · · ⊗ v k ) = P γ ( t ) v ⊗ · · · ⊗ P γ ( t ) v k . In case γ ( t ) is a loop based at p , this essentially realizes every element of h ∈ Hol ( E ⊗ k ) as g ⊗ k for g ∈ Hol ( E ) . So the map g (cid:55)→ g ⊗ k gives a natural surjective homomorphism Hol ( E ) → Hol ( E ⊗ k ) . (cid:3) Lemma 3.5.
The holonomy group of E ⊗ k is determined byHol ( E ⊗ k ) = (cid:40) SO (2 , R ) if k is odd ,SO (2 , R ) / {± } if k is even . Proof.
We analyze the kernel of the homomorphism SO (2 , R ) → Hol ( E ⊗ k ) . Fix p ∈ M . For thematrix M ( α ) = (cid:20) cos α − sin α sin α cos α (cid:21) , let v ± ∈ E C p be an eigenvector of M ( α ) with eigenvalue e ± iα . The conclusion then follows from M ( α )( v i ⊗ · · · ⊗ v i k ) = e i ( m − n ) α ( v i ⊗ · · · ⊗ v i k ) , where m = { j : i j = 1 } , n = { j : i j = − } . In greater detail, if k is odd then there is no wayto make e i ( m − n ) α = 1 for every choice of i , ..., i k unless α = 0 ; if k is even, then there is no wayto make e i ( m − n ) α = 1 for every choice of i , ..., i k unless α = 0 or α = π . (cid:3) Parallel sections of the tensor powers of the complexified tangent bundle.Lemma 3.6. Π( E ⊗ k ) = 0 if k is odd. Proof.
It follows directly from − ∈ Hol ( E ⊗ k ) . (cid:3) To describe the parallel sections of the even tensor powers of E , we introduce the followingnotations: Definition 3.7.
Fix p ∈ M , let { e , e } be an orthonormal basis of T p M . Then for someneighborhood V of p , we define local sections X , X : V → T V by parallel transporting e , e to every q ∈ V . Finally, we introduce the following local sections of E : h + = X − iX , h − = X + iX . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 10
Example 3.8.
Let M be the unit sphere M = { ( x, y, z ) ∈ R : x + y + z = 1 } . For p = (1 , , ,we take the polar coordinate x = cos θ sin φ, y = sin θ sin φ, z = cos φ, where θ ∈ [0 , π ) , φ ∈ (0 , π ) . Then on the neighborhood V = M \ { (0 , , ± } of p , the metric isof the form ds = dφ + sin φdθ . We then take X = ∂ φ , X = 1sin φ ∂ θ . In this case, h + = ∂ φ − √− φ ∂ θ , h − = ∂ φ + √− φ ∂ θ . Example 3.9.
Let M be a complete hyperbolic surface of genus g . From the discussion in [JS], M is realized as the orbit space H/ Γ , where H is the Poincaré disk H = { ( x, y ) ∈ R : x + y < } and Γ is a Fuchsian group (with no fixed points on H ). For p = (0 , , let V be the interior of thefundamental region of Γ containing p . In other words, V is the interior of a hyperbolic polygonwith g sides. Viewed as a coordinate chart of M near p , V can be endowed with a metric ofthe form ds = 4( dx + dy )(1 − x − y ) . We then take X = 1 − x − y ∂ x , X = 1 − x − y ∂ y . In this case, h + = 1 − x − y ∂ x − √− ∂ y ) , h − = 1 − x − y ∂ x + √− ∂ y ) . Proposition 3.10.
Each element in the following set (cid:40) h i ⊗ · · · ⊗ h i k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) i , ..., i k ∈ { + , −} , { j : i j = + } = { j : i j = −} (cid:41) extend to global sections M → E ⊗ k and is parallel. The set form a basis of Π( E ⊗ k ) . Proof.
Fix p ∈ M . Let M ( α ) ∈ SO (2 , R ) be the matrix as in Lemma 3.5, which acts on e , e ∈ T p M by M ( α ) e = cos( α ) e + sin( α ) e ,M ( α ) e = − sin( α ) e + cos( α ) e . With this action, M ( α ) acts on h + | p , h − | p in the fiber E p by M ( α ) h + | p = e iα h + | p , M ( α ) h − | p = e − iα h − | p . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 11
In other words, ( h ± ) p is an eigenvector of M ( α ) with eigenvalue e ± iα .To determine the parallel sections Π( E ⊗ k ) , it suffices to study the fixed point subspace of ( E p ) ⊗ k under the action of SO (2 , R ) / {± } . By the same computation shown in Lemma 3.5, weknow that M ( α )( h i | p ⊗ · · · ⊗ h i k | p ) = h i | p ⊗ · · · ⊗ h i k | p if and only if the number of + appearing in i , ..., i k coincides with the number of − . Then bysimple linear algebra, the set of vectors { h i | p ⊗ · · · ⊗ h i k | p : { j : i j = + } = { j : i j = −}} form a basis of the fixed point subspace (( E C p ) ⊗ k ) Hol ( E ⊗ k ) .Since M is complete, every point on M are connected to p by some geodesic path. Everyelement in this fixed point subspace E Hol ( E ⊗ k ) p thus defines a parallel section globally on M viaparallel transport along geodesic paths. Moreover, the parallel section defined by h i | p ⊗· · ·⊗ h i k | p coincides with the local section h i ⊗ · · · ⊗ h i k on V , since the latter is also defined via paralleltransports. Thus the latter extends to global sections. (cid:3) Parallel sections of the tensor algebra of the affinization.
Consider the followingaffinization of E : (cid:98) E = T M ⊗ R ( M × C [ t, t − ]) ⊕ ( M × C k )= E ⊗ ( M × C [ t, t − ]) ⊕ ( M × C k )= (cid:98) E − ⊕ (cid:98) E ⊕ (cid:98) E + , where (cid:98) E ± = E ⊗ ( M × t ± C [ t ± ])= ∞ (cid:77) k =1 E ⊗ ( M × C t ± k ) , (cid:98) E = E ⊗ ( M × C t ) ⊕ ( M × C k ) . We will need to use the tensor algebra bundle T ( ˆ E ) = ( M × C ) ⊕ ˆ E ⊕ ˆ E ⊗ ⊕ · · · . For each p ∈ M , the fiber T ( (cid:98) E ) p is simply the tensor algebra of the vector space E p : T ( (cid:98) E ) p = C ⊕ (cid:98) E p ⊕ ( (cid:98) E p ) ⊗ ⊕ · · · . Also note that for two smooth sections
X, Y ∈ Γ( T ( (cid:98) E )) , ( X ⊗ Y ) p = X p ⊗ Y p , p ∈ M (3)defines an associative algebra structure on Γ( T ( (cid:98) E )) . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 12
We construct the bundles T ( ˆ E ± ) , T ( E ) and T ( M × C k ) similarly. We will rewrite these bundlesin the following way: T ( (cid:98) E ± ) = ∞ (cid:77) n =0 n (cid:77) k =1 (cid:77) m + ··· + m k = nm ,...,m k ∈ Z + ( E ⊗ C ( M × C t ± m )) ⊗ C · · · ⊗ C ( E ⊗ C ( M × C t ± m k )) .T ( E ) = ∞ (cid:77) n =0 ( E ⊗ C t ) ⊗ n = ∞ (cid:77) n =0 E ⊗ n .T ( M × C k ) = ∞ (cid:77) n =0 ( M × C k ) ⊗ n = ∞ (cid:77) n =0 ( M × C k n ) . We will need to use the parallel sections of all these bundles. Note that essentially T ( (cid:98) E ± ) and T ( E ) are direct sums of E ⊗ n . The following lemma will then apply to determine the parallelsection of these bundles. Lemma 3.11.
Let B , B , ... be a sequence of vector bundles on M . Let B = (cid:76) ∞ i =1 B i . Then theparallel sections of B is the direct sum of the parallel sections of B i , i.e., Π( B ) = (cid:76) ∞ i =1 Π( B i ) . Proof.
Obviously (cid:76) ∞ i =1 Π( B i ) ⊆ Π( B ) . We show the inverse inclusion here. Let X be a parallelsection of B . Fix any p ∈ M and piecewise smooth loop γ based on p . Consider X p = (cid:80) i finite ( X i ) p ,which is a finite sum of components in ( B ) p , ( B ) p , ... The parallel transport T Bγ (1) applied on X amounts to the sum of the action of T B i γ (1) on X i . Since it is a direct sum, we necessarily have T B i γ (1) ( X i ) p = ( X i ) p . Since γ is arbitrarily chosen, we see that ( X i ) p ∈ ( B i ) Hol ( B ) p . That is to say, X p is a finite sum of elements in ( B i ) Hol ( B ) p . Thus X is a finite sum of parallel sections in B i . Sowe proved that Π( B ) ⊆ (cid:76) ∞ i =1 Π( B i ) . (cid:3) The lemma thus determines the parallel sections of these bundles:
Proposition 3.12. Π( T ( (cid:98) E ± )) = ∞ (cid:77) n =0 n (cid:77) k =1 (cid:77) m + ··· + m k = nm ,...,m k ∈ Z + Π(( E ⊗ R ( M × C t ± m )) ⊗ C · · · ⊗ C ( E ⊗ R ( M × C t ± m k ))= ∞ (cid:77) n =0 n (cid:77) k =1 (cid:77) m + ··· + m k = nm ,...,m k ∈ Z + span (cid:40) ( h i ⊗ t ± m ) ⊗ · · · ⊗ ( h i k ⊗ t ± m k ) : i , ..., i k ∈ { + , −} , { j : i j = + } = { j : i j = −}} (cid:41) Π( T ( E )) = ∞ (cid:77) n =0 Π(( E ⊗ ( M × C t )) ⊗ n ) . = ∞ (cid:77) n =0 span (cid:40) ( h i ⊗ t ) ⊗ · · · ⊗ ( h i n ⊗ t ) : i , ..., i n ∈ { + , −} , { j : i j = + } = { j : i j = −}} (cid:41) . Π( M × C k ) = ∞ (cid:77) n =0 Π( M × C k n ) = ∞ (cid:77) n =0 C k n . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 13
Proof.
The structure of Π( T ( (cid:98) E ± )) and Π( T ( E )) follows directly from the lemma. The structureof Π( M × C k ) follows from the fact that any smooth function f satisfying ∇ ˙ γ f = 0 for everypath γ is constant. (cid:3) Structure of the MOSVA.
With the knowledge of the parallel sections, we now explicitlyidentify the MOSVA constructed by Huang in [H2].
Theorem 3.13.
Let V ( l, ) be the MOSVA constructed by Huang in [H2] (cf. Theorem 2.5).Then the vectors h i ( − m ) · · · h i k ( − m k ) , i , ..., i k ∈ { + , −} , { j : i j = + } = { j : i j = −} form a basis for V U ( l, ) . Together with the following vertex operator action Y ( h i ( − m ) · · · h i k ( − m k ) , x )= ◦◦ m − d m − dx m − h i ( x ) · · · m k − d m k − dx m k − h i k ( x ) ◦◦ defines a MOSVA structure on V U ( l, ) . Proof.
For l = 1 the theorem has been proved in [H2], Proposition 3.3. The generalization to l ∈ C is a trivial modification of the whole process. For exposition purposes, we will sketch adirect proof using the computational results in [H1] modified by the general central charge.For convenience, we use V to denote the space V U ( l, ) . The grading of V is given by specifying V n to be the span of the vectors h i ( − m ) · · · h i k ( − m k ) , with i , ..., i k satisfying the conditionsin the statement, and m + · · · + m k = n . From Proposition 3.12, n ≥ . So the grading is lowerbounded.Let u = a ( − m ) · · · a k ( − m k ) and v = b ( − n ) · · · b k ( − n k ) , a , ..., a k , b , ..., b k ∈{ h + , h − } , m , ..., m k , n , ..., n k ∈ Z + , the coefficient of each power of x in Y ( u, x ) v is in V .In fact, the coefficients of each power of x is a sum of elements of the form ◦◦ a ( p ) · · · a k ( p k ) ◦◦ b ( − n ) · · · b k ( − n k ) . For every such p , ..., p k , let σ be the unique element in S k satisfying the condition σ (1) < · · · < σ ( α ) , σ ( α + 1) < · · · < σ ( β ) , σ ( β ) < · · · < σ ( k ) ,p σ (1) , ..., p σ ( α ) < , p ( σ α +1 ) , ..., p ( σ β ) > , p ( σ β +1 ) , ..., p ( σ k ) = 0 , so that the element can be written as a σ (1) ( p σ (1) ) · · · a σ ( α ) ( p σ ( α ) )) a σ ( α +1) ( p σ ( α +1) ) · · · a σ ( β ) ( p σ ( β ) )) · a σ ( β +1) (0) · · · a σ ( k ) (0)) b ( − n ) · · · b k ( − n k ) . From the relations of the algebra, for i, j ∈ { + , −} , n > , h i (0) commutes with all h j ( − n ) , while h i (0) = 0 . Thus if there exists some j such that p j = 0 , the element is simply zero, which iscertainly in V . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 14
Otherwise, if p , ..., p k (cid:54) = 0 , then β = k . The element in question would then be a σ (1) ( p σ (1) ) · · · a σ ( α ) ( p σ ( α ) )) a σ ( α +1) ( p σ ( α +1) ) · · · a σ ( β ) ( p σ ( β ) )) b ( − n ) · · · b k ( − n k ) . From the relation of the algebra, for i, j ∈ { + , −} , p, q > , the commutator of h i ( p ) and h j ( − n ) is lpδ p,n ( h i , h j ) . Notice that every time we swap the position of h i ( p ) and h j ( − n ) , the numberof + and − in the commutator term are both lowered by 1. So at the end of the day when all h i ( p ) with p > are positions before , the number of + and − in all the extra commutatorterms are still kept the same. This shows that the elements are all in V . Thus we proved that Y ( u, x ) v ∈ V [[ x ]] .Now we argue the weak associativity. From Corollary 4.9 in [H1], for every a , ..., a k , b , ..., b k ∈{ h + , h − } , m , ..., m k , n , ..., n k ∈ Z + , Y ( a ( − m ) · · · a k ( − m k ) , x ) Y ( b ( − n ) · · · b k ( − n k ) , x )= min { k ,k } (cid:88) i =0 (cid:88) k ≥ p > ··· >q i ≥ ≤ q < ··· ··· >q i ≥ ≤ q < ···
. Thus all such a p ( s p ) would have the priority acting v , resulting in (cid:89) p (cid:54) = p ,...,p i a p ( s p ) c ( − r ) · · · c k ( − r k ) , which is zero when (cid:88) p (cid:54) = p ,...,p i s p > r + · · · + r k . So the power of x is (cid:88) p (cid:54) = p ,...,p i ( − s p − m p ) ≥ − r − · · · − r k − (cid:88) p (cid:54) = p ,...,p i m p ≥ − r − · · · − r k − m − · · · − m k . The lower bound we obtained at the right-hand-side works for all possible choices of i and p , ..., p i . Moreover, it depends only on the element a ( − m ) · · · a k ( − m k ) and c ( − r ) · · · c k ( − r k ) .So the pole-order condition is verified.Other axioms are verified similarly as in [H1]. (cid:3) Corollary 3.14.
The graded dimension of the MOSVA V U ( l, ) is ∞ (cid:88) n =2 n − · F (cid:18) − n , − n (cid:19) q n . Proof.
It suffices to argue the formula for V U ( l, ) . Let n ∈ Z + . We compute dim V (2 n ) and dim V (2 n +1) separately.For each fixed k ∈ Z + , the set of ordered k -tuples ( m , ..., m k ) of positive numbers such that m + · · · + m k = 2 n is well-known to be (cid:0) n − k − (cid:1) . Also within i , ..., i k , the number of + must bethe same as the number of − . Thus k must be an even number. Write k = 2 p . Then the numberof possible assignments of + and − to i , ..., i p is simply (cid:0) pp (cid:1) . Summing up all possible choicesof p , we have dim V (2 n ) = n (cid:88) p =1 (cid:18) pp (cid:19)(cid:18) n − p − (cid:19) = n (cid:88) p =1 n − p !( p − n − p )! Recall that in general, F ( a, b ; c ; z ) = ∞ (cid:88) q =0 a ( a + 1) · · · ( a + q − b ( b + 1) · · · ( b + q − c ( c + 1) · · · ( c + q − z q q ! Putting in a = 1 − n, b = 3 / − n, c = 2 , z = 4 , we have F (1 − n, − n ; 2; 4) OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 16 = ∞ (cid:88) q =0 (1 − n )(1 − n + 1) · · · (1 − n + q − · ( − n )( − n + 1) · · · ( − n + q − · · · (2 + q −
1) 2 q q != n − (cid:88) q =0 ( − q ( n − n − · · · ( n − q ) · (3 − n )(3 − n + 2) · · · (3 − n + 2 q − q + 1)! 2 q q != n − (cid:88) q =0 (2 n − n − · · · (2 n − q ) · (2 n − n − · · · (2 n − q − q + 1)! q != n − (cid:88) q =0 (2 n − n − q − q + 1)! q ! = n (cid:88) p =1 (2 n − n − p )! p !( p − Thus dim V (2 n ) = 2(2 n − · F (1 − n, − n ; 2; 4) Similarly, we compute that dim V (2 n +1) = 2(2 n ) · n (cid:88) p =1 (2 n − n − p + 1)! p !( p − n ) · F ( 12 − n, − n ; 2; 4) The conclusion then follows. (cid:3)
Remark 3.15.
Note that in particular the weight-1 subspace is zero. If we understand h + ( − m ) and h − ( − m ) as the creation operators of certain physical objects (particles or strings), then thetheorem simply says that these physical objects are always created in a bulk (in pairs only if dim M = 2 ). There does not exist one-object states in the MOSVA V ( l, ) . Remark 3.16.
Note also that V ( l, ) does not distinguish manifolds with different curvatures.Indeed, for any manifold with holonomy group SO (2 , R ) , their MOSVAs are isomorphic.4. Modules generated by eigenfunctions of the Laplace-Beltrami operator
Let V U ( l, f ) be the V ( l, ) -module generated by a smooth function f : U → C in [H2] (cf.Section 2.4). In addition, we assume that the function f satisfies − ∆ f = λf, where ∆ is the Laplace-Beltrami operator on U , λ ∈ C . In this section, we study the module V U ( l, f ) . In physics, eigenfunctions correspond to quantum states. So the module V U ( l, f ) canbe understood as string-theoretic excitements of the quantum state corresponding to f . We willfirst deduce a lemma on covariant derivatives, then use it to show that every covariant derivativeof eigenfunctions is a scalar multiple of the eigenfunction. Using this lemma, we identify a basisfor V U ( l, f ) . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 17
Fundamental lemma of covariant derivatives.
In order to study the actions of Π( T ( E )) on f , we will use the following theorem: Theorem 4.1.
Let f : U → C be a C -valued smooth function. Then for n ≥ , we have ( ∇ n f )( Z , ..., Z n − , Z n ) − ( ∇ n f )( Z , ..., Z n , Z n − ) = 0 . And for i = 1 , ..., n − , ( ∇ n f )( Z , ..., Z i , Z i +1 , ..., Z n ) − ( ∇ n f )( Z , ..., Z i +1 , Z i , ..., Z n )= n (cid:88) j = i +2 ( ∇ n − f )( Z , ..., − R ( Z i , Z i +1 ) Z j , ..., Z n )=( ∇ n − f )( Z , ..., − R ( Z i , Z i +1 ) Z i +2 , Z i +3 ..., Z n )+ ( ∇ n − f )( Z , ..., Z i +2 , − R ( Z i , Z i +1 ) Z i +3 , ..., Z n )++ · · · · · · + ( ∇ n − f )( Z , ..., Z i +2 , Z i +3 , ..., − R ( Z i , Z i +1 ) Z n ) . Proof.
We prove the first equation by induction on n . For n = 3 , we have ( ∇ f )( Z , Z , Z ) = ( ∇ Z ( ∇ f ))( Z , Z )= ∇ Z (( ∇ f )( Z , Z )) − ( ∇ f )( ∇ Z Z , Z ) − ( ∇ f )( Z , ∇ Z Z ) (note that ∇ f ( X, Y ) = ∇ f ( Y, X ) ) = ∇ Z (( ∇ f )( Z , Z )) − ( ∇ f )( Z , ∇ Z Z ) − ( ∇ f )( ∇ Z Z , Z ) = ( ∇ f )( Z , Z , Z ) . Assume the equation holds for n − : ( ∇ n f )( Z , ..., Z n − , Z n ) = ( ∇ Z ( ∇ n − f ))( Z , ..., Z n − , Z n )= ∇ Z (( ∇ n − f )( Z , ..., Z n − , Z n )) − ( ∇ n − f )( ∇ Z Z , ..., Z n − , Z n ) − ( ∇ n − f )( Z , ..., ∇ Z Z n − , Z n ) − ( ∇ n − f )( Z , ..., Z n − , ∇ Z Z n ) (by induction hypothesis) = ∇ Z (( ∇ n − f )( Z , ..., Z n , Z n − )) − ( ∇ n − f )( ∇ Z Z , ..., Z n , Z n − ) − ( ∇ n − f )( Z , ..., Z n , ∇ Z Z n − ) − ( ∇ n − f )( Z , ..., ∇ Z Z n , Z n − )= ( ∇ n f )( Z , ..., Z n , Z n − ) . So the first equation is proved.For the second equation, we first consider the case i = 1 : ( ∇ n f )( Z , Z , Z , · · · , Z n ) = ( ∇ Z ( ∇ n − f ))( Z , Z , · · · , Z n )= ∇ Z (( ∇ n − f )( Z , Z ..., Z n )) − ( ∇ n − f )( ∇ Z Z , Z , ..., Z n ) − n (cid:88) j =3 ( ∇ n − f )( Z , ..., ∇ Z Z j , ..., Z n ) OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 18 = ∇ Z ∇ Z (( ∇ n − f )( Z , ..., Z n )) − n (cid:88) j =3 ∇ Z (( ∇ n − f )( Z , ..., ∇ Z Z j , ..., Z n )) (4) − ∇ ∇ Z Z (( ∇ n − f )( Z , ..., Z n )) + n (cid:88) j =3 ( ∇ n − f )( Z , ..., ∇ ∇ Z Z Z j , ..., Z n ) (5) − n (cid:88) j =3 (cid:32) ∇ Z (( ∇ n − f )( Z , ..., ∇ Z Z j , ..., Z n ) − j − (cid:88) k =3 ( ∇ n − f )( Z , ..., ∇ Z Z k , ..., ∇ Z Z j , ..., Z n ) (cid:33) (6) − n (cid:88) j =3 − ( ∇ n − f )( Z , ..., ∇ Z ∇ Z Z j , ..., Z n ) − n (cid:88) k = j +1 ( ∇ n − f )( Z , ..., ∇ Z Z j , ..., ∇ Z Z k , ..., Z n ) . (7)Similarly, ( ∇ n f )( Z , Z , Z , · · · , Z n ) = ( ∇ Z ( ∇ n − f ))( Z , Z , · · · , Z n )= ∇ Z ∇ Z (( ∇ n − f )( Z , ..., Z n )) − n (cid:88) j =3 ∇ Z (( ∇ n − f )( Z , ..., ∇ Z Z j , ..., Z n )) (8) − ∇ ∇ Z Z (( ∇ n − f )( Z , ..., Z n )) + n (cid:88) j =3 ( ∇ n − f )( Z , ..., ∇ ∇ Z Z Z j , ..., Z n ) (9) − n (cid:88) j =3 (cid:32) ∇ Z (( ∇ n − f )( Z , ..., ∇ Z Z j , ..., Z n ) − j − (cid:88) k =3 ( ∇ n − f )( Z , ..., ∇ Z Z k , ..., ∇ Z Z j , ..., Z n ) (cid:33) (10) − n (cid:88) j =3 − ( ∇ n − f )( Z , ..., ∇ Z ∇ Z Z j , ..., Z n ) − n (cid:88) k = j +1 ( ∇ n − f )( Z , ..., ∇ Z Z j , ..., ∇ Z Z k , ..., Z n ) . (11)Then in the difference, the second sum in (4) cancels out with the first term in the sum of (10);the first term in the sum of (6) cancels out with the second sum in (8); the second term in thesum of (6), together with second term in the sum of (7), cancel out those in (10) and (11). Sothe difference is ( ∇ n f )( Z , Z , Z , ..., Z n ) − ( ∇ n f )( Z , Z , Z , ..., Z n )= ( ∇ Z ∇ Z − ∇ Z ∇ Z )(( ∇ n − f )( Z , ..., Z n )) − ∇ ∇ Z Z −∇ Z Z (( ∇ n − f )( Z , ..., Z n )+ n (cid:88) j =3 ( ∇ n − f )( Z , ..., ∇ ∇ Z Z −∇ Z Z Z j , ..., Z n ) + n (cid:88) j =3 ( ∇ n − f )( Z , ..., ( ∇ Z ∇ Z − ∇ Z ∇ Z ) Z j , ..., Z n )= n (cid:88) j =3 ( ∇ n − f )( Z , ..., ( ∇ Z ∇ Z − ∇ Z ∇ Z + ∇ ∇ Z Z −∇ Z Z ) Z j , ..., Z n )= n (cid:88) j =3 ( ∇ n − f )( Z , ..., − R ( Z , Z ) Z j , ..., Z n ) . So the case i = 1 is proved for arbitrary n . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 19
We proceed by induction of i . The base case has been proved above. Now we proceed withthe inductive step. ( ∇ n f )( Z , ..., Z i , Z i +1 , ..., Z n ) = ( ∇ Z ( ∇ n f ))( Z , ..., Z i , Z i +1 , ..., Z n )= ∇ Z (( ∇ n − f )( Z , ..., Z i , Z i +1 , ..., Z n )) − i − (cid:88) k =2 ( ∇ n − f )( Z , ..., ∇ Z Z k , ..., Z i , Z i +1 , ..., Z n ) − ( ∇ n − f )( Z , ..., ∇ Z Z i , Z i +1 , ..., Z n ) − ( ∇ n − f )( Z , ..., Z i , ∇ Z Z i +1 , ..., Z n ) − n (cid:88) k = i +2 ( ∇ n − f )( Z , ..., Z i , Z i +1 , ..., ∇ Z Z k , ..., Z n ) . Similarly, ( ∇ n f )( Z , ..., Z i +1 , Z i , ..., Z n ) = ( ∇ Z ( ∇ n f ))( Z , ..., Z i +1 , Z i , ..., Z n )= ∇ Z (( ∇ n − f )( Z , ..., Z i +1 , Z i , ..., Z n )) − i − (cid:88) k =2 ( ∇ n − f )( Z , ..., ∇ Z Z k , ..., Z i +1 , Z i , ..., Z n ) − ( ∇ n − f )( Z , ..., ∇ Z Z i +1 , Z i , ..., Z n ) − ( ∇ n − f )( Z , ..., Z i +1 , ∇ Z Z i , ..., Z n ) − n (cid:88) k = i +2 ( ∇ n − f )( Z , ..., Z i +1 , Z i , ..., ∇ Z Z k , ..., Z n ) . We use the induction hypothesis to see that the difference is expressed as ∇ Z n (cid:88) j = i +2 ( ∇ n − f )( Z , ..., − R ( Z i , Z i +1 ) Z j , ..., Z n ) − n (cid:88) j = i +2 i − (cid:88) k =2 ( ∇ n − f )( Z , ..., ∇ Z Z k , ..., − R ( Z i , Z i +1 ) Z j , ..., Z n ) − n (cid:88) j = i +2 ( ∇ n − f )( Z , ..., − R ( ∇ Z Z i , Z i +1 ) Z j , ..., Z n ) − n (cid:88) j = i +2 ( ∇ n − f )( Z , ..., − R ( Z i , ∇ Z Z i +1 ) Z j , ..., Z n ) − n (cid:88) k = i +2 k − (cid:88) j = i +2 ( ∇ n − f )( Z , ..., − R ( Z i , Z i +1 ) Z j , ..., ∇ Z Z k , ..., Z n ) − n (cid:88) k = i +2 ( ∇ n − f )( Z , ..., Z i +2 , ..., − R ( Z i , Z i +1 ) ∇ Z Z k , ..., Z n ) − n (cid:88) k = i +2 n (cid:88) j = k +1 ( ∇ n − f )( Z , ..., Z i +2 , ..., ∇ Z Z k , ..., − R ( Z i , Z i +1 ) Z j , ..., Z n ) , which is equal to the right-hand-side. (cid:3) Remark 4.2.
We call Theorem 4.1 the fundamental lemma of covariant derivatives, as it is offundamental importance in this paper and has lots of important consequences.
OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 20
Zero-mode actions.
Now we use the lemma to compute Π( T ( E )) f . Recall the definitionof the sectional curvature K = ( R ( U, V ) V, U )( U, U )( V, V ) − ( U, V ) , where U, V are any vector fields. Then by our assumption in Definition 3.7 ( X , X ) = ( X , X ) = 1 , ( X , X ) = 0 . We compute directly that ( R ( X , X ) X , X ) = K, ( R ( X , X ) X , X ) = 0 , ( R ( X , X ) X , X ) = 0 , ( R ( X , X ) X , X ) = − K. In other words, R ( X , X ) X = KX , R ( X , X ) X = − KX . Therefore, R ( h + , h − ) h + = 2 Kh + , R ( h + , h − ) h − = − Kh − . Proposition 4.3.
Let f be an eigenfunction for the Laplace-Beltrami operator of eigenvalue λ .Then for every r ∈ Z + , ( ∇ r f )( h ⊗ r + ⊗ h ⊗ r − ) = r (cid:89) α =1 ( − λ + α ( α − K ) = ( − λ )( − λ + 2 K ) · · · ( − λ + r ( r − K ) , ( ∇ r f )( h ⊗ r − ⊗ h ⊗ r + ) = r (cid:89) α =1 ( − λ + α ( α − K ) = ( − λ )( − λ + 2 K ) · · · ( − λ + r ( r − K ) . Proof.
We apply induction. If r = 1 , then ( ∇ f )( h + ⊗ h − ) = ( ∇ f )(( X − √− X ) ⊗ ( X + √− X ))= ( ∇ f )( X ⊗ X + X ⊗ X ) + √− ∇ f )( X ⊗ X − X ⊗ X ) . From Theorem 4.1, the second term is zero. The first term is simply ∆ f . Thus we have ( ∇ f )( h + ⊗ h − ) = − λf. So the base case is proved.Assume the conclusion holds for all smaller r . We use Theorem 4.1 to shift the h + and h − inthe middle position ∇ r ( h ⊗ r + ⊗ h ⊗ r − ) = ( ∇ r f )( h ⊗ ( r − ⊗ h + ⊗ h − ⊗ h ⊗ ( r − − )= ( ∇ r f )( h ⊗ ( r − ⊗ h − ⊗ h + ⊗ h ⊗ ( r − − ) − r − (cid:88) p =0 ( ∇ r − f )( h ⊗ ( r − ⊗ h ⊗ p − ⊗ R ( h + , h − ) h − ⊗ h ⊗ ( r − − p ) − )= ( ∇ r f )( h ⊗ ( r − ⊗ h − ⊗ h + ⊗ h ⊗ ( r − − ) + 2 K r − (cid:88) p =0 ( ∇ r − f )( h ⊗ ( r − ⊗ h ⊗ ( r − − ) OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 21 = ( ∇ r f )( h ⊗ ( r − ⊗ h − ⊗ h + ⊗ h ⊗ ( r − − ) + 2 K ( r − ∇ r − ) f ( h ⊗ ( r − ⊗ h ⊗ ( r − − ) . In other words, the price for passing h + from r -th position to ( r + 1) -th position is K ( r − ∇ r − f ( h ⊗ ( r − ⊗ h ⊗ ( r − − ) . Similarly, the price of passing h + from ( r + 1) -th position to ( r + 2) -th position is K ( r − ∇ r − f ( h ⊗ ( r − ⊗ h ⊗ ( r − − ) . We continue the process until h + arrives at (2 r − -position and sum up the total price, to seethat ( ∇ r f )( h ⊗ r + ⊗ h ⊗ r − ) = ( ∇ r f )( h ⊗ ( r − ⊗ h ⊗ r − − ⊗ h + ⊗ h − )+ 2 K (( r −
1) + ( r −
2) + · · · + 1)( ∇ r − f )( h ⊗ ( r − ⊗ h ⊗ ( r − − ) . (12)Recall Theorem 4.1 in [H2]: if X and Y are two parallel tensors of degree m and n , then ( ∇ m + n f )( X, Y ) = ( ∇ n ( ∇ m f ( Y )))( X ) . Thus with the conclusion of the base case, the first term on the right-hand-side of (12) is simply ∇ r − [( ∇ f )( h + ⊗ h − )]( h ⊗ ( r − ⊗ h ⊗ ( r − − ) = − λ ( ∇ r − f )( h ⊗ ( r − ⊗ h ⊗ ( r − − ) . Computing the second term and combine it back to (12), we see that ( ∇ r f )( h ⊗ r + ⊗ h ⊗ r − ) = ( − λ + Kr ( r − ∇ r − f )( h ⊗ ( r − ⊗ h ⊗ ( r − − ) . (13)The first conclusion then follows from induction hypothesis.The second conclusion follows from an almost identical argument. We shall not repeat thedetails here. (cid:3) Proposition 4.4.
Let f be an eigenfunction for the Laplace-Beltrami operator of eigenvalue λ .Fix any i , ..., i r ∈ { + , −} such that { j : i j = + } = { j : i j = −} = r .(1) There exists a single-variable polynomial P , such that ( ∇ r f )( h i ⊗ · · · ⊗ h i r ) = P ( λ ) f. (2) The roots of P ( λ ) are contained in { p ( p − K, p = 1 , ..., r } . (14)(3) If i r − t +1 = · · · = i r , then for every p ≤ t , p ( p − K are roots of P ( λ ) . Proof.
We argue by induction on r . For r = 1 , there are only two cases of ( i , i ) : (+ , − ) and ( − , +) . P ( λ ) = − λ works for both cases. (2) and (3) are obvious.Assume that all conclusions hold for strictly smaller r . Without loss of generality, we assumethat i r = i r − = · · · = i r − t +1 = − , i r − t = + for some t ∈ [1 , r ] . In other words, there are t OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 22 consecutive h − at the rear, precede by a h + . Theorem 4.1 allows us to move the h + from the (2 r − t ) -th position to the (2 r − -position. By a computation similar to that for (13), we have ( ∇ r f )( h i ⊗ · · · ⊗ h i r − t − ⊗ h + ⊗ h − ⊗ h − ⊗ · · · ⊗ h − )= ( − λ + Kt ( t − ∇ r − f )(( h i ⊗ · · · ⊗ h i r − t − ⊗ (cid:99) h + ⊗ (cid:99) h − ⊗ h − ⊗ · · · ⊗ h − ) . Here the hat notation is introduced to show the removed terms. By the induction hypothesis,there exists a polynomial Q ( λ ) satisfying (1), (2) and (3). So ( ∇ r f )( h i ⊗ · · · ⊗ h i r − t − ⊗ h + ⊗ h − ⊗ h − ⊗ · · · ⊗ h − ) = ( − λ + Kt ( t − Q ( λ ) f. Therefore (1) holds with P ( λ ) = ( − λ + Kt ( t − Q ( λ ) . The roots of P ( λ ) , by inductionhypothesis, are contained in { p ( p − K : p = 1 , ..., r − } ∪ { t ( t − K } . Since t ∈ [1 , r ] , (2)holds for P ( λ ) . For j = i , ..., j r − t = i r − t , j r − t +1 = · · · = j r − = − , the induction hypothesisshows that for every α = 1 , ..., t − , α ( α − K are roots for Q ( λ ) . Therefore (3) holds, withthe additional root t ( t − K for P ( λ ) . (cid:3) Since Π( T ( E )) is spanned by h i ⊗ · · · ⊗ h i r in Proposition 4.4, we have thus proved thefollowing theorem: Theorem 4.5.
Let f be an eigenfunction of the Laplace-Beltrami operator. Then as a vectorspace, Π( T ( E )) f = C f. In other words, the action of every parallel tensor on an eigenfunction f generates only scalarmultiples of f . Remark 4.6.
In case f is an eigenfunction with real eigenvalue, then the same argument aboveshows that Π( T ( T M )) f ∈ R f . Remark 4.7.
While eigenfunctions defined globally on a manifold are known to have real andnonpositive eigenvalues, we would like to note that eigenfunctions with imaginary eigenvalues doexist locally. Indeed, let M be the two-dimensional unit sphere with coordinates as in Example3.8. Then the Laplace-Beltrami operator is represented by ∆ = ∂ ∂φ + cot φ ∂∂φ + 1sin φ ∂ ∂θ . Consider a function f ( φ, θ ) = u ( φ ) + √− v ( φ ) , where u, v , together with their derivative u (cid:48) , v (cid:48) ,satisfies following linear system of ODE ddφ uu (cid:48) vv (cid:48) = a − cot φ − b
00 0 0 1 b − cot φ a uu (cid:48) vv (cid:48) , with some initial conditions specified φ = π/ . Since all the coefficients are smooth near π/ , thesolution exists smoothly in ( π/ − (cid:15), π/ (cid:15) ) for some (cid:15) > . Thus f is a smooth complex-valued OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 23 function defined in the open subset { ( φ, θ ) : π/ − (cid:15) < φ < π/ (cid:15), ≤ θ < π } . It is routine tocheck that ∆ f = ( a + √− b ) f . Remark 4.8.
Theorem 4.5 can be generalized to higher dimensional orientable and non-orientablespace forms. See [Q2].
Remark 4.9.
We call an eigenvalue λ specific if λ = α ( α − for some α ∈ Z + , generic if other-wise. Proposition 4.4 in particular shows if f is an global eigenfunction over the space form withnegative section curvature, then the action of individual parallel tensors h i ⊗ · · · ⊗ h i r are non-vanishing. On the other hand, it is well known that the eigenvalues of global eigenfunctions overthe two-dimensional unit sphere are all specific. The difference of generic eigenvalues and specificeigenvalues will also be reflected on the irreducible V -modules generated by eigenfunctions, aswill be seen later in the paper.4.3. Structure of the module.
Having determined the action of Π( T ( E )) on f , we now ex-plicitly identify the module for the MOSVA constructed by Huang in [H2] (cf. Section 2.4. Thefollowing theorem by Dong, Li, and Mason will be needed in the discussion: Lemma 4.10 ([LL], Proposition 4.5.6, [DLM]) . Let W be a V -module and let T be a subset of W . Then the submodule generated by T is spanned by { v n w : v ∈ V, n ∈ Z , w ∈ T } . Though the lemma was formulated for vertex algebras, the proof uses only weak associativityand thus applies to modules for MOSVAs (see [LL], Proposition 4.5.7).Now we state a general theorem regarding the spanning set of a module generated by a lowestweight element.
Theorem 4.11.
Let W be a module for V ( l, ) generated by an element w of lowest weight µ ,i.e., for every n ∈ Z + , W [ µ − n ] = 0 . Then W has the following spanning set (cid:88) i X (1) i ( − t ) · · · X ( k ) i ( − t k ) X ( k +1) i (0) · · · X ( r ) i (0) w : r ≥ , k = 1 , ..., r, t , .., t k > (cid:88) i X (1) i ⊗ · · · ⊗ X ( r ) i ∈ Π( E ⊗ r ) . (15)Here the ⊗ symbol is omitted for convenience. Proof.
Consider now the action of (cid:80) i X (1) i ( − m ) · · · X ( r ) i ( − m r ) . We see that Y (cid:32)(cid:88) i X (1) i ( − m ) · · · X ( r ) i ( − m r ) , x (cid:33) w = (cid:88) n ,...,n r ∈ Z r (cid:89) j =1 ( − n j − · · · ( − n j − m j + 1)( m j − (cid:32) ◦◦ (cid:88) i X (1) i ( n ) · · · X ( r ) i ( n r ) ◦◦ w (cid:33) x (cid:80) rj =1 ( − n j − m j ) . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 24
Since w is of lowest weight in W , if one of the n , ..., n r is positive, then after taking the normalordering, the coefficient is zero. So it suffices to focus on nonpostive n , ..., n r . Then W isspanned by the following elements (cid:88) n + ··· + n r = − p,n ,...,n r ≤ r (cid:89) j =1 ( − n j − · · · ( − n j − m j + 1)( m j − (cid:32) ◦◦ (cid:88) i X (1) i ( n ) · · · X ( r ) i ( n r ) ◦◦ w (cid:33) (16)for all r ≥ , p ≥ , m , ..., m r > , and all (cid:80) i X (1) i ⊗ · · · ⊗ X ( r ) i ∈ Π( E ⊗ r ) .For each fixed n , ..., n r ≤ , ◦◦ (cid:88) i X (1) i ( n ) · · · X ( r ) i ( n r ) ◦◦ f = (cid:88) i X ( σ (1)) i ( n σ (1) ) · · · X ( σ ( k )) i ) ( n σ ( k ) ) X ( σ ( k +1)) i (0) · · · X ( σ ( r )) i (0) f, where σ is the unique permutation such that σ (1) < · · · < σ ( k ) , σ ( k + 1) < · · · < σ ( r ); n σ (1) , ..., n σ ( k ) < , n σ ( k +1) = · · · = n σ ( r ) = 0 . Note that (cid:80) i X ( σ (1)) i ⊗ · · · ⊗ X ( σ ( r )) i stays in Π( E ⊗ r ) , thus the summand of (16) for each fixed n , ..., n r is a vector in (15). Therefore, (16) is a sum of vectors in (15). So W is a subset of thelinear span of (15).Now we argue that every vector in (15) is in W . First we notice that for each j = 1 , ..., r ( − n j − · · · ( − n j − m j + 1)( m j − (cid:54) = 0 ⇒ n j = 0 or n j ≤ − m j . This allows us to represent each vector in (15) as a linear combination of elements of the form(16) by choosing m , ..., m r and p appropriately. We show this by induction on k . In case k = 1 ,we pick m = t , m = · · · = m r = t + 1 and p = t . Then every nonzero summand in (16) isgiven by ( n , ..., n r ) such that n + · · · + n r = − t ,n = 0 or n ≤ − t ,n i = 0 or n i ≤ − ( t + 1) , i = 2 , ..., r. The only solution is ( n , ..., n r ) = ( − t , , ..., . Thus (cid:80) i X (1) i ( − t ) X (2) i (0) · · · X ( r ) i (0) f is anelement of W . The base case is proved.Now assume every element in (15) of smaller k is represented by a linear combination ofelements of the form (16). We pick m = t , ..., m k = t k , m k +1 = · · · = m r = t + · · · + t k + 1 and p = t + · · · + t k . Then every nonzero summand in (16) is given by ( n , ..., n r ) such that n + · · · + n r = − ( t + · · · + t k ) ,n i = 0 or n i ≤ − t i , i = 1 , ..., k,n i = 0 or n i ≤ − ( t + · · · + t k + 1) , i = k + 1 , ..., r. Necessarily, n k +1 = · · · = n r = 0 . For n , ..., n k , aside from the choice n = − t , ..., n k = − t k , allother choices would involve some zeros. By induction hypothesis, the summands in (16) given OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 25 by these extra choices are all contained in W . Subtracting (16) by these summands, we obtain (cid:80) i X (1) i ( − t ) · · · X ( k ) i ( − t k ) X ( k +1) i (0) · · · X ( r ) i (0) , which is then in W . (cid:3) Corollary 4.12.
Let f : U → C be an eigenfunction for the Laplace-Beltrami operator overan open subset U of M . Let V U ( l, f ) be the V ( l, ) -module constructed by Huang in [H2] (cf.Section 2.4) with wt f = 0 . Then the following set h i ( − t ) · · · h i k ( − t k ) h i k +1 (0) · · · h i r (0) f : r ≥ , k = 1 , ..., r, t , .., t k > { j ∈ [1 , r ] : i j = + } = { j ∈ [1 , r ] : i j = −} ; ∀ p = k + 1 , ..., r, { j ∈ [ p, r ] : i j = + } (cid:54) = { j ∈ [ p, r ] : i j = −} . (17)forms a basis for V U ( l, f ) . Proof.
With the same argument above, we see that elements in (17) without the third and fourthrequirements forms a spanning set of V U ( l, f ) . The third and fourth requirements are introducedto exclude the relation brought by Π( T ( E )) f = C f . It is clear that vectors in (17) are linearlyindependent in T ( (cid:99) E p − ) ⊗ T ( E p ) ⊗ Π( T ( E )) C ∞ ( U ) . (cid:3) Remark 4.13.
Obviously, V U ( l, f ) is not grading-restricted, as one can insert arbitrarily manyparallel sections between the k -th and ( k + 1) -th positions. The resulted set of new tensors areall linearly independent to each other in T ( (cid:98) E − ) p ⊗ T ( E ) p ⊗ Π( T ( E )) C ∞ ( U ) . Nevertheless, theirreducible quotients of V U ( l, f ) are grading-restricted, as shown later.4.4. Isomorphic relations.
We know that two orientable space forms with different sectionalcurvatures cannot be distinguished by the MOSVAs. Now we show that they can be distinguishedby the modules.
Theorem 4.14.
Let M , M be orientable space forms of the same dimension. Let K and K be their sectional curvatures. Let U and U be open subsets of M and M , f : U → C , f : U → C be eigenfunctions for the Beltrami-Laplace operator of eigenvalues λ and λ . Then(1) C f and C f are isomorphic as Π( T ( E )) -modules if and only if λ = λ , K = K .(2) V U ( l, f ) and V U ( l, f ) are isomorphic as V ( l, ) -modules if and only if λ = λ , K = K . Proof.
For Part (1), without loss of generality, let
T f = f . We argue that T is an Π( T ( E )) -isomorphism if and only if λ = λ , K = K .The only if part can be seen by the action of h + ⊗ h + ⊗ h − ⊗ h − and h + ⊗ h + ⊗ h − ⊗ h − .Recall that for any eigenfunction f of eigenvalue λ over an open subset of an orientable spaceform with sectional curvature K ( h + ⊗ h − ) f = ( ∇ f )( h + ⊗ h − ) = − λf OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 26 ( h + ⊗ h + ⊗ h − ⊗ h − ) f = ( ∇ f )( h + ⊗ h + ⊗ h − ⊗ h − ) = − λ ( − λ + 2 K ) f Thus T (( h + ⊗ h − ) f ) = ( h + ⊗ h − ) T ( f ) ⇒ λ f = λ f ⇒ λ = λ T (( h + ⊗ h + ⊗ h − ⊗ h − ) f ) = ( h + ⊗ h + ⊗ h − ⊗ h − ) T ( f ) ⇒ λ − λ K (dim M −
1) = λ − λ K (dim M − ⇒ K = K . The if part follows from an argument similar to the process of Proposition 4.3 and Proposition4.4, showing that T (( h i ⊗ · · · ⊗ h i r ) f ) = ( h i ⊗ · · · ⊗ h i r ) T ( f ) for every i , ..., i r ∈ { + , −} with { j : i j = + } = { j : i j = −} . We shall not repeat here.For Part (2), if λ = λ and K = K , then C f is isomorphic to C f as Π( T ( E )) -modules.We then see that the induced module T ( E ) p ⊗ Π( T ( E )) f is isomorphic to the induced mod-ule T ( E p ) ⊗ Π( T ( E )) f as N p ( E ) -modules. From the construction of the modules V U ( l, f ) and V U ( l, f ) in Section 2.4 and [H2], they are isomorphic as V ( l, ) -modules.Conversely, if V U ( l, f ) is isomorphic to V U ( l, f ) , we can similarly consider the actions of h + ( − h + ( − and h + ( − h + ( − h − ( − h − ( − . The zero-modes of these action are pre-cisely h + (0) ⊗ h − (0) and h + (0) ⊗ h + (0) ⊗ h − (0) ⊗ h − (0) discussed in Part (1). The same discussiongives λ = λ and K = K . (cid:3) Irreducible modules generated by eigenfunctions
In this section, we study quotients of V U ( l, f ) , where U is an open subset of an orientablespace form, f : U → C is an eigenfunction for the Beltrami-Laplace operator of eigenvalue λ .Since V U ( l, f ) is generated by the unique (up to a scalar multiple) lowest weight element f ,any submodule is proper if and only if it does not contain f . Since the homogeneous subspaceof weight λ is of 1-dimensional, the sum of two proper submodules does not contain f and thusstays as a proper submodule. Thus we conclude the following proposition: Proposition 5.1.
There exists a unique maximal proper submodule of V U ( l, f ) . Thus V U ( l, f ) has a unique irreducible quotient. This irreducible quotient is called the irreducible modulegenerated by f .5.1. Lowest weight projection formula.
The main tool to locate the irreducible quotient isan explicit formula of the projection Y ( v, x ) w to the lowest weight space of V U ( l, f ) for everyhomogeneous v ∈ V ( l, and w ∈ V U ( l, f ) . In other words, the formula gives the coefficient of x − n − in Y ( v, x ) where v n w is of the same weight of f . This coefficient will be called the lowestweight projection of Y ( v, x ) w . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 27
Notation 5.2.
To avoid using quadruple subscripts, in this section, we use | + (cid:105) to denote thevector field h + , |−(cid:105) to denote the vector field h − . Let i , ..., i s ∈ { + , −} . We denote each h i j by | i j (cid:105) . For m , ..., m s ∈ Z , the operator h i ( m ) · · · h i s ( m s ) will be denoted by | i j ( m ) · · · i s ( m s ) (cid:105) ,or | i ( m ) · · · i k ( m k ) (cid:105) · | i k +1 ( m k +1 ) · · · i s ( m s ) (cid:105) , or | i ( m ) · · · i k ( m k ) (cid:105)| i k +1 ( m k +1 ) · · · i s ( m s ) (cid:105) , forevery k = 1 , ..., s . In case m = · · · = m s = 0 , we may also use | i · · · i r (cid:105) to denote theoperator h i (0) · · · h i r (0) , omitting the zero indicator. But for some situations we will still keep | i (0) · · · i r (0) (cid:105) without the abbreviation. The inner product ( h i j , h k s ) will simply be denoted as (cid:104) i j , k s (cid:105) . It follows from a direct computation that (cid:104) + , + (cid:105) = ( h + , h + ) = 0 , (cid:104)− , −(cid:105) = ( h − , h − ) = 0 (cid:104) + , −(cid:105) = ( h + , h − ) = 2 , (cid:104)− , + (cid:105) = ( h − , h + ) = 2 . Proposition 5.3.
Let m , ..., , m s , t , ..., t k ∈ Z + . Then for v = | i ( − m ) · · · i s ( − m s ) (cid:105) ,w = | j ( − t ) · · · j k ( − t k ) j k +1 (0) · · · j r (0) (cid:105) f. Then for n = m + · · · + m s + t + · · · + t k − ,v n w is the lowest weight component in Y ( v, x ) w . If s < k , then v n w = 0 ; if s ≥ k , then v n w isequal to (cid:88) ≤ c k < ··· Since wt v = m + · · · m s , wt w = t + · · · + t k , we know that wt v n w = wt f if and onlyif m + · · · + m s + t + · · · + t k − n − . Thus n = m + · · · + m s + t + · · · + t k − . By definition, Y ( v, x ) w = (cid:88) n , ··· n s ∈ Z s (cid:89) p =1 ( − n p − · · · ( − n p − m p + 1)( m p − ◦◦ | i ( n ) · · · i s ( n s ) (cid:105) ◦◦ wx − n − m −···− n s − m s . Thus for n specified above, the coefficient x − n − in the series can be simplified as (cid:88) n + ··· + n s = t + ··· + t k n ,...,n s ∈ Z + ∪{ } s (cid:89) p =1 ( − n p − · · · ( − n p − m p + 1)( m p − ◦◦ | i ( n ) · · · i s ( n s ) (cid:105) ◦◦ w. (19)Here n , ..., n s ∈ Z + ∪ { } because any occurrence of negative number will result in a zerosummand. The normal ordering originally pushes all the zero-modes to the right. However, OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 28 since the zero-modes commute with positive-modes, we can instead have all the positive-modesfirst act on w , then compose with the zero-modes.Now for any n > and any i ∈ { + , −} , we study the action of i ( n ) on w . Using the fact that | i ( n ) j ( − t ) (cid:105) = | j ( − t ) i ( n ) (cid:105) + tδ nt l (cid:104) i, j (cid:105) , we see that | i ( n ) w (cid:105) = | i ( n ) j ( − t ) · · · j k ( − t k ) j k +1 (0) · · · j r (0) (cid:105) f = k (cid:88) q =1 t q δ n,t q (cid:104) i, j q (cid:105) (cid:12)(cid:12)(cid:12) j ( − t ) · · · (cid:92) j q ( − t q ) · · · j k ( − t k ) j k +1 (0) · · · j r (0) (cid:69) f. In particular, the action is nonzero only when n coincides with one of the t q ’s. For all otherchoices of n , we simply get zero.Based on the above observations, we see that (19) is nonzero only when s ≥ k . To evaluate(19), we first choose k elements c > ... > c k in { , ..., s } and specify i c ( n c ) , ..., i c k ( n c k ) aspositive-modes. Then (19) becomes (cid:88) ≤ c k < ··· Remark 5.4. Since for every t > , ( − t − · · · ( − t − m + 1)( m − − m − (cid:18) t + m − t (cid:19) , formula (18) can also be written as (cid:88) ≤ c k < ··· We first locate a submodule large enough that the quotient by which isgrading-restricted. Proposition 5.5. For every j , ..., j r , there exists a constant C = C j ,...,j r ( λ ) depending onlyon j , ..., j r and the eigenvalue λ , such that for every i , ..., i s ∈ { + , −} satisfying (cid:80) sα =1 i α + (cid:80) rβ =1 j β = 0 , | i · · · i s j · · · j r (cid:105) f − C | i · · · i s (cid:105) (cid:12)(cid:12) j (cid:48) · · · j (cid:48) N (cid:11) f = 0 , (22)here N = (cid:12)(cid:12)(cid:12)(cid:16)(cid:80) rβ =1 j β (cid:17)(cid:12)(cid:12)(cid:12) , j (cid:48) = · · · = j (cid:48) N = sgn (cid:16)(cid:80) rβ =1 j β (cid:17) . Proof. Without loss of generality, assume that (cid:80) rβ =1 j β < . Let P be the number of + ’s in j , ..., j r . We perform induction on P . If P = 0 , then N = r , and j k +1 = · · · = j r = − . Wesimply take C = 1 and j (cid:48) = · · · = j (cid:48) N = − . (22) holds.Assume the conclusion holds when the number of +’s is strictly less than P . In case j r = − ,we further assume that j r = · · · = j γ +1 = − , j γ = + for some γ ∈ [1 , r ] . Using the samecomputation as in Proposition 5.5, we see that | i · · · i s j · · · j r (cid:105) f = | i · · · i s j · · · j γ − + − · · · −(cid:105) f = | i · · · i s j k +1 · · · j t − − · · · −(cid:105)| + −(cid:105) f + K ( r − γ − r − γ ) (cid:12)(cid:12) i · · · i s j · · · j γ − (cid:100) + − − · · · − (cid:11) = ( − λ + K ( r − γ − r − γ )) (cid:12)(cid:12) i · · · i s j · · · j γ − (cid:100) + − − · · · − (cid:11) . Here we use the hat notation to indicated the removed terms. Since (cid:12)(cid:12) i · · · i s j · · · j γ − (cid:100) + − − · · · − (cid:11) contains one less + and one less − compared to the original | i · · · i s j · · · j γ − + − − · · · −(cid:105) , theinduction hypothesis gives a constant C (0) = C (0) j ... (cid:98) j γ (cid:100) j γ +1 ...j r ( λ ) depending only on j , ..., j γ − , j γ +2 , ..., j r ,such that (cid:12)(cid:12) i · · · i s j · · · j γ − (cid:100) + − − · · · − (cid:11) = C (0) | i · · · i s (cid:105) (cid:12)(cid:12) j (cid:48) · · · j (cid:48) N (cid:11) , with j (cid:48) = · · · = j (cid:48) N = − . Thus (22) holds with C = ( − λ + K ( r − t − r − t )) C (0) j ··· j t − j t +2 ...j r ( λ ) .In case j r = + , there exists some γ such that (cid:80) rβ = γ j β = 0 . Then | i · · · i s j · · · j r (cid:105) = | i · · · i s j · · · j γ − (cid:105)| j γ · · · j r (cid:105) f = C (1) | i · · · i s j · · · j γ − (cid:105) f for some constant C (1) = C (1) j γ ··· jr ( λ ) depending only on j γ , ..., j r and λ . By induction hypothesis, | i · · · i s j · · · j γ − (cid:105) f = C (2) | i · · · i s j (cid:48) · · · j (cid:48) N (cid:105) f for some C (2) = C (2) j γ ··· jr ( λ ) depending only on j , ..., j γ − and λ . Thus (22) holds with C ( λ ) = C (1) = C (1) j γ ··· jr ( λ ) C (2) j γ ··· jr ( λ ) . (cid:3) Remark 5.6. In natural language, we will describe conclusion of Proposition 5.5 as expressing | i · · · i s j · · · j r (cid:105) f uniformly as C | i · · · i s j (cid:48) · · · j (cid:48) N (cid:105) with respect to i , ..., i s . This turns out to beconvenient when we study the general criterion for an irreducible V ( l, ) -module to be grading-restricted later in this paper. OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 30 Theorem 5.7. Let j , ..., j r ∈ { + , −} satisfying (cid:80) rp =1 j p = 0 , t , ..., t k ∈ Z + . Set N = (cid:12)(cid:12)(cid:12)(cid:80) rp = k +1 j p (cid:12)(cid:12)(cid:12) . Then there exists a constant C = C j k +1 ,...,j r ( λ ) depending only on j k +1 , · · · j r and the eigenvalue λ , such that | j ( − t ) · · · j k ( − t k ) j k +1 (0) · · · j r (0) (cid:105) f − C ( λ ) (cid:12)(cid:12) j ( − t ) · · · j k ( − t k ) j (cid:48) (0) · · · j (cid:48) N (0) (cid:11) f (23)generates a proper submodule. Here j (cid:48) = · · · = j (cid:48) N = sgn (cid:16)(cid:80) rp = k +1 j p (cid:17) . Proof. Without loss of generality, assume (cid:80) rp = k +1 j p < . This means among j k +1 , ..., j r , thereare more − ’s than + ’s, with N being the difference. Now consider the lowest weight projectionof Y ( v, x ) w , with v = | i ( − m ) · · · i s ( − m s ) (cid:105) , w = | j ( − t ) · · · j k ( − t k ) j k +1 (0) · · · j r (0) (cid:105) f . FromProposition 5.3, the projection is a sum of elements of the form (cid:89) ≤ p ≤ sp (cid:54) = c ,...,c k ( − m p − · k (cid:89) p =1 ( − t σ ( p ) − · · · ( − t σ ( p ) − m c p + 1)( m c p − · k (cid:89) p =1 t σ ( p ) l (cid:104) i c p , j σ ( p ) (cid:105)· (cid:12)(cid:12)(cid:12) i (0) · · · (cid:92) i c k (0) · · · (cid:92) i c (0) · · · i s (0) j k +1 (0) · · · j r (0) (cid:69) f From Proposition 5.5, we see that for every fixed c , ..., c k and σ ∈ S k , (cid:89) ≤ p ≤ sp (cid:54) = c ,...,c k ( − m p − · k (cid:89) p =1 ( − t σ ( p ) − · · · ( − t σ ( p ) − m c p + 1)( m c p − · k (cid:89) p =1 t σ ( p ) l (cid:104) i c p , j σ ( p ) (cid:105)· (cid:0)(cid:12)(cid:12)(cid:12) i (0) · · · (cid:92) i c k (0) · · · (cid:92) i c (0) · · · i s (0) j k +1 (0) · · · j r (0) (cid:69) f − (cid:12)(cid:12)(cid:12) i (0) · · · (cid:92) i c k (0) · · · (cid:92) i c (0) · · · i s (0) (cid:69)(cid:12)(cid:12) j (cid:48) (0) · · · j (cid:48) N (0) (cid:11) f (cid:1) = 0 . with respect to any choice of i , ..., i s and m , ..., m s . Thus for every homogeneous v ∈ V and w given by (23), the lowest weight projection of Y ( v, x ) w does not contain f . Thus (23) generatesa proper submodule. (cid:3) Remark 5.8. From the proof, it is clear that the constant C is a polynomial in λ whose rootsdepend on the positioning of + and − in j k +1 , ..., j r and are contained in the set (14). So theconstant C is nonzero for generic eigenvalues λ . On the other hand, if the eigenvalue λ happens tomake C = 0 , then the element | j ( − t ) · · · j k ( − t k ) j k +1 (0) · · · j r (0) (cid:105) f itself generates a submodulethat can be quotiented out. Theorem 5.9. Let V (1) U ( l, f ) be the quotient of V U ( l, f ) by the submodule generated by elementsof the form (23).(1) The set h i ( − t ) · · · h i k ( − t k ) h i k +1 (0) · · · h i r (0) f : r ≥ , t , ..., t k > { j : i j = + } = { j : i j = −} i k +1 = · · · = i r ∈ { + , −} forms a basis for V (1) U ( l, f ) . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 31 (2) V (1) U ( l, f ) is grading-restricted. The graded dimension of V (1) U ( l, f ) is ∞ (cid:88) n =1 · n − q n . Proof. Let C be the constant in (23). In case C (cid:54) = 0 , the relation (23) allows us to identify h i ( − t ) · · · h i k ( − t k ) h i k +1 (0) f · · · h i r (0) in (17) with either h i ( − t ) · · · h i k ( − t k ) h + (0) · · · h + (0) f or h i ( − t ) · · · h i k ( − t k ) h − (0) · · · h − (0) f . The former happens when (cid:80) rp = k +1 i p > , with thenumber of + ’s being N = (cid:12)(cid:12)(cid:12)(cid:80) rp = k +1 i p (cid:12)(cid:12)(cid:12) . The latter happens when (cid:80) rp = k +1 i p < , with thenumber of − ’s being N as well. In case C = 0 , h i ( − t ) · · · h i k ( − t k ) h i k +1 (0) f · · · h i r (0) is simplyidentified with zero and will also be quotiented out. In both cases, the first conclusion follows.To see V (1) U ( l, f ) is grading-restricted, we fix n > and consider the homogeneous subspaceof weight λ + n . Then t + · · · + t k = n . Thus k is at most n . Also, i + · · · + i k is bounded by [ − k, k ] . Since i k +1 + · · · + i r = − ( i + · · · + i k ) , r and i k +1 = · · · = i r , r is bounded above by k .Thus for fixed n , there are only finitely many choices for k, t , ..., t k , r, i , ..., i r .To determine exactly how many choices, we first fix k > . Then the choices for t , ..., t k is (cid:0) n − k − (cid:1) . We can choose i , ..., i k freely. But once i , ..., i k are fixed, the choice for r and i k +1 , ..., i r is unique. Thus the total number of choices is n (cid:88) k =1 (cid:18) n − k − (cid:19) k = 2 · n − (cid:88) k =0 (cid:18) n − k (cid:19) k = 2 · n − . The second conclusion then follows. (cid:3) Remark 5.10. One can also assign the weight of f to be other numbers. A natural choice forwt f is the eigenvalue λ of f . In this case, the graded dimension will then be shifted by a factor q λ . Since we are studying the modules generated by one eigenfunction at this moment, we willstick to the choice that wt f = 0 for the module V U ( l, f ) for convenience. Remark 5.11. A more conceptual way to obtain V (1) U ( l, f ) is to start from the T ( E ) -module ( T ( E ) ⊗ Π( T ( E )) C f ) /N , where N is the T ( E ) -submodule generated by ( X ⊗ Y ⊗ Z ⊗ · · · ⊗ Z n ) f − ( Y ⊗ X ⊗ Z ⊗ · · · ⊗ Z n ) f + n (cid:88) i =1 ( Z ⊗ · · · ⊗ R ( X, Y ) Z i ⊗ · · · ⊗ Z n ) f (24)and form the induced module T ( (cid:99) E p − ) ⊗ (( T ( E ) ⊗ Π( T ( E )) C f ) /N ) . The V ( l, ) -submodule gener-ated by ⊗ (1 ⊗ f ) is indeed isomorphic to V (1) U ( l, f ) . Conceptually, the submodule generated by f amounts to requires nonparallel tensors in T ( E p ) to satisfy similar relations as the covariantderiviatves on C ∞ ( U ) . But we are not going to actually act these nonparallel tensors on C ∞ ( U ) .5.3. Second quotient. OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 32 Theorem 5.12. Let j , ..., j r ∈ { + , −} satisfy (cid:80) rp =1 j p = 0 . Let t , ..., t k ∈ Z + . Then for every τ ∈ S k , | j ( − t ) · · · j k ( − t k ) j k +1 (0) · · · j r (0) (cid:105) f − (cid:12)(cid:12) j τ (1) ( − t τ (1) ) · · · j τ ( k ) ( − t τ ( k ) ) j k +1 (0) · · · j r (0) (cid:11) f (25)generates a proper submodule. Proof. We apply Proposition 5.3 to w = (cid:12)(cid:12) j τ (1) ( − t τ (1) ) · · · j τ ( k ) ( − t τ ( k ) ) j k +1 (0) · · · j r (0) (cid:11) f . Then(18) becomes (cid:88) ≤ c k < ··· Let V (2) U ( l, f ) be the quotient of V (1) U ( l, f ) by the submodule generated byelements of the form (25). Then the union of the following three sets (cid:40) h + ( − t ) · · · h + ( − t r/ ) h − ( − t r/ ) · · · h − ( − t r ) f : r ≥ even, t ≥ · · · ≥ t r/ ≥ ,t r/ ≥ · · · ≥ t r ≥ (cid:41) , h + ( − t ) · · · h + ( − t r/ ) h − ( − t r/ ) · · · h − ( − t k ) h − (0) · · · h − (0) f : r ≥ even, r/ ≤ k ≤ r − t ≥ · · · ≥ t r/ ≥ ,t r/ ≥ · · · ≥ t k ≥ , h − ( − t ) · · · h − ( − t r/ ) h + ( − t r/ ) · · · h + ( − t k ) h + (0) · · · h + (0) f : r ≥ even, r/ ≤ k ≤ r − t ≥ · · · ≥ t r/ ≥ ,t r/ ≥ · · · ≥ t k ≥ , forms a basis of V (2) U ( l, f ) . Proof. The relation (25) allows us to permute all the negative modes arbitrarily in the quotient.Note that { j : i j = + } = { j : i j = −} . In case there is no zero modes, all the h + canbe placed to the front and all the h − to the rear, forming two groups. In each group, higherweights can then be arranged to the front, lowers in the rear. In case there exists zero modes,by Theorem 5.9, either all zero modes are h + (0) , or are h − (0) . We arrange the terms similarlyaccording to the choices of zero modes. (cid:3) OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 33 Remark 5.14. It is not difficult to see that the graded-dimension of V (2) U ( l, f ) is ∞ (cid:88) n =1 (cid:88) r even, ≤ r ≤ n n − r/ (cid:88) m = r/ p r/ ( m ) p r/ ( n − m ) + 2 (cid:88) r even, ≤ r ≤ n p r/ ( n ) + n − (cid:88) m =1 r/ (cid:88) k =1 p r/ ( m ) p k ( n − m ) q n , where p k ( n ) is the number of unordered partitions of n into exactly k parts, i.e., the number of λ , ..., λ k such that λ ≥ · · · ≥ λ k ≥ , λ + · · · + λ k = n . Whether or not this series can befurther simplified in terms of some special functions remains a problem.We will proceed to show that V (2) U ( l, f ) is irreducible if the eigenvalue of f is generic (recallRemark 4.9). The proof will need the following fact in the polynomial algebra. Lemma 5.15. (1) The following set of polynomials (cid:88) σ ∈ S k (cid:18) t σ (1) + x − t σ (1) (cid:19) · · · (cid:18) t σ ( n ) + x n − t σ ( n ) (cid:19) : t ≥ · · · t n ≥ (26)is a linearly independent subset in C [ x , ..., x n ] .(2) The following set of polynomials (cid:88) σ ∈ S n (cid:18) t σ (1) + x − t σ (1) (cid:19) · · · (cid:18) t σ ( n ) + x n − t σ ( n ) (cid:19) · (cid:88) τ ∈ S m (cid:18) s τ (1) + y − s τ (1) (cid:19) · · · (cid:18) s τ ( m ) + y − s τ ( m ) (cid:19) : t ≥ · · · t n ≥ , s ≥ · · · ≥ s m ≥ is a linearly independent subset in C [ x , ..., x n , y , ..., y n ] . Proof. For (1), we consider the linear map from C [ x , ..., x n ] to itself defined by x t · · · x t n n (cid:55)→ (cid:18) t + x − t (cid:19) · · · (cid:18) t n + x n − t n (cid:19) . Since the highest degree term of the image is x t · · · x t n n , it is clear that the matrix of the linearmap with respect to the basis x t · · · x t n n is upper-triangular with non-vanishing diagonal entries.Thus the linear map is invertible. So the set (26) is linearly independent if and only if (cid:40) (cid:88) σ ∈ S n x t σ (1) · · · x t σ ( n ) n : t ≥ · · · ≥ t n ≥ (cid:41) is linearly independent, which holds from the linear independence of the set of monomial sym-metric polynomials (modified by a positive integer).For (2), note that C [ x , ..., x n , y , ..., y m ] = C [ x , ..., x n ] ⊗ C [ y , ..., y m ] . The conclusion thenfollows from the following general fact: if S is a linearly independent subset in V , T is a linearlyindependent subset in W , then { s ⊗ t : s ∈ S, t ∈ T } is a linearly independent subset in V ⊗ W . (cid:3) OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 34 Theorem 5.16. Let λ be a generic eigenvalue, then V (2) U ( l, f ) is irreducible. Proof. We first consider the case when w ∈ V (2) U ( l, f ) is homogeneous of weight ω . We show thatif the lowest weight projection of Y ( v, x ) w is zero for every v ∈ V , then w = 0 .Set w = (cid:88) a t ··· t k i ··· i r (cid:12)(cid:12) i ( − t ) · · · i r/ ( − t r/ ) i r/ ( − t r/ ) · · · i k ( − t k ) i k +1 (0) · · · i r (0) (cid:11) f, here the sum is over all possible choices of r ≥ even, i , .., i r ∈ { + , −} , r/ ≤ k ≤ r , and t , ..., t k such that t + · · · + t k = ω , t ≥ · · · ≥ t r/ , t r/ ≥ · · · ≥ t k . We proceed by inductionof r that every a t ··· t k i ··· i r = 0 From the first part of Proposition 5.3, if we choose v = h + ( − m ) h − ( − m ) , then except forthose terms with k ≤ , all other terms in the lowest weight projection of Y ( v, x ) w are zero. Soit suffices to consider only the following part in w : a ω + − h + ( − ω ) h − (0) f + a ω − + h − ( − ω ) h + (0) f + ω − (cid:88) t =1 a t,ω − t + − h + ( − t ) h − ( − ω + t ) f. By assumption and (21), a ω + − ( − m + m − (cid:18) ω + m − ω (cid:19) ωl | + −(cid:105) f + a ω − + ( − m + m − (cid:18) ω + m − ω (cid:19) ωl |− + (cid:105) f + ω − (cid:88) t =1 a t,ω − t + − ( − m + m − (cid:18) t + m − t (cid:19)(cid:18) ω − t + m − ω − t (cid:19) t ( ω − t ) l f = 0 . Note that the left-hand-side can be regarded as a polynomial in m , m . Since the equality holdsfor every m , m ∈ Z + , thus left-hand-side, as a polynomial in m , m , has to be zero. Sincepolynomials involving different variables are linearly independent, it follows that a ω + − ωl | + −(cid:105) f = a ω − + ωl |− + (cid:105) f = 0 . Using Lemma 5.15 (with n = 1 , m = 1 ), we also see that a t,ω − t + − t ( ω − t ) l f = 0 , t = 1 , ..., ω − . Since l (cid:54) = 0 , ω ∈ Z + , and λ is generic, it forces that a ω + − = a ω − + = 0 , a t,ω − t + − = 0 , t = 1 , ..., ω − . This finishes the proof of the base case r = 2 .Now assume that a t ...t k j ...j r = 0 for smaller r . Then we apply h i ( − m ) · · · h i r ( − m r ) to w . By(21), (cid:88) a t ··· t k j ··· j r (cid:88) ≤ c k < ··· We have shown that V (2) U ( l, f ) is irreducible if the eigenvalue λ is generic.For λ = p ( p − K , we have the following conclusion, OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 36 Theorem 5.17. Let λ = p ( p − K be the eigenvalue of f for some p ∈ Z + . Then for everyeven number r ≥ k + p , every t ≥ · · · ≥ t r/ , t r/ ≥ · · · ≥ t k , the element (cid:12)(cid:12) j ( − t ) · · · j r/ ( − t r/ ) j r/ ( t − r/ ) · · · j k ( − t k ) j k +1 (0) · · · j r (0) (cid:11) f (31)generates a proper submodule in V (2) U ( l, f ) . Proof. Note that j k +1 = · · · = j r . Thus in (18), the term (cid:12)(cid:12)(cid:12) i (0) · · · (cid:99) i c k (0) · · · (cid:99) i c (0) · · · i s (0) j k +1 (0) · · · j r (0) (cid:69) f contains at least p consecutive + or − . From Proposition 4.4 Part (2), λ = p ( p − K annihilates(18). The conclusion then follows. (cid:3) Theorem 5.18. Let λ = p ( p − K be the eigenvalue of f for some p ∈ Z + . Let V (3) U ( l, f ) bethe quotient of V (2) U ( l, f ) by the submodule generated by elements of the form (31). Then thethe union of the following three sets (cid:40) h + ( − t ) · · · h + ( − t r/ ) h − ( − t r/ ) · · · h − ( − t r ) f : r ≥ even, t ≥ · · · ≥ t r/ ≥ ,t r/ ≥ · · · ≥ t r ≥ (cid:41) , h + ( − t ) · · · h + ( − t r/ ) h − ( − t r/ ) · · · h − ( − t k ) h − (0) · · · h − (0) f : r ≥ even; max( r − p + 1 , r/ ≤ k ≤ r − t ≥ · · · ≥ t r/ ≥ ,t r/ ≥ · · · ≥ t k ≥ , h − ( − t ) · · · h − ( − t r/ ) h + ( − t r/ ) · · · h + ( − t k ) h + (0) · · · h + (0) f : r ≥ even; max( r − p + 1 , r/ ≤ k ≤ r − t ≥ · · · ≥ t r/ ≥ ,t r/ ≥ · · · ≥ t k ≥ , forms a basis of V (3) U ( l, f ) . In other words, there are at most p − consecutive h − (0) (resp., h − (0) ) at the rear of the second (resp., the third) set of basis. Theorem 5.19. Let λ = p ( p − K be the eigenvalue of f . Then V (3) U ( l, f ) is irreducible. Proof. Observe that for every basis element in Theorem 5.18, there exists i , ..., i s and a choiceof c , ..., c k such that (cid:12)(cid:12)(cid:12) i (0) · · · (cid:92) i c k (0) · · · (cid:92) i c (0) · · · i s (0) j k +1 (0) · · · j r (0) (cid:69) f (cid:54) = 0 . With this observation, we can formulate an argument similarly as Theorem 5.16. Details shallnot be repeated here. (cid:3) OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 37 Remark 5.20. In case f is a global eigenfunction over a compact space form of negative sectionalcurvature, then the eigenvalue of f must be a positive real number and will never coincide with p ( p − K . Thus, V (2) M ( l, f ) is irreducible. Remark 5.21. In case M = S , the two-dimensional unit sphere in R , then K = 1 . It is alsowell known that if f is an eigenfunction over M , then eigenvalue of f is p ( p − for some p ∈ Z + .Thus V (2) M ( l, f ) is not irreducible. One has to take the third quotient to obtain the irreducible V (3) M ( l, f ) . Notation 5.22. One easily sees from Theorem 4.14 that if f , f are two functions with thesame eigenvalue λ , then the irreducible quotient of V U ( l, f ) and V U ( l, f ) are isomorphic.6. Some Classification Results on Lowest Weight Modules In this section, we will set up a correspondence between irreducible V ( l, ) -modules and ir-reducible Π( T ( E )) -modules. Then we introduce a geometrically interesting condition on the Π( T ( E )) -modules, called the covariant derivative condition. We will then classify all irreducible V ( l, ) -modules and all V ( l, ) -modules of finite length, whose lowest weight subspaces satisfythis covariant derivative condition and generate the whole module.For convenience, we will use V to denote the MOSVA V ( l, ) hereafter.6.1. Lowest weight V -modules.Definition 6.1. A V -module W is a lowest weight V -module if there exists some µ ∈ C , suchthat W = (cid:96) n ∈ N W [ µ + n ] , and W is generated by the lowest weight space W [ µ ] .Obviously W [ µ ] is a Π( T ( E )) -module. On the other hand, given a Π( T ( E )) -module Φ , weconsider the V -submodule W (0) (Φ , [ µ ]) of the induced T ( (cid:99) E p − ) -module T ( (cid:99) E p − ) ⊗ T ( E ) ⊗ Π( T ( E )) Φ . where we assign the subspace ⊗ ⊗ Φ the weight µ ∈ C . Proposition 6.2. W (0) (Φ , [ µ ]) is universal in the following sense that every lowest weight V -module with Φ as the lowest weight subspace is a quotient of W (0) (Φ , [ µ ]) . Proof. It follows from Theorem 4.11 that both W (0) (Φ , [ µ ]) and W are spanned by h i ( − t ) · · · h i k ( − t k ) h i k +1 (0) · · · h i r (0) f : f ∈ Φ; r ≥ , k = 1 , ..., r, t , .., t k > { j ∈ [1 , r ] : i j = + } = { j ∈ [1 , r ] : i j = −} ; ∀ p = k + 1 , ..., r, { j ∈ [ p, r ] : i j = + } (cid:54) = { j ∈ [ p, r ] : i j = −} . . ‘ OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 38 Note also that the set forms a basis for W (0) (Φ , [ µ ]) . The map sending the basis vectors of W (0) (Φ , [ µ ]) to the spanning vectors of W is obviously a homomorphism of V -modules. Then W is the quotient of W (0) (Φ , [ µ ]) by the kernel of the map. (cid:3) Proposition 6.3. Let W be an irreducible V -module. Then W is a lowest weight V -module.The lowest weight subspace of W is an irreducible Π( T ( E )) -module. Proof. Fix m ∈ C such that W [ m ] (cid:54) = 0 . Then from the axiom of V -modules, there exists a smallestnumber µ in the congruence class m + Z ∈ C / Z , such that W [ µ ] (cid:54) = 0 . Since W is irreducible,then from Theorem 4.11, we know that W = (cid:96) n ∈ N W [ µ + n ] . From the definition of the vertexoperator Y ( v, x ) , we see that the lowest weight subspace W [ µ ] is also a Π( T ( E )) -module. Since W is irreducible, for any w ∈ W [ µ ] , the submodule generated by w must also coincide with W .In particular, the Π( T ( E )) -submodule generated by any w ∈ W [ µ ] coincides with W [ µ ] . Thus W [ µ ] is irreducible. (cid:3) Proposition 6.4. Let Φ be an irreducible Π( T ( E )) -module. Then for any µ ∈ C , there exists aunique irreducible V -module W (Φ , [ µ ]) = (cid:96) n ∈ N W [ µ + n ] such that the lowest weight space W [ µ ] is Φ . Proof. Consider the universal lowest weight V -module W (0) (Φ , [ µ ]) . It is clear that a V -submoduleof W (0) (Φ , [ µ ]) is proper if and only if it intersects the lowest weight subspace trivially. So thesum of two proper submodules stays proper. Thus there exists a unique maximal submodule.The quotient of W (0) (Φ , [ µ ]) by this unique maximal submodule is thus the irreducible V -modulesatisfying the conditions. (cid:3) Remark 6.5. It follows from the uniqueness in Proposition 6.4 that there is a bijective corre-spondence between irreducible Π( T ( E )) -modules and the irreducible V -modules. Remark 6.6. If the irreducible V -module is grading-restricted, then its lowest weight subspaceis certainly a finite-dimensional. But the converse does not necessarily hold. Indeed, usingthe lowest weight projection formula, one can show that an irreducible V -module is grading-restricted, if and only if the lowest weight subspace is finite-dimensional, and the action of Π( T ( E )) satisfies the following technical condition:For every w ∈ W [ µ ] , every ζ ∈ Z \ { } , there exists R ∈ Z + , such that for every r > R , j , ..., j r ∈ { + , −} satisfying (cid:80) rp =1 j p = ζ , there exists C j (cid:48) ··· j (cid:48) r (cid:48) for every r (cid:48) ≤ R, j (cid:48) , ..., j (cid:48) r (cid:48) ∈ { + , −} satisfying (cid:80) r (cid:48) q =1 j (cid:48) q (cid:48) = ζ , such that for every i , ..., i s satisfying (cid:80) sp =1 i p = − ζ , | i · · · i s j · · · j r (cid:105) w = (cid:88) C j (cid:48) ··· j (cid:48) r (cid:48) (cid:12)(cid:12) i · · · i k j (cid:48) · · · j (cid:48) r (cid:48) (cid:11) w, where the sum is over all r (cid:48) ≤ R and all choices of j (cid:48) , ..., j (cid:48) r (cid:48) satisfying (cid:80) r (cid:48) q =1 j (cid:48) q = ζ .In other words, | i · · · i k j · · · j r (cid:105) w can be expressed as a finite linear combination of terms whosetail length is bounded above, and the expression is uniform with respect to i , ..., i s . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 39 Remark 6.7. Though the technical condition in the previous remark reduces the problem ofclassifying irreducible grading-restricted V -modules to classifying irreducible finite-dimensional Π( T ( E )) -modules satisfying the technical condition, in practice it is not restrictive enough for usto work out an actual classification. We should emphasize that Π( T ( E )) , regarded as a subalgebraof T ( E p ) , contains infinitely generators that are algebraically independent. Thus to identifyeven a one-dimensional irreducible Π( T ( E )) -module requires a specification of infinitely manyconstants representing the actions of these infinitely many generators. Specifying all possiblechoices of these infinitely many constants satisfying the technical condition is highly nontrivial.6.2. Covariant derivative condition. Instead of working on the technical necessary and suf-ficient condition in Remark 6.6, we will instead work on the following sufficient condition that isgeometrically interesting. Definition 6.8. Let N be a Π( T ( E )) -module. We say N satisfies the covariant derivativecondition if for every (cid:80) i X (1) i ⊗ · · · X ( r ) i ∈ Π( T ( E )) , (cid:88) i (cid:0) X (1) i ⊗ · · · ⊗ X ( r − i ⊗ X ( r ) i − X (1) i ⊗ · · · ⊗ X ( r ) i ⊗ X ( r − i (cid:1) (32)act by zero, and for i = 1 , ..., r − , (cid:88) i (cid:0) X (1) i ⊗ · · · ⊗ X ( j ) i ⊗ X ( j +1) i ⊗ · · · ⊗ X ( r ) i − X (1) i ⊗ · · · ⊗ X ( j +1) i ⊗ X ( j ) i ⊗ · · · ⊗ X ( r ) i (cid:1) + (cid:88) i r (cid:88) k = j +2 (cid:0) X (1) i ⊗ · · · ⊗ X ( j +2) i ⊗ · · · R ( X ( j ) i , X ( j +1) i ) X ( k ) i ⊗ · · · ⊗ X ( r ) i (cid:1) (33)act by zero. Remark 6.9. Π( T ( E )) -modules satisfying the covariant derivative condition automatically sat-isfies the condition in Remark 6.6. For every w ∈ N , and ζ ∈ Z \ { } , R can simply be chosenas | ζ | . Then for every r > | ζ | , and every j , ..., j r ∈ { + , −} satisfying (cid:80) rp =1 j p = ζ , there existsa constant C , such that for every i , ..., i s ∈ { + , −} , | i · · · i s j · · · j r (cid:105) = C | i · · · i k (cid:105)| α (cid:105) R . where α = sgn ( ζ ) (cf. Theorem 5.7). Remark 6.10. The covariant derivative condition is just one of the possible candidates forgeometrically interesting conditions. Another option of such a condition comes from the actionof parallel tensors on certain differential forms and shall be discussed in future work.The covariant derivative condition can be used to express the actions of all generators of Π( T ( E )) in terms of the action of | + −(cid:105) . Proposition 6.11. Let M be a finite-dimensional Π( T ( E )) -module satisfying the covariantderivative condition. Then for every i , ..., i r ∈ { + , −} with (cid:80) rp =1 i r = 0 , the action of | i · · · i r (cid:105) is uniquely determined by the action of | + −(cid:105) , and can be expressed as a polynomial in | + −(cid:105) . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 40 Proof. In a nutshell, we can process the contraction of + − pairs similarly as in Proposition 4.4.More elaborately, we show the conclusion by induction on r . For r = 2 , we see from (32) that |− + (cid:105) = | + −(cid:105) . Now assume the conclusion for all smaller r . Without loss of generality, let i r = i r − = · · · = i k +1 = − , i k = + . Then from (33), | i · · · i k i k +1 · · · i r (cid:105) = | i · · · i k + − · · · −(cid:105) = | i · · · i k − + · · · −(cid:105) − r (cid:88) j = k +2 | i · · · i k − · · · − , R (+ − ) − , − · · · −(cid:105) = | i · · · i k − + · · · −(cid:105) + 2 K ( r − k − | i · · · i k − · · · − − · · · −(cid:105) . In the first term, the + in is now moved to ( k + 2) -th position. The second term, by inductionhypothesis, is a polynomial of | + −(cid:105) . Repeating the process to move + further right and handlethe extra term similarly by induction hypothesis, until when we have | i · · · i k − · · · − + −(cid:105) = | i · · · i k − · · · −(cid:105)| + −(cid:105) . The induction hypothesis implies that first factor is a polynomial in | + −(cid:105) , and the second factoris precisely | + −(cid:105) . Thus we proved the conclusion for r . (cid:3) Proposition 6.12. Let M be a finite-dimensional Π( T ( E )) -module. If M is irreducible, then M is one-dimensional. Proof. Let m ∈ M be an eigenvector for | + −(cid:105) . Then from Proposition 6.11, for every i , ..., i r ∈{ + , −} with (cid:80) rp =1 i p = 0 , m is an eigenvector for | i · · · i r (cid:105) . Thus C m is a submodule of M andhas to coincide with M . (cid:3) This result, combined with Proposition 6.3, 6.4 and Theorem 4.11, yields the following classi-fication theorem of irreducible V -modules satisfying the covariant derivative condition. Theorem 6.13. Let W be an irreducible V -module such that the lowest weight subspace W [ µ ] is a finite-dimensional Π( T ( E )) -module satisfying the covariant derivative condition. Then W isisomorphic to the irreducible quotient of V U ( l, f )[ µ ] for some eigenfunction f . Here V U ( l, f )[ µ ] isthe V -module generated by f with the weight of f specified as µ . In other words, every irreducible V -module satisfying the covariant derivative condition is generated by an eigenfunction (locallyover an open subset of the manifold M ). Notation 6.14. We will use the notation W ( λ, [ µ ]) to denote the irreducible quotient of V U ( l, f )[ µ ] .6.3. V -modules of class C : indecomposable case.Definition 6.15. A V -modules W is of class C , if it satisfies the following two conditions: OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 41 (1) W is a lowest weight module. The lowest weight subspace of W , as a Π( T ( E )) -module,satisfies the covariant derivative condition.(2) W admits a composition series W = W ⊇ W ⊇ · · · ⊇ W n ⊇ W n +1 = 0 of finite length, such that for i = 1 , ..., n , the composition factor W i /W i +1 isomorphic to W ( λ i , [ µ ]) for some λ i ∈ C and some µ ∈ C .To classify V -modules of class C . We first classify the indecomposable modules with identicalcomposition factors, i.e., λ = · · · = λ n = λ. Proposition 6.16. Let Φ be a n -dimensional Π( T ( E )) -module satisfying the covariant derivativecondition. If Φ is indecomposable, then Φ is isomorphic to the n -dimensional Π( T ( E )) -modulewhere the Jordan canonical form of | + −(cid:105) is a single Jordan block λ · · · λ . . . ... . . . . . . . . . ... . . . λ 00 0 · · · λ of dimension n . Proof. Choose a basis of Φ such that | + −(cid:105) is represented by its Jordan canonical form. FromProposition 6.11, under this basis, every | i · · · i r (cid:105) is represented by a block-diagonal lower-triangular matrix consistently with the block-decomposition of that for | + −(cid:105) . Since M is in-decomposable, the Jordan canonical form of | + −(cid:105) has to contain only one Jordan block. ByProposition 6.11 again, the matrice of all other Π( T ( E )) elements are then uniquely determinedby that of | + −(cid:105) . (cid:3) Proposition 6.17. Let λ ∈ C be generic, i.e., λ (cid:54) = p ( p − K for every p ∈ Z + . Then there existsan indecomposable module (unique up to isomorphism) of class C with identical compositionfactors. Proof. Let Φ be the indecomposable Π( T ( E )) -module given in Proposition 6.16. Let f , ..., f n be a basis such that | + −(cid:105) is represented by the single Jordan block. Let Φ i be the submoduleof Φ spanned by f i , f i +1 , ..., f n . We use the simplified notation W (0) i for W (0) (Φ i , [ µ ]) . Then thespace Φ i forms the lowest weight subspace in W (0) i . We now take quotients of certain submodulessimilarly as in Section 5. Instead of going through every detail, we just sketch the process withcomments on necessary modification:(1) If w is an element of W (0) i such that the lowest weight projection of Y ( v, x ) w is constantlyzero, then w generates a proper submodule. OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 42 (2) The lowest weight projection formula in Proposition 5.3 stays in effect, where the f in(18) and (21) can be taken as f i , f i +1 , ..., f n .(3) The statement of Theorem 5.7 should be modified as follows: Let j , ..., j r ∈ { + , −} satisfying (cid:80) rp =1 j p = 0 , t , ..., t k ∈ Z + . Set N = (cid:12)(cid:12)(cid:12)(cid:80) rp = k +1 j p (cid:12)(cid:12)(cid:12) . Then there exists apolynomial P ( x ) ∈ C [ x ] depending only on j k +1 , · · · j r , such that for every α = i, i +1 , ..., n , | j ( − t ) · · · j k ( − t k ) j k +1 (0) · · · j r (0) (cid:105) f α − (cid:12)(cid:12) j ( − t ) · · · j k ( − t k ) j (cid:48) (0) · · · j (cid:48) N (0) (cid:11) ( P ( | + −(cid:105) ) f α ) generates a proper submodule. Here j (cid:48) = · · · = j (cid:48) N = sgn (cid:16)(cid:80) rp = k +1 j p (cid:17) . Note that theterm P ( | + −(cid:105) ) f α involves only f β for β ≥ α , and the coefficient of f α coincides with C = C j k +1 ...j r ( λ ) given in Theorem 5.7.(4) Let W (1) i be the quotient of W (0) i by elements in (4), we see that W (1) i has the basis h i ( − t ) · · · h i k ( − t k ) h i k +1 (0) · · · h i r (0) f α : α = i, i + 1 , ..., nr ≥ , t , ..., t k > { j : i j = + } = { j : i j = −} i k +1 = · · · = i r ∈ { + , −} . (5) The statement of Theorem 5.12 stays in effect, where f in (25) can be taken as f α , α = i, i + 1 , ..., n .(6) let W i be the quotient of W (1) i by all the elements discussed in (5). We see that W i hasthe following basis h + ( − t ) · · · h + ( − t r/ ) h − ( − t r/ ) · · · h − ( − t r ) f α : α = i, i + 1 , , ..., n,r ≥ even, t ≥ · · · ≥ t r/ ≥ ,t r/ ≥ · · · ≥ t r ≥ , h + ( − t ) · · · h + ( − t r/ ) h − ( − t r/ ) · · · h − ( − t k ) h − (0) · · · h − (0) f α : α = i, i + 1 , , ..., n,r ≥ even, r/ ≤ k ≤ r − t ≥ · · · ≥ t r/ ≥ ,t r/ ≥ · · · ≥ t k ≥ , h − ( − t ) · · · h − ( − t r/ ) h + ( − t r/ ) · · · h + ( − t k ) h + (0) · · · h + (0) f α : α = i, i + 1 , , ..., n,r ≥ even, r/ ≤ k ≤ r − t ≥ · · · ≥ t r/ ≥ ,t r/ ≥ · · · ≥ t k ≥ . Thus we have constructed the modules W ⊇ W ⊇ · · · ⊇ W n ⊇ . OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 43 Take W = W . We show that the lowest weight V -module W satisfies the conditions (1) and(2). Condition (1) holds because none of the above quotients have changed the lowest weightsubspace. For condition (2), we first note that the quotient W i /W i +1 is generated by the image ¯ f i of f i . Note also that for every polynomial P ( x ) ∈ C [ x ] , P ( | + −(cid:105) ) f i is a linear combination of f i , f i +1 , ..., f n with coefficient of f i being P ( λ ) . So the quotient is isomorphic to W ( λ, [ µ ]) .Before arguing the uniqueness, we first make some observations. Fix any i = 1 , ..., n . Notice if U, V are two submodules of W (0) i intersecting lowest weight subspace trivially, then so is U + V .Thus, there exists a unique maximal submodule of W (0) i intersecting the lowest weight subspacetrivially. One can argue similarly as Theorem 5.16 that the module W i we constructed above isobtained from taking the quotient of W (0)1 by the unique maximal submodule intersecting thelowest weight subspace trivially. More precisely, Formula (27) in the proof of Theorem 5.16 canthen be modified as (cid:88) a t ··· t k j ··· j r ,α (cid:88) ≤ c k < ··· We use the notation W ( λ, [ µ ] , n ) to denote the lowest weight V -module weidentified in Proposition 6.17 OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 44 On the other hand, the following proposition is certainly surprising. Proposition 6.19. Let λ = p ( p − K for some p ∈ Z + . Then there does not exists any inde-composable lowest weight V -modules satisfying (1) and (2) with identical composition factors. Proof. It suffices the nonexistence for n = 2 . The case for any larger n can be reduced to the n = 2 case by taking the quotient of W by the submodule W .Suppose the contrary that such a V -module W exists. Let W [ µ ] be the lowest weight subspace.Let f , f be a basis of the W [ µ ] such that | + −(cid:105) f = λf + f , | + −(cid:105) f = λf . Then as a vector space, W has the basis h + ( − t ) · · · h + ( − t r/ ) h − ( − t r/ ) · · · h − ( − t r ) f α : α = 1 , r ≥ even ; t ≥ · · · ≥ t r/ ≥ ,t r/ ≥ · · · ≥ t r ≥ , h + ( − t ) · · · h + ( − t r/ ) h − ( − t r/ ) · · · h − ( − t k ) h − (0) · · · h − (0) f α : α = 1 , r > even; max( r − p + 1 , r/ ≤ k ≤ r − t ≥ · · · ≥ t r/ ≥ ,t r/ ≥ · · · ≥ t k ≥ . h − ( − t ) · · · h − ( − t r/ ) h + ( − t r/ ) · · · h + ( − t k ) h + (0) · · · h + (0) f α : α = 1 , r > even; max( r − p + 1 , r/ ≤ k ≤ r − t ≥ · · · ≥ t r/ ≥ ,t r/ ≥ · · · ≥ t k ≥ . From Theorem 4.11, the element w = h + ( − p h − (0) p f is in W . We now use the lowest weight projection formula to show that w = 0 .First we note that w is a linear combination of basis vectors of weight p . Thus t + · · · + t k = p .This in particular means that k can be at most as large as p .Second, we note that for every v = | i ( − m ) · · · i s ( − m s ) (cid:105) , if s < p , then the lowest pro-jection of Y ( v, x ) w is zero. Indeed, if we write w = | j ( − · · · j p ( − j p +1 (0) · · · j p (0) (cid:105) with j = · · · = j p = + , j p +1 = · · · = j p = − , then the factor (cid:81) pα =1 (cid:104) i c α , j σ ( α ) (cid:105) appearing in eachsummand in (18) is always zero if s < p , as any choice of p elements in i , ..., i s has to containat least one positive.With the above two observations, we can use an argument identical to that Proposition 6.17(cf. Theorem 5.16), to see that the coefficients of the basis vector with k < p are all zero. Sincethe argument is identical, we should not repeat it here. OSVA AND MODULES OVER 2D ORIENTABLE SPACE FORMS 45 Thus, we show that in the expression of w as a linear combination of basis vectors of W , allcoefficients are zero. Thus h + ( − p h − (0) p f = 0 in W .We interpret the vanishing of h + ( − p h − (0) f as follows: if we view the lowest weight subspace W [ µ ] as a Π( T ( E )) -module and denote it by Φ , then W is the quotient of W (0) (Φ , [ µ ]) by somesubmodule that contains h + ( − p h − (0) p f . However, if we consider the lowest weight projectionof Y ( h + ( − p h − ( − p , x ) h + ( − p h − (0) p f , which, by Theorem 5.3, is equal to p l p ( p !) h + (0) · · · h + (0) h − (0) · · · h − (0) f . Here, by Proposition 6.11, Proposition 4.3 and the assumption λ = p ( p − K , h + (0) · · · h + (0) h − (0) · · · h − (0) f = p (cid:89) β =1 ( − h + ⊗ h − + β ( β − K ) f = p − (cid:89) β =1 ( − h + ⊗ h − + β ( β − K ) · ( − λf + f + p ( p − Kf )= p − (cid:89) β =1 ( − h + ⊗ h − + β ( β − K ) f = p − (cid:89) β =1 ( − p ( p − K + β ( β − K ) f . In other words, the submodule generated by h + ( − p h − (0) p f contains a nonzero multiple of f and thus have nontrivial intersection to the lowest weight subspace. Then W , as a quotient of W (0) (Φ , [ µ ]) , have a one-dimensional lowest weight subspace, contradictory to our assumption. (cid:3) Remark 6.20. Proposition 6.19 implies that for any V -module W of class C with identicalcomposition factors W ( p ( p − K, [ µ ]) for some p ∈ Z + , then the Jordan canonical form of | + −(cid:105) has to be the diagonal matrix p ( p − K · I . It is clear that in this case W is a direct sum of W ( p ( p − K, [ µ ]) .6.4. V -modules of class C : general case. We are now ready for the classification theoremfor V -modules of class C : Theorem 6.21. Let W be a V -module of class C with lowest weight µ ∈ C . Then W is a directsum of W ( λ, [ µ ] , n ) with generic λ ∈ C , and W ( λ (cid:48) , [ µ ]) with specific λ (cid:48) ∈ { p ( p − K } . Proof. 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