Conformal maps in higher dimensions and derived geometry
aa r X i v : . [ m a t h . AG ] F e b Conformal maps in higher dimensions and derivedgeometry
Mikhail KapranovFebruary 24, 2021
Abstract
By Liouville’s theorem, in dimensions 3 or more conformal transformations form afinite-dimensional group, an apparent drastic departure from the 2-dimensional case.We propose a derived enhancement of the conformal Lie algebra which is an infinite-dimensional dg-Lie algebra incorporating not only symmetries but also deformationsof the conformal structure. Our approach is based on (derived) deformation theory ofthe ambitwistor space of complex null-geodesics.
Introduction
The classical theorem of Liouville (1850) says that the behavior of conformal maps in di-mensions ě
3, as compared to dimension 2, is drastically different. While in 2 dimensions,holomorphic functions give an infinite-dimensional supply of local conformal maps, any con-formal map between connected domains
U, V Ă R n , n ě
3, comes from a global M¨obiustransformation of the conformal sphere S n , i.e., from an element of the finite-dimensionalgroup O p n ` , q . This apparent discontinuity can be rather puzzling.The goal of this note is to propose a resolution to this apparent puzzle by using thepoint of view of derived geometry, i.e., of the homological, derived category-style approachto algebraic and differential geometry [5, 11, 17, 24]. More precisely, we recover the missinginfninite-dimensional part of the conformal group (it is technically easier to start with theLie algebra) in a different cohomological degree. For n “ n -dimensional complex analytic conformal manifold M of appropriate type weintroduce a differential graded (dg-) Lie agebra R conf p M q which is of infinite-dimensionalnature regardless of n . Its cohomology includes: • H “ conf p M q , the usual Lie algebra of conformal Killing vector fields (locally, infinite-dimensional for n “
2, finite-dimensional for n ą H being the space of infinitesimal deformations of the conformal structure (locally,zero for n “
2, infinite-dimensiomal for n ą M “ C n with flat metric, the total size of H ‚ R conf p M q varies “continuously”with n . To illustrate this point, we identify H ‚ R conf p C n q as a representation of SO p n, C q inTheorem 5.3.Our approach is based on the analogy with behavior of holomorphic functions on C n vs. C n zt u . While passing from C to C zt u increases the supply of holomorphic functions,passing from C n to C n zt u , n ě
2, does not (Hartogs’ theorem). But if we look at the totalcohomology H ‚ p C n zt u , O q of the sheaf of holomorpic functions, we find the missing singularparts in the cohomological degree n ´ L p M q ,the space of complex null-geodesics in a holomorphic conformal manifold M , with its naturalcontact structure. Its holomorphic contact geometry completely encodes the holomorphicconformal geometry of M . In particular, the space of (holomorphic) conformal Killing fieldson M is found as the space of sections(0.1) conf p M q “ H p L p M q , κ q , where κ is the sheaf of holomorphic contact vector fields on L p M q . Already in the local case( M is a small geodesically convex neighborhood of a point x ), L p M q is a complex manifoldwhich (for n ě
3) is not compact but has compact directions: it contains p n ´ q -dimensionalcomplex projective quadrics L x formed by null-geodesics passing through various points x P M . For manifolds of this type, coherent sheaves such as κ , can have finite-dimensional H but infinite-dimensional higher cohomology. We define R conf p M q “ R Γ p L p M q , κ q , the dg-Lie algebra of derived global sections of the sheaf of Lie algebras κ .There can be other approaches to defining R conf p M q , for example, in the C , rather thanholomorphic case. The holomorphic approach adopted here provides a natural way to arriveat the idea of a derived extension of the conformal algebra. It also leads to a conceptuallytransparent proof of (the infinitesimal, holomorphic version of) Liouvuille’s theorem, in theform of finite-dimensionality of the H -space in (0.1).Since for n ě
3, the infinite-dimensionality of the dg-Lie algebra R conf p M q is situated inthe odd cohomological degree 1, one can integrate it, in a purely algebraic way, cf. [3, 18], toa derived group R Conf p M q . The classical truncation’ of R Conf p M q is Conf p M q , the usualLie group of (holomorphic) conformal diffeomorphisms, and the whole R Conf p M q can beseen as an infinite-dimensional formal derived thickening of Conf p M q . Such derived groups,and their analogs for conformal superspaces [4, 19] should be of importance for the studyof (super)conformal quantum field theories in dimensions n ě
3. See [22] for a somewhatdifferent recent appearance of derived geometry constructions in that context.2 am grateful to C. Schweigert and M. Yamazaki for interest in this work and usefulsuggestions. This research was supported by the World Premier International ResearchCenter Initiative (WPI Initiative), MEXT, Japan.
A. The classical theorems.
The classical theorem of Hartogs says:
Theorem 1.1.
For n ě , every holomorphic function on C n ´ t u extends holomorphicallyto C n . In the algebro-geometric version, over the base field C and over the Zariski topology, weconsider the affine space A n (i.e., “ C n ‘considered as an algebraic variety”). The correspond-ing statement is that for n ě H p A n zt u , O q “ H p A n , O q “ C r z , ¨ ¨ ¨ , z n s , (no increase of the ring of regular functions), while for n “ C r z, z ´ s “ H p A zt u , O q Ľ H p A , O q “ C r z s . with tne new part being z ´ C r z ´ s (space of polar parts of functions with poles at 0).This phenomenon looks like discontinuity: something seemingly disappears as we pass tohigher dimensions.There is an even more classical result in geometry where the situation changes drasticallyin passing to higher dimensions: the Liouville theorem. We consider the flat Euclideanspace p R n , x´ , ´yq and look at conformal transformations between domains U , V in R n . If n “
2, then locally, we have an infinite-dimensional supply of such transformations, as anyholomorphic function on a domain in C “ R is conformal. However, the Liouville theoremsays: Theorem 1.4.
Let n ě , let U, V be connected open domains in R n and ϕ : U Ñ V be aconformal diffeomorphism. Then ϕ extends to a Moebius-type conformal map (compositionof rigid motions, dilations and inversions) defined on the entire R n with the possible exceptionof one point, and belonging to the standard conformal group O p n ` , q . Let us recall the geometric meaning of the group O p n ` , q in this case. For this, andfor further analysis, it is convenient to work in the complex analytic situation. B. The complex setting.
Let M be a complex manifold of dimension n . We can speakabout holomorphic Riemannian metrics on M . Such a metric is a holomorphic section of thevector bundle S p T ˚ M q which is non-degenerate at each point. In holomorphic coordinates itis gives as a symmetric matrix g p z q “ } g ij p z q} of holomorphic functions.3 (holomorphic) conformal metric on M is, naively, a “Riemannian metric defined up toa scalar”. This means that locally we have a representative g ij p z q which is a holomorphicRiemanian metric, with the understanding that we identify g ij p z q „ λ p z q g ij p z q , λ P O ˚ M , i.e., that λ p z q g ij p z q represents the same conformal metric as g ij p z q .If we fix a point z P M , then a complete invariant of a “non-degenerate symmetric formon T z M defined modulo scalars”, is the null-cone C z M Ă T z M . Thus, globally: Definition 1.5. (a) A holomorphic conformal metric on a complex manifold M is a holomor-phic family of non-degenerate quadratic cones C z Ă T z M . A complex conformal manifold isa complex manifold equipped with a holomorphic conformal metric.(b) Let p M, p C z qq and p M , p C z qq are two n -dimensional complex conformal manifolds A conformal mapping ϕ : M Ñ M is a biholomorphic map whose differential takes null-conesto null-cones: d z ϕ p C z q “ C ϕ p z q .An alternative definition would be that a conformal metric is given by a holomorphic linebundle Λ on M an Λ-valued scalar product g P Hom p S T M , Λ q on T M . It is easily seen to beequivalent to the above. Example 1.6 (Conformal quadric). (a) Let Q Ă P n ` “ P p C n ` q be a non-degeneratequadric hypersurface. It has a canonical conformal structure defined as follows. For z P Q let T z Q Ă P n ` be the projective tangent space to Q at z . It is an algebraic variety isomorphicto P n , and we have a canonical identification of the (usual) tangent spaces T z T z Q “ T z Q .The intersection C z “ Q X T z Q (inside P n ` ) is a quadratic hypersurface in T z Q with onesingular point, namely z . It is called the projective tangent cone to Q . The (intrinsic) tangentcone to this intersection is a nondegenerate quadratic cone C z in T z T z Q , i.e., in T z Q . Thisdefines the conformal structure on Q .(b) Thus any automorphism of P n ` preserving Q defines a conformal mapping Q Ñ Q .Such automorphisms form the group O p n ` , C q .(c) If we fix one point Q , then Q z C Q is isomorphic to A n as an algebraic varietyso to C n as a complex manifold. The induced conformal structure on C n is the standard flatconformal structure. Thus any g P O p n ` , C q defines a conformal mapping from C n (minus,possibly, a quadratic cone hypersurface) to itself.(d) If we want to restrict to real points, then Q gives the n -sphere S n , with its standardconformal structure. The real group O p n ` , q acts by conformal mappings of S n to itself.Each projective tangent cone C z S n gives just the point z , and S n ´ t8u is identified with R n via the stereographic projection. In this way any g P O p n ` , q defines a conformal mapof R n (minus, possibly, a single point) to itself.The holomorphic version of Liouville’s theorem can be formulated as follows.4 heorem 1.7. Let n ě , let U, V Ă Q be open domains and ϕ : U Ñ V be a holomorphicconformal mapping. Then ϕ extends to an isomorphism of algebraic varieties Φ : Q Ñ Q . Let us concentrate on the Lie algebra (infinitesimal) version. Let M be a complex man-ifold with conformal structure. A conformal Killing field on M is a holomorphic vectorfield preserving the conformal structure. We denote by conf p M q the Lie algebra formed byconformal Killing fields. The Lie algebra version of the Liouville theorem is: Theorem 1.8.
Let Q Ă P n ` be the n -dimensional quadric.(a) For any n ě we have conf p Q q “ so p n ` , C q .(b) Let n ě . For any connected open domain U Ă Q the restriction map so p n ` , C q “ conf p Q q ÝÑ conf p U q is an isomorphism. As we shall see, this fact is a manifestation of the same phenomenon as the Hartogstheorem and can be overcome in a similar way.
A way to recover the missing polar parts in Hartogs’ theorem is by using the full cohomology H ‚ p A n zt u , O q , not just H . We have the following elementary fact. Proposition 2.1.
For any n ě we have H i p A n zt u , O q » $’&’% C r z , ¨ ¨ ¨ , z n s , if i “ ,z ´ ¨ ¨ ¨ z ´ n ¨ C r z ´ , ¨ ¨ ¨ , z ´ n s , if i “ n ´ , , otherwise . Here the new space, formed by the polar parts(2.2) H n ´ p A n zt u , O q “ H n t u p A n , O q “ z ´ ¨ ¨ ¨ z ´ n ¨ C r z ´ , ¨ ¨ ¨ , z ´ n s appears in cohomological degree n ´ n th cohomology with support at0. In fact, the second identification in (2.2) holds for any n ě
1. Thus passing to thecohomology restores the continuity.The easiest way to establish Proposition 2.1 is by using the ˇCech complex associatedto the covering of A n zt u by the affine open sets U i “ t z i ‰ u , i “ , ¨ ¨ ¨ , n . The spaceΓ p U i , ¨¨¨ ,i p , O q of regular functions on each p -fold intersection is realized inside the Laurentpolynomial ring C r z ˘ , ¨ ¨ ¨ , z ˘ n s , and the p n ´ q th cohomology will appear as the span ofthe Laurent monomials which will not appear in any of the Γ p U i , ¨¨¨ ,i p , O q . This approach is5quivalent to the classical computation of the cohomology of the sheaves O p d q on P n ´ dueto Serre, see [10].If we consider the complex manifiold C n instead of the algebraic variety A n and the sheaf O hol of holomorphic functions, we have a statement similar to Proposition 2.1, but with H n ´ p C n zt u , O hol q “ H n t u p C n , O hol q “ z ´ ¨ ¨ ¨ z ´ n ¨ C rr z ´ , ¨ ¨ ¨ , z ´ n ss ent being the space of Taylor series representing entire functions. This space is known as thespace of holomorphic hyperfunctions on C n with support at 0, see [23].For any sheaf F of C -vector spaces on a topological space X we denote by R Γ p X, F q thederived functor of sections of F , i.e., “the” complex of C -vector spaces whose cohomologyis H ‚ p X. F q . Such a complex is defined uniquely up to unique isomorphism in the derivedcategory. If F has some additional algebraic structure (commutative algebra, Lie algebraetc.), then it is well known that R Γ p X, F q can be defined in such a way as to inherit thisstructure. Examples 2.3. (a) Let X is a complex manifold and F “ O X be the sheaf of holomorphicfunctions. It is a sheaf of commutative algebras. The Dolbeault complex Ω , ‚ p X q , Bq is amodel for R Γ p X, O X q which has the structure of a commutative dg-algebra.The sheaf F “ T X of holomorphic vector fields on X is a sheaf of Lie algebras. TheDolbeault complex p Ω , ‚ p X, T X q , Bq is a dg-Lie algebra model for R Γ p X, T X q , with the Liestructure given by the Schouten bracket.(b) Let X be an algebraic variety with Zariski topology and F “ O X ibe the sheaf ofregular functions, A commutative dg-aglebra model for R Γ p X, O X q can be obtained as thegoobal relative de Rham complex Γ p J, Ω ‚ J { X q where J Ñ X is a Jouianolou torsor , i.e., anaffine variety which is made into a Zariski ocally trivial bundle over X with fibers being affinespaces and transition functions being affine transformations. See [2] for a general discussionand [6] for a concrete example with X “ A n zt u .(c) If X is any topological space and F is any sheaf of commutative (resp. Lie, etc.) C -algebras, then the ˇCech model for R Γ p X, F q produces a cosimplicial commutative (resp.Lie, etc.) C -algebra. There is a general procedure of converting a cosimplicial algebra ofany given type into a dg-algebra of the same type using the Thom-Sullivan constructioninvolving polynomial differential forms on simplices. It provides the most general way tomake R Γ p X, F q to inherit the algebra structure present on F . We refer to [11] § n -dimensional replacement of the algebra C r z, z ´ s of Laurent polyno-mials is the commutative dg-algebra A r n s “ R Γ p A n zt u , O q defined as in Example 2.3(b). In particular, tensoring A r n s with a finite-dimensional reductiveLie algebra g leads to interesting higher-dimensional derived generalizations of Kac-Moodyalgebras [6, 8]. 6 Ambitwistor description of conformal metrics
We want to show that Liouville’s theorem, at least in its complex, infinitesimal form (1.8),can be seen as a Hartogs-type phenomenon and therefore can be “overcome” by introducingcohomological degrees of freedom. For this, we recall the main points of the ambitwistorapproach [16, 19, 20] to holomorphic conformal metrics (in any dimension, in particularwithout assuming self-duality in dimension 4).
A. The space of null-geodesics.
Let p M, g q be an n -dimensional complex manifoldwith a holomorphic Riemannian metric. We can then speak about null-geodesics in M which are parametrized holomorphic curves γ : U Ñ M , U Ă C open, satisfying the complexversion of the geodesic equation and such that γ p t q is isotropic everywhere. The elementarybut fundamental fact is, see [16] § II.2:
Proposition 3.1.
For two conformally equivalent metrics g p z q and λ p z q g p z q , the null-geodesics are the same up to a re-parametrizaton. Put differently, let
QT M Ă P p T M q be the quadric bundle formed by the null-directionsin the tangent spaces T x M , x P M . It is a complex manifold of dimension 2 n ´
2. The“complex geodesic flow” for g p z q is the 1-dimensional complex foliation L on QT M whoseleaves are the tangent lifts of null-geodesics for g p z q . Note that QT M depends only on theconformal class of g p z q . Proposition 3.1 says that so does L .Let now p M, p C x qq be a holomorphic conformal manifold of dimension n . We then havethe quadric bundle QT M Ă P p T M q with fibers Q p T x M q “ P p C x q Ă P p T x M q . By the above, QT M carries a canonical 1-dimensional holomorphic foliation L whose leaves, are, locally,the lifts of complex null-geodesics for any holomorphic metric representing p C x q .The space of null-geodesics L “ L p M q is defined as the space of leaves of the foliation L .In the sequel we assume that this space of leaves exists, i.e., intuitively, the global behavior ofcomplex null-geodesics in not too wild. More precisely, following [16], we make the following Definition 3.2.
A holomorphic conformal manifold p M, p C x qq of dimension n is called civi-lized , if:(1) There is a Hausdorff complex manifold L of dimension 2 n ´ ρ : QT M Ñ L whose fibers are precisely the leaves of L , with the property:(2) The restriction of ρ to any quadric Q p T x M q , x P M , is a holomorphic embedding (thatis, no complex null-geodesic passes through the same point twice).For a civilized M the manifold L “ L p M q is defined uniquely up to a unique isomorphism.In the sequel we will assume that M is civilized. Here are some examples. Examples 3.3. (a) (Flat case, noncompact) M “ C n with a flat conformal metric. Inthis case L p C n q is a closed subvariety in the space of all straight lines in C n . It is an7lgebraic variety which, for n ě
3, is not affine and not projective. More precisely, let Q n ´ Ă P n ´ “ P p M q be the quadric formed by null-lines in M through 0. Any null-line in M can be seen, in a unique way, as a translation of a line passing through 0. This meansthat L p M q is the total space of the vector bundle l ÞÑ M { l on Q , i.e., of the bundle whosefiber over the point r l s represented by a null-line l through 0, is M { l . The bundle l ÞÑ M { l is in fact defined on the entire P p M q “ P n ´ and as such, is identified with T P p M q p´ qq ,because of the “Euler sequence” [7]:0 Ñ O P p M q p´ q ÝÑ M b O P p M q ÝÑ T P p M q p´ q ÝÑ . (We recall that O P p M q p´ q is the tautological line bundle, i.e., l ÞÑ l in the above notation.)So we conclude that(3.4) L p C n q “ Tot ` T P n ´ p´ q| Q n ´ ˘ . (b) (Flat case, compact) M “ Q n Ă P n ` is the n -dimensional projective quadric. In thiscase L p Q n q consists of all straight lines in P n ` which lie on Q n . It is a projective algebraicvariety identified with G is p , C n ` q , the Grassmannian of 2-dimensional subspaces in C n ` which are isotropic with respect to the quadratic form defining Q n .(c) (Local case) We can always replace M by a sufficiently small neighborhood arounda fixed point x (small with respect to the curvature data of the metric near x ). Then thesituation will be similar to the flat case (a), so we get a civilized manifold. See [16], § II.1.For each x P M we define(3.5) L x “ γ P L ˇˇ x P γ ( Ă L to consist of null-geodesics that pass through x . The condition (2) of Definition 3.2 impliesthat L x is identified with the quadric Q p T x M q , i.e., is isomorphic to Q n ´ . The conformalgeometry of M is encoded by the system of subvarieties L x . That is, x and y are null-separated, if and only if L x X L y ‰ H . Further, comparison with the flat case identifies thenormal bundle of each L x in L . That is, with respect to any identification L x » Q n ´ wehave(3.6) N L x { L » T P n ´ p´ q| Q n ´ . The bundle in the RHS of (3.6) is homogeneous (equivariant under automorphisms of Q n ´ ),so one can write (3.6) without reference to a particular way of identifying L x with Q n ´ . B. L p M q as a contact manifold. Let X be a complex manifold of odd dimension2 m `
1. We recall [1, 16, 21] that a (holomorphic) contact structure on X is a (holomorphic)vector subbundle Θ Ă T X of rank 2 m which is maximally non-integrable in the followingsense. Let κ “ T X { Θ be the quotient line bundle. Then Θ is given by the vanishing of the8autological κ -valued contact form θ : T X Ñ κ . A local trivialization of κ makes θ into ausual holomorphic 1-form. The maximal non-integrability conditions means that(3.7) θ ^ dθ ^ ¨ ¨ ¨ ^ dθ loooooomoooooon m ‰ . This condition is known to be independent on the way we represent θ as a usual form bychoosing a trivialization of κ . More precisely, since θ is, intrinsically, a 1-form with values ina line bundle, dθ is not invariantly defined, but it is invariantly defined modulo (the wedgeideal generated by) θ . Therefore θ ^ p dθ q ^ n is invariantly defined as a volume form withvalues in κ bp m ` q , becuase it involves only dθ modulo θ , as θ ^ θ “
0. Since it is nowherevanishing, we get a canonical identification of line bundles(3.8) κ bp m ` q » ω bp´ q X , where ω X is the line bundle of volume forms.Another consequence of the same remark is that dθ is invariantly defined and non-degenerate on Ker p θ q “ Θ, that is, we have a non-degenerate skew-symmetric form(3.9) dθ : Λ p Θ q ÝÑ κ . If now Z Ă X is a smooth hypersurface which is transversal to Θ everywhere, then it carriesthe 1-dimensional bicharacteristic foliation B , with tangent spaces to the leaves being the1-dimensional subspaces B z “ Ker p dθ | T z Z X Θ z q Ă T z Z, z P Z. It is classical (the “contact reduction”) that the space of leaves of B (the space of bicharac-teristics), if it exists, is again a contact manifold, now of dimension 2 m ´ p M, p C x qq is a holomorphic conformal manifold of dimension n . Then T ˚ M is asymplectic manifold, so P p T ˚ M q is a contact manifold [1] which is identified with P p T M q bythe conformal structure. The hypersurface QT M Ă P p T M q of null-directions is transversalto Θ P p T M q and its bicharacteristic foliation B is just the null-geodesic foliation L . This showsthat the space L of null-geodesics carries a canonical contact structure Θ “ Θ L .Explicitly, Θ can be defined as follows. Let γ P L be a null-geodesic, considered as a1-dimensional complex submanifold in M . At any x P γ , the line T x γ Ă T x M is isotropic, soits orthogonal p T x γ q K is a hyperplane in T x M containing T x γ . Now, the contact hyperplaneΘ γ Ă T γ L consists of infinitesimal displacements of γ which, for each x P γ , move x inside p T x γ q K .We further recall that an m -dimensional smooth submanifold W of a 2 m ` p X, Θ q is called Legendrian , if the tangent spaces of W lie in Θ. In our case X “ L , it is clear from the above explicit definition of Θ that any subvariety L x , see (3.5), isLegendrian. The main result of the ambitwistor description of conformal metrics [15, 16, 20]can be summarized as follows. 9 heorem 3.10. (a) Any local Legendrian deformation of any L x inside L is of the form L y for some y .(b) Let M , M be two civilized holomorphic conformal manifolds of dimension n . Let x i P M i , i “ , , be two points. Holomorphic conformal diffeomorphisms M Ñ M taking x ÞÑ x are in bijection with holomorphic contact diffeomorphisms L p M q Ñ L p M q taking L x Ñ L x . Remark 3.11.
In dimensions ě L x is automatically Legendrian, and any holomorphic diffeomorphism L p M q Ñ L p M q is automatically contact, see [16]. For special consideration of the case ofdimension 3 see [14, 15]. Nevertheless, it seems natural to keep track of the contact structureis all dimensions, since it is the natural geometric structure present in the problem. A. Contact Hamiltonians and conformal Killing fields.
Let p X, Θ q be a holomor-phic contact manifold of dimension 2 m `
1. A contact vector field on X is a holomorphicvector field ξ preserving the distribution Θ. That is, if we trivialize κ “ T X { Θ and viewthe contact form θ as a usual 1-form, then we should have Lie ξ p θ q “ f ¨ θ for some function f . It is well known [21] that such a ξ is uniquely determined by the contact Hamiltonian (4.1) H “ θ p ξ q P H p X, κ q , where we now view θ as a κ -valued 1-form. In this way the sheaf of contact vector fieldsis identified with the sheaf of holomorphic sections of the line bundle κ . The Lie algebrastrucure on contact vector fields translates to a canonical bi-differential operator (Poisson-Jacobi bracket) κ ˆ κ Ñ κ .We now specialize to X “ L p M q where M is a civilized holomorphic conformal manifoldof dimension n . Theorem 3.10(b) gives, as the infinitesimal version, the following. Corollary 4.2.
We have an identification of Lie algebras conf p M q » H p L p M q , κ q . B. The derived conformal algebra.
The above suggests the following definition.
Definition 4.3.
Let p M, p C x qq be a civilized holomorphic conformal manifold. The derivedconformal algebra of M is the dg-Lie algebra R conf p M q : “ R Γ p L p M q , κ q .
10n particular, the 1st cohomology of this dg-Lie algebra is H p L p M q , κ q which is thespace of infinitesimal deformations of L p M q as a contact manifold, i.e., by Theorem 3.10,of infinitesimal deformations of M as a conformal manifold. We now see that the infinite-dimensionality of the 2-dimensional conformal group does not “disappear” in dimensions ě
3, but is transformed into the infinite-dimensionality of the moduli space of local confor-mal metrics. Indeed, symmetries and deformations are, from the point of view of derivedgeometry [11], always governed by the same algebraic structure: an appropriate differentialgraded Lie algebra.
A. Statement of the result.
We now analyze the cohomology of the derived conformalalgebra of the n -dimensional flat space, n ě
3. We will be interested in the algebraic skeletonof the problem, i.e., in dealing with polynomials rather than power series. Therefore we willwork with the algebraic variety A n instead of the complex manifold C n , and understand L p A n q as an algebraic variety as well. So we form the dg-Lie algebra R conf p A n q “ R Γ p L p A n q , κ q , considering κ as the sheaf of regular sections on the Zariski topology of L p A n q . We will iden-tify the cohomology if this dg-Lie algebra, i.e., H ‚ p L p A n q , κ q as a module over the orthogonalgroup SO p n, C q .More precisely, we denote by M “ C n the standard n -dimensional complex vector spaceand think of A n as “ M considered an an algebraic variety”, i.e., as the spectrum of thealgebra S ‚ p M ˚ q . Fixing a nondegenerate quadratic form q P S p M ˚ q , we get a flat metricon A n and the variety L p A n q .Recall the basics of representation theory of GL p n, C q , see [9]. Given a sequence of integers a “ p a ě ¨ ¨ ¨ ě a n q (a dominant weight for GL p n q ), we have the Schur functor Σ a from thecategory of n -dimensional C -vector spaces and their isomorphisms to the category of finite-dimensional C -vector spaces, with Σ a p V q being “the” space of irreducible representation of GL p V q with highest weight a . If all a i ě
0, we think of a as a Young diagram with rowsof lengths a , ¨ ¨ ¨ , a n . If a “ p a , ¨ ¨ ¨ , a p , , ¨ ¨ ¨ , q , we write Σ a , ¨¨¨ ,a p for Σ a , dropping thezeroes at the end. We also write 1 p “ p p hkkkikkkj , ¨ ¨ ¨ , , , ¨ ¨ ¨ , q , p ď n . Note the particular casesand properties: Σ d p V q “ Σ d, , ¨¨¨ , p V q “ S p p V q , Σ p p V q “ Λ p p V q , Σ a , ¨¨¨ ,a n p V q ˚ » Σ a , ¨¨¨ ,a n p V ˚ q » Σ ´ a n , ¨¨¨ , ´ a p V q . Given two weights a “ p a , ¨ ¨ ¨ , a n q and b “ p b , ¨ ¨ ¨ , b n q , the decomposition of the tensorproduct Σ a p V q b Σ b p V q » à c Σ c p V q ‘ N cab N cab known as the Littlewood-Richardson coefficients. There are two important cases when N cab “ Examples 5.1. (a) (Horizontal Young multiplication) c “ a ` b , i.e., c i “ a i ` b i . If all a i , b i ě
0, then c is the Young diagram obtained by “adding” a and b in the horizontaldirection (row by row). The resulting projection Σ a p V q b Σ b p V q Ñ Σ a ` b p V q is induced, viathe Borel-Weil theorem, by tensor multiplication of line bundles on the flag variety.(b) (Vertical Young multiplication). Dually, suppose that a, b are nonnegative and c isthe Young diagram obtained by adding a and b in the vertical direction, column by column.Then N cab “ y : Σ a p V q b Σ b p V q Ñ Σ c p V q can be calledthe vertical Young multiplication . For instance, if a “ r , b “ s , then c “ r ` s and we getthe exterior multiplication. We will be particularly interested in the projection(5.2) y d : S d p V q b S p V q ÝÑ Σ d, p V q , d ě . We now specialize to V “ M ˚ where p M, q q is as above and write SO p n q “ SO p n, C q for the group of automorphisms of p M, q q with determinant 1. Note that M » M ˚ as an SO p n q -module. The projection (5.2) gives an SO p n q -equivariant map y d,q “ y p´ b q q : S d p M ˚ q ÝÑ Σ d, p M ˚ q . Theorem 5.3.
The dg-Lie algebra R conf p A n q has the following cohomology spaces: • H “ Λ p M ˚ q ‘ M ˚ ‘ M ˚ ‘ C “ Λ p M ˚ ‘ C q “ so p n ` q (the usual conformalalgebra). • H “ À d ě Coker p y d,q q , with each y d,q , d ě , being injective. • H i “ for i ě . B. Moduli space interpretation.
We now explain why the space H in Theorem 5.3can be seen as the space of local deformations of the conformal class of the flat metric. Forthis we think of the components of a Riemannian metric g ij p z q as a formal Taylor series on C n near 0 and view the symmetric algebras below as the spaces of polynomials dense in thespaces of power series.The symmetric algebra S ‚ p M ˚ q is (after completion) the space of formal germs of func-tions on M “ C n near 0. So the corresponding space of germs of the metric itself is the tensorproduct S p M ˚ q b S ‚ p V ˚ q . As this is a linear space, we view it as the space of infinitesimaldeformations of the flat metric. The Pieri formula [9] gives(5.4) S b S d » S d ` ‘ Σ d ` , ‘ Σ d, . Let us now quotient by changes of coordinates (understood infinitesimally, as vector fields).The space of vector fields (understood in the same sense as above) is M b S ‚ p M ˚ q . Weidentify M with M ˚ as a SO p n q -module. Again, the Pieri formula gives(5.5) M ˚ b S d p M ˚ q » S d ` p M ˚ q ‘ Σ d, p M ˚ q .
12o the “moduli space” of metrics modulo coordinate changes has, as the tangent space atthe trivial metric, the result of subtracting the contributions from (5.5) for all d from thecontributions from (5.4) for all d , which gives À d ě Σ d, p M ˚ q . For instance, the lowest sum-mand here, Σ , p M ˚ q , is precisely the space of all possible values of the Riemann curvaturetensor at the origin.Further, let us look at the effect of passing to conformal classes, i.e., quotienting bymultiplication by scalar functions, on the tangent space to the moduli space. The space offunctions is S ‚ p V ˚ q . So taking the cokernel of the map y q : S ě p V q ÝÑ à d ě Σ d, p V q has the effect of passing to the tangent space of the moduli space of conformal classes. Remark 5.6.
Finally, it is instructive to compare the situation with the 2-dimensional casewhen we have an infinite-dimensional conformal algebra in homological degree 0. The dif-ference is that for dim p M q “ y d,q : S d p M ˚ q ÝÑ Σ d, p M ˚ q “ S d ´ p M ˚ q b Λ p M ˚ q b is surjective, not injective. The kernel of y d,q has dimension 2, it is the space of tracelesssymmetric tensors in 2 variables. So in each degree we have two basis vectors contributingto the kernel. This matches the identification conf p A q “ C r z sB z ‘ C r z sB z . A. Identifying the bundle κ . We first identify the line bundle κ , the target ofthe contact form, starting from the compact flat case. That is, let V “ C n ` with a non-degenerate scalar product x´ , ´y and let Q n Ă P p V q “ P n ` be the quadric of null-directions.Then L p Q n q is the isotropic Grassmannian G is p , V q Ă G p , V q . We denote by S the tauto-logical rank 2 bundle on both G p , V q and G is p , V q and put O p q “ Λ p S ˚ q . Lemma 6.1.
The line bundle κ G is p ,V q is identified with O p q . Proof:
Let E Ă V be a 2-dimensional isotropic subspace and r E s P G is p , V q be the corre-sponding point. Then it is standard that T r E s G p , V q » Hom p E, V { E q . Inside this, T r E s G is p , V q consists of linear maps f : E Ñ V { E such that(6.2) x f p e q , e y “ e P E. E is isotropic, x f p e q , e y is well defined.) This is a codimension 3 subspace in Hom p E, V { E q .Further, the contact hyperplane Θ E Ă T r E s G is p , V q is Hom p E, E K { E q (a codimension 4 sub-space in Hom p E, V { E q ), see the general discussion in § f satisfying (6.2), we have x f p e q , e y “ ´x f p e q , e y , for any e , e P E. Therefore the expression x f p e q , e y is a linear map Λ p E q Ñ C . Vanishing of this map meansthat f : E Ñ E K { E , i.e., f P Θ E . This gives an identification of vector spaces T r E s G is p , V q{ Θ E » Λ p E ˚ q , and so an identification of line bundles κ » Λ p S ˚ q “ O p q .We now pass from L p Q n q to the Zariski open part L p A n q which is, by (3.4), the totalspace of an algebraic vector bundle whose projection we denote by π : L p A n q “ Tot ` T P n ´ p´ q| Q n ´ ˘ π ÝÑ Q n ´ . Let us write for short Q : “ Q n ´ , G : “ p T P n ´ p´ qq ˚ “ Ω P n ´ p q . In a more algebro-geometric language the identification of L p A n q with the total space reads: L p A n q “ Spec à d ě S d p G | Q q . Lemma 6.1 implies that κ L p A n q » π ˚ O Q p q , and therefore H i p L p A n , κ q “ à d H i p Q, S d p G qp q| Q q . B. Cohomology on P p M q using Borel-Weil-Bott. We invoke the short exact se-quence of sheaves on P n ´ “ P p M q (6.3) 0 Ñ S d p G qp´ q ¨ q ÝÑ S d p G qp q ÝÑ S d p G qp q| Q Ñ S d p G qp˘ q on P p M q . Lemma 6.4. On P p M q , p´ q The sheaf S d p G qp´ q has H “ S d ´ p M ˚ q (understood as for d “ ) and no othercohomology. p` q – The sheaf S p G qp q has H “ M ˚ and no other cohomology. – The sheaf S p G qp q has H “ Λ p M ˚ q and no other cohomology. The sheaf S p G qp q has no cohomology. – The sheaf S d p G qp q , d ě , has H “ Σ d ´ , p M ˚ q and no other cohomology. Proof:
We use the Borel-Weil-Bott theorem for flag varieties, see [19], Ch.1, § F “ F p M q be the space of complete flags M Ă M Ă ¨ ¨ ¨ Ă M n “ M “ C n , dim p M i q “ i, with the natural projection p : F Ñ P p M q “ t M Ă M u . We denote by M i the tautologicalbundle on F of rank i . To a weight a “ p a , ¨ ¨ ¨ , a n q P Z n (not necessarily dominant) weassociate the line bundle O F p a q “ p M { M n ´ q b a b p M n ´ { M n ´ q b a b ¨ ¨ ¨ b p M { M q b a n ´ b M a n on F . As mentioned in Example 3.3(a), G ˚ “ T P p M q p´ q is the universal quotient bundlewhose fiber at M Ă M is M { M . This implies that S d p G q “ p ˚ O F p , ¨ ¨ ¨ , , ´ d, q . Indeed, taking the space of sections of the line bundle p M { M q bp´ d q “ O P p M { M q p d q on theprojective space P p M { M q or, equivalently, of the pullback of this line bundle to the full flagvariety of M { M , gives S d p M { M q ˚ . This implies that for any b P Z S d p G qp b q “ p ˚ O F p , ¨ ¨ ¨ , , ´ d, b q , and so(6.5) H ‚ p P p M q , S d p G qp b qq “ H ‚ p F, O F p , ¨ ¨ ¨ , , ´ d, b qq . We now recall the procedure of finding H ‚ p F, O F p a qq for a P Z n given by Bott’s theorem.That is, if λ “ p λ ě ¨ ¨ ¨ ě λ n q is a dominant weight, w P S n is a permutation of length ℓ p w q and ρ “ p n, n ´ , ¨ ¨ ¨ , q , then H i ` F, O F p w p λ ` ρ q ´ ρ q ˘ “ Σ λ p M q , if i “ ℓ p w q , , otherwise.Thus to find H ‚ p F, O F p a qq we need to represent a “ w p λ ` ρ q ´ ρ with λ dominant. If sucha representation is impossible, i.e., if a ` ρ has repetitions, then O F p a q has no cohomology.After these preparations, let us establish part p` q of Lemma 6.4. From (6.5) we see thatwe need to find H ‚ p F, O F p a qq , where a “ p , ¨ ¨ ¨ , , ´ d, q , so a ` ρ “ p n, n ´ , ¨ ¨ ¨ , , ´ d, q . d “ d ě
1, a single elementary transposition(length 1) takes a ` ρ to p n, n ´ , ¨ ¨ ¨ , , , ´ d q , then subtracting ρ we get p , ¨ ¨ ¨ , , ´ d q .So in this case the only non-trivial cohomology is H p F, O F p a qq “ Σ , ¨¨¨ , , ´ d p M q “ S d ´ p M ˚ q as claimed.Let us now establish part p´ q of Lemma 6.4. We have a “ p , ¨ ¨ ¨ , , ´ d, ´ q , so a ` ρ “ p n, n ´ , ¨ ¨ ¨ , , ´ d, q . Now, • If d “
0, then a is dominant so we have only H p F, O F p a qq “ Σ , ¨¨¨ , , ´ p M q “ M ˚ . • If d “
1, then a is still dominant, so we have only H p F, O F p a qq “ Σ , ¨¨¨ , , ´ p M q “ Λ p M ˚ q . • If d “
2, we get a ` ρ “ p¨ ¨ ¨ , , , q , a repetition so no cohomology. • If d ě
3, then a ` ρ “ p¨ ¨ ¨ , , ´ d, q which is ordered, by an elementary transposition,to p¨ ¨ ¨ , , , ´ d q . Subtracting ρ , we get p , ¨ ¨ ¨ , , ´ , ´ d q , so the only cohomologyis H p F, O F p a qq “ Σ , ¨¨¨ , , ´ , ´ d p M q “ Σ d ´ , p M ˚ q . Lemma 6.4 is proved.
C. Cohomology on Q . We now finish the proof of Theorem 5.3. Let us display thecohomology (known from Lemma 6.4) of the first two sheaves S d p G qp˘ q in (6.3) in a table(Fig. 1), under these sheaves. Under the third sheaf, S d p G qp q| Q , let us write the conclusionabout its cohomology preceded by the sign “ ñ ”. We note that in the last row, the map H p P p M q , S d p G qp´ qq Ñ H p P p M q , S d p G qp qq induced by multiplication with q , is propor-tional to y d ´ ,q . This follows by invariance, by letting q P S p M ˚ q vary and using the fact(Example 5.1(b)) thatdim Hom GL p M q ` S d ´ p M ˚ q b S p M ˚ q , Σ d ´ , p M ˚ q ˘ “ . The fact that the coefficient of proportionality is non-zero, is implied by the next lemma.
Lemma 6.6.
The map r q : H p P p M q , S d p G qp´ qq ÝÑ H p P p M q , S d p G qp qq induced by multiplication with q , is injective. / / S d p G qp´ q ¨ q / / S d p G qp q / / S d p G qp q| Q / / d “ H ‚ “ H “ M ˚ ñ H “ M ˚ d “ H “ C H “ Λ p M ˚ q ñ H “ C ‘ Λ p M ˚ q d “ H “ M ˚ H ‚ “ ñ H “ M ˚ d ě H “ S d ´ p M ˚ q c ¨ y d ´ ,q / / H “ Σ d ´ , p M ˚ q ñ H “ Coker p y d ´ ,q q . Figure 1: Calculating cohomology on Q Ă P p M q . Proof of Lemma 6.6:
Let ̟ : F Ñ G p , M q be the projection. We denote by M thetautological rank 2 bundle on G p , M q . If E Ă M is a 2-dimensional subspace and r E s P G p , M q is the corresponding point, then ̟ ´ p E q “ F p M { E q ˆ P p E q . By applying theBorel-Weil-Bott theorem to the fibers of ̟ , we find that r q is identified with the morphism S d ´ p M ˚ q “ H ` G p , M q , S d ´ p M q ˚ ˘ ÝÑ H ` G p , M q , Σ d ´ , p M ˚ q ˘ “ Σ d ´ , p M ˚ q induced by the morphism of vector bundles on G p , M q y d ´ ,q | M ´ : S d ´ p M q ˚ q ÝÑ , Σ d ´ , p M ˚ q which, on each fiber, i.e., on each E Ă M as above, is the morphism y d ´ ,q | E : S d ´ p E ˚ q ÝÑ Σ d ´ , p E ˚ q corresponding to the 2-dimensional space E and the quadratic form q | E . This morphismhas been discussed in Remark 5.6, and its kernel is the subspace in S d ´ p E ˚ q formed bypolynomials harmonic (traceless) with respect to q | E . So we are reduced to the followingfact. Lemma 6.7.
Let M be a complex vector space of dimension ě and q P S p M ˚ q be anon-degenerate quadratic form. If f P S d ´ p M ˚ q , d ě , is such that for any -dimensionalsubspace E Ă M , the restriction f | E is harmonic with respect to q | E , then f “ . Proof:
We can assume q to come from a positive definite quadratic form on a real form M R “ R n of M . Then it is enough to prove the lemma under the assumptions that f is17 real homogeneous polynomial of degree d ´ R n and the restriction of f to any realsubspace E is harmonic with respect to q | E . If E is a 2-dimensional real space with a positivedefinite quadratic form, then we can use Euclidean geometry in E . In particular, a harmonicpolynomial homogeneous of degree m is, in polar coordinates p R, φ q a linear combinationof R m cos p mφ q and R m sin p mφ q , and therefore it is invariant under Euclidean rotations by2 π { m in E . So our assumptions on f : R n Ñ R imply that the restriction of f to any 2-plane E Ă R n is invariant under rotations by 2 π {p d ´ q in this plane. If d ě
4, this implies that f p x q depends only on the radius } x } “ q p x q { , so by homogeneity f p x q “ const ¨} x } d ´ ,which contradicts the above trigonometric shape of f | E , so f “ d “
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