Kaluza-Klein theory revisited: projective structures and differential operators on algebra of densities
aa r X i v : . [ m a t h - ph ] D ec KALUZA-KLEIN THEORY REVISITED: PROJECTIVE STRUCTURESAND DIFFERENTIAL OPERATORS ON ALGEBRA OF DENSITIES.
H. M. KHUDAVERDIAN
Abstract.
We consider differential operators acting on densities of arbitrary weights onmanifold M identifying pencils of such operators with operators on algebra of densities of allweights. This algebra can be identified with the special subalgebra of functions on extendedmanifold c M . On one hand there is a canonical lift of projective structures on M to affinestructures on extended manifold c M . On the other hand the restriction of algebra of allfunctions on extended manifold to this special subalgebra of functions implies the canonicalscalar product. This leads in particular to classification of second order operators with useof Kaluza-Klein-like mechanisms. Algebra of densities
In mathematical physics it is very useful to consider differential operators acting ondensities of various weights on a manifold M (see [7] and citations there). We say that s = s ( x ) | Dx | λ is a density of weight λ on M if under change of local coordinates s = s ( x ) | Dx | λ = s ( x ( x ′ )) (cid:12)(cid:12)(cid:12)(cid:12) det (cid:18) ∂x∂x ′ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) λ | Dx ′ | λ , ( λ is an arbitrary real number). We denote by F λ ( M ) the space of densities of weight λ onmanifold M . (The space of functions on M is F ( M ), densities of weight λ = 0.)Densities can be multiplied. If s = s ( x ) | Dx | λ and s = s ( x ) | Dx | λ are densities ofweights λ , λ respectively then s = s · s = s ( x ) s ( x ) | Dx | λ + λ is a density of weight λ + λ . We come to the algebra F ( M ) = ⊕ λ F λ ( M ) of finite linear combinations of densitiesof arbitrary weights. Use a formal variable t instead volume form | Dx | . Thus an arbitrarydensity s = s ( x ) | Dx | λ + · · · + s k | Dx | λ k can be written as a function on x, t which is quasipolynomial on t , s ( x, t ) = s ( x ) t λ + · · · + s k ( x ) t λ k . An arbitrary density s ∈ F ( M ) canbe identified with function P s r ( x ) t λ r on the extended manifold c M , which is quasipolynomialon ‘vertical‘ variable t . There is a natural fibre bundle structure c M → M . Extended manifold c M is the frame bundle of the determinant bundle of M , ( x i , t ) are local coordinates on c M .Changing of local coordinates is:( x i ′ , t ′ ) : x i ′ = x i ′ ( x i ) , t ′ = t ′ ( x i , t ) = det (cid:18) ∂x i ′ ∂x i (cid:19) t . The fibre bundle c M → M can be used for studying projective geometry on M since thereis a canonical construction which assigns to an arbitrary projective connection on manifold Mathematics Subject Classification.
Key words and phrases. differential operator, algebra of densities, pencil of operators, self-adjoint opera-tors, equivariant maps on operators. M an usual affine connection on c M (see the last section). This affine connection on c M canbe used for describing the ‘projective geometry’ on the base manifold M . Such investigationcan be traced to H.Weyl, Weblen and Thomas. On the other hand we will come to additionalgeometrical structures on fibre bundle c M → M if instead algebra of all smooth functionson extended manifold c M we consider only the subalgebra of functions on c M , which arequasipolynomial on vertical variable t , i.e. algebra F ( M ) of densities on M . This algebracan be equipped with the canonical scalar product: If s = s ( x ) | Dx | λ and s = s ( x ) | Dx | λ are two densities with a compact support then h s , s i = R M s ( x ) s ( x ) | Dx | , if λ + λ = 1 , λ + λ = 1 . (1)This construction turns out to be very important tool to study geometry of differentialoperators on M [4], [5]. We give short exposition of these results. Here we come to theseresults and formulate new ones based on alternative approach of Kaluza-Klein-like reduction.2. Differential operators on algebra of densities
Consider linear operator b w such that b w ( s ) = λ s in the case if s is a density of weight λ ,( s ∈ F λ ( M )). If s is a density of weight λ and s is a density of weight λ then b w ( s · s ) = ( λ + λ ) s · s = ( b w s ) · s + s · ( b w s ) . Leibnitz rule is obeyed, b w is first order differential operator on the algebra of densities. Inlocal coordinates ( x i , t ) on c M , b w = t ∂∂t . A differential operator b ∆ on algebra of densities hasappearance b ∆ = b ∆ (cid:0) x, ∂∂x , t, b w (cid:1) in local coordinates. An arbitrary operator b ∆ on algebra ofdensities defines the pencil of operators: b ∆
7→ { ∆ λ } : ∆ λ = b ∆ (cid:12)(cid:12) b w = λ . E.g. the operator b ∆ = A ( b w ) S ik ∂ i ∂ k + B ( b w ) T i ∂ i + C ( b w ) R on algebra F ( M ) defines the pencilof operators { ∆ λ } : ∆ λ = A ( λ ) S ik ∂ i ∂ k + B ( λ ) T i ∂ i + C ( λ ) R . Here A, B, C are polynomialson b w . If for example A = 1 + b w , B = b w and C = 1, then b ∆ is third order operator onthe algebra of densities, which defines the pencil of second order operators. Operators onalgebra of densities can be identified with operator pencils which depend polynomially onpencil parameter λ . Remark 1.
Here we consider only operators which do not change the weight of densities: b ∆ = b ∆( x, ∂∂x , b w ), i.e. for corresponding pencil { ∆ λ } , ∆ λ : F λ ( M ) → F λ ( M ).Canonical scalar product (1) defines adjointness of linear operators. Linear operator b ∆acting on the algebra of densities has an adjoint b ∆ ∗ : h b ∆ s , s i = h s , b ∆ ∗ s i . One can seethat ( x i ) ∗ = x i , ∂ ∗ i = − ∂ i and b w ∗ = 1 − b w .To consider self-adjoint and anti-self-adjoint operators on extended manifold c M is veryilluminating for studying geometry of operators on base manifold M . (See for details [4],[5],[1] and [6].) First order operators and Kaluza-Klein mechanism
Consider an arbitrary first order operator b K such that it does not change the weight ofdensities (see Remark 1) and obeys normalisation condition b K (1) = 0. In local coordinatesit has the following appearance b K = K i ( x ) ∂ i + K ( x ) b w . One can see that its adjoint is equal to b K ∗ = − K i ( x ) ∂ i − ∂ i K i ( x ) + K ( x )(1 − b w ). b K is avector field on extended manifold c M . One can define divergence of this vector field:div b K = − ( K + K ∗ ) = ∂ i K i ( x ) − K ( x ) . (2)Notice that the divergence is defined in spite of the absence of well-defined volume form on c M (see for details [4] and [6]).We see that vector field b K is divergenceless iff it is anti-self-adjoint: b K = − b K ∗ ⇔ div b K = 0 ⇔ b K = K i ( x ) ∂ i + b w∂ i K i ( x ) . One can see that divergenceless vector field b K is a Lie derivative of densities along itsprojection, vector field K on M . An arbitrary vector field X = X i ( x ) ∂ i can be lifted toanti-self-adjoint (i.e. divergenceless) vector field on extended manifold c M , which is nothingbut Lie derivative of densities: X b L X such that for arbitrary s = s ( x ) | Dx | λ , b L X ( s ) = b X ( s ) = (cid:0) X i ( x ) ∂ i + b w∂ i X i ( x ) (cid:1) s ( x ) | Dx | λ = (cid:0) X i ∂ i s ( x ) + λ∂ i X i s ( x ) (cid:1) | Dx | λ . It is useful to consider a connection on a bundle c M → M . It assigns to every vector field X = X i ( x ) ∂ i on M its lifting, the horizontal vector field b X hor = X i ( x ) ∂ i + γ i ( x ) X i ( x ) b w on c M . Connection defines derivation ∇ X on algebra of densities: for s = s ( x ) | Dx | λ ∇ X ( s ) = b X hor ( s ) = (cid:0) X i ( x ) ∂ i s ( x ) + λγ i ( x ) X i ( x ) s ( x ) (cid:1) | Dx | λ . (3)Under changing of local coordinates x i → x i ′ = x i ′ ( x i ) components γ i of connection aretransforming in the following way: γ i ′ = ∂x i ∂x i ′ (cid:18) γ i + ∂∂x i log (cid:18) det ∂x j ′ ∂x j (cid:19)(cid:19) , ( γ i ( x ) | Dx | = ∇ i | Dx | ) . Connection γ i ( x ) defines divergence div γ of vector fields on M , which is equal to divergence(2) of horizontal lifting of this vector field: div γ X = div b X hor = ∂ i X i ( x ) − γ i ( x ) X i ( x ). Remark 2.
Let X = X i ∂ i be a projection on M of a vector field b X = X i ∂ i + b wX on M ,and b X hor be a horizontal lifting of X . Then b X − b X hor = b w ( X − γ i X i ) is a vertical vectorfield and X − γ i X i is a scalar field. Remark 3.
A volume form ρ = ρ ( x ) | Dx | on M naturally defines a connection γ i = − ∂ i log ρ ( x ). A Riemannian metric G = g ik dx i dx k on M naturally defines a volume form ρ = √ det g | Dx | . The corresponding connection γ i = − Γ kik , where Γ ikm are Cristoffel symbolsof Levi-Civita connection of the metric. In this case div γ is a standard divergence operator(with respect to a volume form). H. M. KHUDAVERDIAN Second order operators and Kaluza-Klein reduction
Let b ∆ be an arbitrary second order operator on algebra of densities F ( M ): b ∆ = S ik ( x ) ∂ i ∂ k + 2 b wB i ( x ) ∂ i + b w C ( x ) | {z } second order derivatives + D i ( x ) ∂ i + ˆ wE ( x ) | {z } first order derivatives + F ( x ) . (4)(As always we consider only operators which do not change weight of densities (see remark1).)Principal symbol of this operator is b S = (cid:18) S ik B i B k C (cid:19) , (in coordinates x i , x = log t ) . b S is a contravariant symmetric tensor field on the extended manifold c M . ‘Space components’ S ik of the tensor field b S are components of symmetric contravariant tensor field on M .Operator b ∆ defines a pencil of second order operators { ∆ λ } , ∆ λ = S ik ∂ i ∂ k + . . . , and allthese operators have the same principal symbol S ik .Put normalisation condition F = b ∆(1) = 0 and consider the operator which is adjoint tooperator (4): b ∆ ∗ = ∂ k ∂ i (cid:0) S ik . . . (cid:1) − b w ∗ ∂ i (cid:0) B i + ( . . . ) (cid:1) + ( b w ∗ ) ( C . . . ) − ∂ i (cid:0) D i . . . (cid:1) + b w ∗ E , ( b w ∗ = 1 − b w ) . The condition that operator b ∆ is self-adjoint, b ∆ ∗ = b ∆ implies that b ∆ = S ik ∂ i ∂ k + ∂ k S ki ∂ i + (2 b w − B i ∂ i + ˆ w∂ i B i + b w ( b w − C . (5)Thus self-adjoint second order operator on algebra of densities, which obeys normalisationcondition b ∆(1) = 0 is uniquely defined by its symbol.The geometry of operator (5) was studied in detail in articles [4], [5] and [1]. Here wepresent and analyze these results, using Kaluza-Klein-like mechanism.Kaluza-Klein mechanism defines a connection (gauge field) and Riemannian metric ona base manifold through Riemannian metric on a total space of fibre bundle c M → M .Connection, i.e. the distribution of horizontal hyperplanes (subspaces which are transversalto the fibres) is defined by the condition that these hyperplanes are orthogonal to the fibreswith respect to Riemannian metric in the bundle space.One can slightly alter this mechanism. Contravariant tensor field b S , principal symbolof operator (5) maps 1-forms (covectors) to vectors on c M . Consider the following Kaluza-Klein-like mechanism: take an arbitrary 1-form Ω on c M such that Ω( b w ) = 0, i.e. Ω isproportional to form dx + . . . ( x = log t ), and the following condition is obeyed: vectorfield b S Ω is proportional to vertical vector field b w ( b w = t ∂∂t = ∂∂x ) . This means that for1-form Ω = a ( x )( dx − γ k ( x ) dx k ) the following condition holds: (cid:18) S ik B i B k C (cid:19) (cid:18) − γ k (cid:19) is proportional to vector (cid:18) (cid:19) . (6)This condition canonically defines distribution of horizontal hyperplanes in c M , which are setsof vectors which annihilate the form Ω. (One can take b S Ω = 0 in the case if b S is degenerate.)Every vector field X = X i ( x )( x ) ∂ i on the base M can be lifted to horizontal vector field which annihilates the connection form Ω (see also equation (3)). This construction works ifcondition (6) is obeyed, i.e. in the case if the equation S ik ( x ) γ k ( x ) = B i ( x ) (7)has a solution. In this case second order operator (4) via its principal symbol b S defines aconnection γ k . In the case if S ik is non-degenerate then an equation (7) has unique solution.In this case operator defines uniquely canonical connection and Riemannian metric on thebase.The field B i ( x ) = S ik ( x ) γ k ( x ) can be considered as an upper connection . It follows from(6) that in this case C − B i γ i is a scalar. and C is related with Brans-Dicke function .In general case if the condition (7) is not obeyed then more detailed analysis shows that B i − S ik Γ k is a vector field and C − B i Γ i + S ik Γ i Γ k is a scalar, where Γ i is an arbitraryconnection.The importance of operator (5) is defined by the following uniqueness Theorem: Theorem 1.
Let ∆ be second order operator acting on densities of weight λ , where λ =0 , , / .Then there exists a unique self-adjoint operator b ∆ ( b ∆ ∗ = b ∆ ) which obeys the followingconditions • b ∆ (cid:12)(cid:12) ˆ w = λ = ∆ , • normalisation condition b ∆(1) = 0 .In other words there exists unique self-adjoint normalised pencil of second order operatorswhich passes through a given operator. This Theorem was formulated and proved in [4] (see also [5]).
Example 4.1.
Consider second order operator ∆ λ = L X ◦ L Y , where X , Y are arbitraryvector fields, L X , L Y are Lie derivatives of densities of weight λ along vector fields X , Y respectively. Construct the following operator on algebra of densities: b ∆ = 12 (cid:16)b L X b L Y + b L Y b L X (cid:17) + 12 (cid:18) b w − λ − (cid:19) (cid:16) b L X b L Y − b L Y b L X (cid:17) . This operator is obviously self-adjoint operator and it passes through operator ∆ λ . One cansee that it is equal to b ∆ = b L X b L Y + (cid:18) b w − λ λ − (cid:19) b L [ X , Y ] . Let ∆ λ = A ij ∂ i ∂ j + A i ∂ i + A be an arbitrary second order operator acting on space ofdensities of a given weight λ , ( λ = 0 , / , λ if for upper connection B i and Brans-Dicke function CB i = A i − ∂ k A ki λ − , C = Aλ ( λ − − ∂ i A i − ∂ i ∂ k A ki ( λ − λ − . The upper connection B i is induced by a connection γ i ( B i = A ik γ k ) iff equation (7) hasa solution (for S ik = A ik ). The condition (7) defines this special property of second orderoperators on densities. It is interesting to analyze its geometrical meaning. H. M. KHUDAVERDIAN Thomas bundle and projective geometry
The canonical constructions studied in the previous sections were successfully performedsince we consider not the algebra of all functions on extended manifold c M , but only functionswhich are quasipolynomial on vertical variable t , since scalar product (1) is not well-definedon algebra of all (smooth) functions on x, t . Nevertheless in general case for fibre bundle c M → M there exists the remarkable construction which assigns to projective class of connectionson M the affine connection on c M . This construction is due to T.Y.Thomas [8]. (See also[2] and [3]). The bundle c M → M sometimes is called Thomas bundle. Now we sketch thisconstruction.We say that two symmetric affine connections ∇ and ˜ ∇ on manifold M belong to the sameprojective class [ ∇ ] = [ ˜ ∇ ] if˜ ∇ − ∇ = ˜Γ ikm − Γ ikm = t k δ im + t m δ ik , ( t i is covector) , where ˜Γ ikm and Γ ikm are Christoffel symbols of connections ˜ ∇ and ∇ respectively. Equivalenceclass of symmetric connections is projective connection. (Projective connection in particulardefines non-parametrised geodesics: two symmetric affine connection belong to the sameclass iff they have the same non-parameterised geodesics.)For affine connection ∇ on n -dimensional manifold M with Christoffel symbols Γ ikm onecan consider symbols Π ikm ( ∇ ) = Π ikm = Γ ikm + 1 n + 1 (cid:0) γ k δ im + γ m δ ik (cid:1) , (8)where γ i = − Γ kik define connection on densities on M (see also remark 3). Two symmetricconnections ∇ , ˜ ∇ belong to the same projective class iff Π ikm ( ∇ ) = Π ikm ( ˜ ∇ ).Let [ ∇ ] be a projective class of symmetric connections on n -dimensional manifold M . ThenThomas construction assigns to this projective class [ ∇ ] the symmetric affine connection b ∇ on the extended manifold c M . with following Christoffel symbols b Γ ikm = Π ikm , b Γ km = 1 n + 1 ( ∂ r Π rkm − Π rsk Π srm ) , b Γ ik = − δ ik n + 1 , b Γ i = b Γ i = 0 , b Γ = − n + 1 . Here Π ikm are symbols (8) corresponding to Christoffel symbols of a connection in the class[ ∇ ]. (We use local coordinates ( x i , x ) = ( x i , log t )) in the extended space.) Acknowledgments
This article is based on my talk at the conference “The ModernPhysics of Compact Stars and Relativistic Gravity” (Yerevan, Septemebr 2013). I am gratefulto organisers of this conference for financial support, and also to Engineering and PhysicalSciences Research Council (EPSRC) for financial support via the Mathematics PlatformGrant (MAPLE) EP/IO1912X/1. I am grateful to T.Voronov for fruitful discussions.
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From Schwarzian Deriv-ative to the Cohomology of Diffeomorphism Groups. (Cambridge University Press, 2005).[8] T.Y.Thomas “A projective theory of affinely connected manifolds” Math. Zeit. , 723-733 (1926) School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL,UK
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