aa r X i v : . [ m a t h - ph ] O c t Spectral geometry of symplectic spinors
Dmitri Vassilevich
Center of Mathematics, Computation and Cognition, Universidade Federal do ABC09210-580, Santo Andr´e, SP, Brazil
Abstract
Symplectic spinors form an infinite-rank vector bundle. Dirac operators on this bundle were con-structed recently by K. Habermann. Here we study the spectral geometry aspects of these operators.In particular, we define the associated distance function and compute the heat trace asymptotics.
According to the noncommutative geometry approach, geometry is defined by spectral triples. Thatis, geometry essentially becomes spectral geometry of natural Dirac type operators. This approach iswelcomed by physicists since it bridges up the differences between classical and quantum geometries. Thisalso moves geometry towards traditional areas of Mathematical Physics. Many links with Quantum FieldTheory and even particle physics have been established, see [4]. The spectral geometry of Riemannianmanifolds has been studied in detail, while the spectral geometry of symplectic manifolds remains alargely uncharted area.Quantization starts with a Poisson structure, that becomes a symplectic structure in the non-degeneratecase. Therefore, from the point of view of Quantum Theory, symplectic manifolds are more importantthan the Riemannian ones. The purpose of this work is to extend the spectral geometry approach to thesymplectic spinors. We address two important aspects, namely the spectral distance function and theheat trace asymptotics.The symplectic spinors were introduced by B. Kostant [15]. The Dirac operator on symplectic spinorsand the corresponding Laplacian were defined by K. Habermann [7, 8], who has also studied their basicproperties. A nice overview is the monograph [9], whose conventions and notations we mostly follow inthis work. The works [7, 8] defined two Dirac operators, D and e D . Since the symplectic spinor bundlehas an infinite rank, neither of these two operators is a Dirac operator in the noncommutative geometrysense. Moreover, the relevant Laplacian P appears to be the commutator of D and e D (rather than thesquare of D or e D ). Therefore, the basic notions of spectral geometry cannot be immediately applied.We shall show that despite the difficulties described above, there exists a modification of the standarddistance formula of noncommutative geometry that reproduces the geodesic distance on base manifold M through symplectic Dirac operators. This is probably the most surprising result of this work.If the base manifold M is almost hermitian, the operator P leaves some subbundles Q Jl of the sym-plectic spin bundle Q invariant. These subbundles are of finite rank. Let P l be a restriction of Laplacian P to Γ( Q Jl ), then P l is also a Laplace type operator. This will allow us to develop the theory of heat traceasymptotic for P l , identify corresponding invariants and compute a couple of leading terms in the asymp-totic expansion. Even more detailed information on the heat trace can be obtained when dim M = 2 andin the particular case M = CP . We shall consider these cases as examples in the last section of thispaper. In this Section we collect some basic facts that will be useful later. In what refers to symplectic spinorswe mostly follow [9].
Consider a symplectic manifold (
M, ω ), dim M = 2 n , with ω being a symplectic form, equipped withan almost complex structure J and a Riemannian metric g related through g ( X, Y ) = ω ( X, JY ) for1 , Y ∈ T M . Let the symplectic connection ∇ be Hermitian, which implies ∇ g = 0 , ∇ J = 0 . (1)The symplectic form ω is also covariantly constant, ∇ ω = 0. This makes M a Fedosov manifold [5].Almost hermitian manifolds admit unitary tangent frames in that both the metric and the symplecticform have a canonical form [14]. Such frames form a principal U ( n ) bundle U ( M ) over M . We shallfollow the notations of [9] and write such frames as ( e , . . . , e n ) = (ˆ e , . . . , ˆ e n , ˆ f , . . . , ˆ f n ). They satisfy g ( e i , e j ) = δ ij ,ω (ˆ e i , ˆ f j ) = δ ij , ω (ˆ e i , ˆ e j ) = ω (ˆ f i , ˆ f j ) = 0 . (2)The almost complex structure J acts on the basis as J ˆ e j = ˆ f j , J ˆ f j = − ˆ e j . (3)It is interesting and useful to follow certain analogies with the Yang-Mills theory. By linearity, thereis u ∈ Γ( T ∗ M × End U ( M )) such that for any vector field X ∇ X e i = ( Xu ) e i = X µ u µ ij e j . (4)Whenever it cannot lead to a confusion we shall use the Einstein conventions of summation over repeatedindices. The Greek indices µ, ν, .. are vector indices corresponding to a local coordinate chart. Theyare introduced to make connections with the Yang-Mills more explicit. It is easy to check that ∇ is asymplectic hermitian connection iff u is antisymmetric and commutes with J , i.e. iff u belongs to the2 n -dimensional real representation of u ( n ). Eq. (4) allows to express the Christoffel symbol through thevectors of unitary frame e i and the U ( n ) connection one-form u . One has the following expression forthe torsion: T ( e i , e j ) = ( e i u ) e j − ( e j u ) e i − [ e i , e j ] . (5)Also the curvature of ∇ can be expressed through u :( R ( e i , e j ) e k ) ρ = e νi e µj F µνkm e ρm , (6)where F µνkm = − ∂ µ u νkm + ∂ ν u µkm + [ u µ , u ν ] km (7)is the Yang-Mills type curvature associated to u µ . We remind that the generators of u ( n ) algebra arelabeled by the pairs of indices ( k, m ).The heat trace asymptotics of Laplace type operators are usually expressed in terms of the Levi-Civitaconnection ∇ LC and corresponding curvatures. This connection is related to ∇ by the text-book formula: g ( ∇ X Y, Z ) = g ( ∇ LC X Y, Z ) + (cid:2) g ( T ( X, Y ) , Z ) − g ( T ( X, Z ) , Y ) − g ( T ( Y, Z ) X (cid:3) (8)The following vector, T , and covector, τ , fields are associated with the torsion: T = n X j =1 T (ˆ e j , ˆ f j ) , τ ( X ) = n X k =1 g ( T ( e k , X ) , e k ) . (9)There is a useful relation which involves the Riemann scalar curvature ¯ ρ : n X j,k =1 g ( R ( e j , e k ) e j , e k ) = − ¯ ρ + 2 n X j =1 ∇ LC e j τ ( e j ) + n X j =1 τ ( e j ) − n X j,l (cid:2) g ( T ( T ( e j , e l ) , e l ) , e j ) + g ( T ( e j , e l ) , T ( e j , e l )) (cid:3) . (10)2 .2 Symplectic spinors and relevant operators Let us remind that the metaplectic group
M p (2 n, R ) is a two-fold covering of the symplectic group Sp (2 n, R ). A metaplectic structure is a principal M p (2 n, R ) bundle (together with a morphism thatensures consistency with the symplectic structure). The metaplectic group is represented in the spaceof square integrable functions L ( R n ). By using this representation one can define a bundle associatedwith the metaplectic structure, which is the symplectic spinor bundle Q . For the sections of this fiberbundle, ϕ, ψ ∈ Γ( Q ), one has a fiber scalar product, h ϕ, ψ i x , x ∈ M , and a fiber norm k ϕ k x = h ϕ, ϕ i x .By integrating h ϕ, ψ i x over M one obtains a scalar product on Γ( Q ).For X ∈ Γ( T M ) and ϕ ∈ Γ( Q ) one defines the symplectic Clifford multiplication X · ϕ satisfying( X · Y − Y · X ) · ϕ = − ı ω ( X, Y ) ϕ (11)From now on, ı ≡ √− Q splits into an orthogonal sum of finite rank subbundles Q Jl , l =0 , , , . . . , rank Q Jl = ( l + n − l !( n − . (12)There is an important operator, H J , acting on ϕ ∈ Γ( Q ) H J ϕ = 12 n X j =1 e j · e j · ϕ . (13)The operator H J equals to a constant q l on each of Q Jl , and q l = − (cid:0) l + n (cid:1) . (14)In the quantum mechanical language, the Clifford multiplication by ˆ e j may be thought of as a canonicalcoordinate operator, while ˆ f j · may be thought of as a conjugate momentum operator. Then H J is minusthe Hamiltonian of n -dimensional harmonic oscillator, while the ladder operators are L ( ± ) j ϕ = (ˆ f j ∓ ıˆ e j ) · ϕ . (15)For any ϕ ∈ Γ( Q J ) L (+) ϕ = 0 . (16)Any symplectic connection ∇ on M induces a covariant derivative on Q that will be denoted by thesame letter ∇ . We shall need the relation ∇ X ( Y · ϕ ) = ∇ X Y · ϕ + Y · ∇ X ϕ (17)between the derivative and the Clifford multiplication and the corresponding curvature R Q ( X, Y ) ϕ ≡ ∇ X ∇ Y ϕ − ∇ Y ∇ X ϕ − ∇ [ X,Y ] ϕ = − i n X j =1 R ( X, Y ) e j · J e j · ϕ . (18)Given a unitary frame ( e , . . . , e n ) one defines a pair of Dirac operators D ϕ = − n X j =1 J e j · ∇ e j ϕ , e D ϕ = n X j =1 e j · ∇ e j ϕ . (19)Note, that the formula for D may be rewritten in the form D ϕ = n X j =1 (cid:0) ˆ e j · ∇ ˆ f j ϕ − ˆ f j · ∇ ˆ e j ϕ (cid:1) (20)3hich does not use the metric or the almost complex structure.An associated second order operator is defined through the commutator P = ı[ e D , D ] . (21)The principal symbol of P is given by the inverse metric. The operator P leaves the subbundles Q Jl invariant, P : Γ( Q Jl ) → Γ( Q Jl ), though neither e D nor D have this property. A restriction of P to Γ( Q Jl )will be denoted by P l We shall need some facts from the theory of asymptotic expansion of the heat trace associated withLaplacians [6] (see also [13, 18]).Let V be a finite rank vector bundle over a compact Riemannian manifold M without boundary. Let P be a Laplace type operator on Γ( V ). Then(1) exist a unique endomorphism ¯ E and a unique connection ¯ ∇ of V such that P = − ( ¯ ∇ + ¯ E ) , (22)where the square in ¯ ∇ is calculated with the Riemannian metric on M and includes the metric Christoffelsymbol.(2) The heat trace K ( P, t ) := Tr (cid:0) exp( − tP ) (cid:1) , t ∈ R + (23)exists and admits a full asymptotic expansion K ( P, t ) ≃ ∞ X k =0 t k − n a k ( P ) (24)as t → +0. Here 2 n = dim M .(3) The heat trace coefficients a k are locally computable. Namely, each a k is given by the integral over M of the bundle trace of a local invariant polynomial constructed from the endomorphism ¯ E , from theRiemann tensor ¯ R ijkl , the curvature ¯Ω ij of ¯ ∇ , and their derivatives. In particular, a ( P ) = (4 π ) − n Z M tr V ( I ) , (25) a ( P ) = (4 π ) − n Z M tr V (cid:0) E + ¯ ρ (cid:1) , (26) a ( P ) = (4 π ) − n Z M tr V (cid:0)
60 ¯ ρ ¯ E + 180 ¯ E + 5 ¯ ρ − ij Ric ij + 2 ¯ R ijkl ¯ R ijkl + 30 ¯Ω ij ¯Ω ij (cid:1) . (27)The Ricci tensor Ric ij = ¯ R lijl and the scalar curvature ¯ ρ = Ric jj are defined in such a way that ¯ ρ = 2on the unit S . Summation over the repeated indices is understood.Note, that the expansion (24) contains even-numbered coefficients and integer powers of t only. Odd-numbered coefficients appear e.g. on manifolds with boundaries.Our purpose is to calculate the coefficients a k and relate them to geometric invariants of the sym-plectic spinor bundle. Let us start with basic definitions related to the distance function in noncommutative geometry, see [17]for a brief introduction. Consider a spectral triple ( A , H, D ) consisting of an algebra A , acting on aHilbert space H by bounded operators, and of a Dirac operator D . The commutator [ D, a ] has to be4ounded for all a ∈ A (or at least the set of a for which [ D, a ] is bounded has to be dense in A ). Onecan define a distance between two states x and y on the algebra A by the formula [3] d ( x, y ) = sup a ∈A {| a ( x ) − a ( y ) | : k [ D, a ] k ≤ } . (28)For a commutative spectral triple, A is the algebra C ( M ) of continuous functions on a compactRiemannian spin c manifold M , H is the space of square-integrable spinors, and D is the canonical Diracoperator. The pure states x correspond to points on M with x : a a ( x ) being the evaluation map. Thedistance (28) coincides with the geodesic distance on M . Since D is the canonical Dirac operator[ D, a ] = − ı γ ( da ) , (29)where γ is a composition of the orthogonal Clifford multiplication and isomorphism between tangent andcotangent bundles made with the inverse Riemannian metric g − . In short, for two one forms α and β , γ ( α ) γ ( β ) + γ ( β ) γ ( α ) = 2 g − ( α, β ). Fiberwise, γ ( α ) is a hermitian matrix. The restriction k [ D, a ] k ≤ | g − ( da ∗ , da ) | ≤ . (30)Complexity or reality of the function a plays no role here. We shall consider real functions in whatfollows.In the context of symplectic spinors one may take the same A = C ( M ), the Hilbert space may beformed by sections of the symplectic spinor bundle. However, the commutators [ D , a ] and [ e D , a ] arepractically never bounded, so that Eq. (28) with D or e D instead of D does not define any interestingdistance on M . Therefore, we return again to the usual spin case and replace the condition k [ D, a ] k ≤ ψ of spin bundle that has a unit fiber norm at each point of M : 1 = k ψ k x = ψ † ( x ) ψ ( x ). Letus take a real a = a ∗ in A and compute k [ D, a ] ψ k x = ψ † ( x ) γ ( da ) γ ( da ) ψ ( x ) = g − ( da, da ) ψ † ( x ) ψ ( x ) = g − ( da, da ) . This shows that the condition k [ D, a ] ψ k x ≤ x on M may be used instead of the originalrestriction on the norm k [ D, a ] k ≤ M .A similar construction in the symplectic case goes as follows. Take a section ϕ of Q J with a constantfiber norm k ϕ k x = 2 everywhere on M . Then,[ e D , a ] ϕ = n X j =1 (cid:16) (ˆ e j a )ˆ e j · ϕ + (ˆ f j a )ˆ f j · ϕ (cid:17) = n X j =1 (cid:16) (ˆ e j a ) (cid:0) − ı2 L ( − ) j (cid:1) + (ˆ f j a ) L ( − ) j (cid:17) ϕ = n X j =1 (cid:16) − ı(ˆ e j a ) + (ˆ f j a ) (cid:17) L ( − ) j ϕ . (31)Furthermore, ( L ( − ) j ϕ , L ( − ) k ϕ ) x = ( ϕ , L (+) j L ( − ) k ϕ ) x = ( ϕ , ( L ( − ) j L (+) j + 2) ϕ ) x δ jk = ( ϕ , ϕ ) x δ jk = δ jk , (32)where we used Eq. (16). Finally, k [ e D , a ] ϕ k x = n X j =1 ( e j a ) = g − ( da, da ) . (33)By collecting everything together we arrive at the following Proposition 3.1.
Let A , ϕ and e D be as defined above. Then d ( x, y ) = sup a ∈A {| a ( x ) − a ( y ) | : k [ e D , a ] ϕ k p ≤ , ∀ p ∈ M } . (34) is the geodesic distance between two points x and y on M . emark The same distance is obtained if one uses everywhere D instead of e D , which can be easilyverified. However, this is true only in the rather restrictive setup used in this work. In general, onedoes not need a Riemannian metric to define D [7–9]. It is enough to take a symplectic manifold witha metaplectic structure and a symplectic connection. Then, ( e , . . . , e n ) = (ˆ e , . . . , ˆ e n , ˆ f , . . . , ˆ f n ) in Eq.(20) may be any symplectic frame. The distance (34) (with D in place of e D ) defines a metric on M . P l . Generic case To compute the heat trace asymptotics one has to rewrite P l in the canonical form (22), compute thecorresponding invariants and calculate the traces. Our starting point is the Weitzenb¨ock formula [9] for P , P ϕ = ∇ ∗ ∇ ϕ + ı n X jk J e j · e k · (cid:0) R Q ( e j , e k ) ϕ − ∇ T ( e j , e k ) ϕ (cid:1) − ∇ J T ϕ . (35)Here ∇ ∗ : Γ( T ∗ M ⊗ Q ) → Γ( Q ) is the formal adjoint operator of the spinor covariant derivative ∇ . It iseasy to see that ∇ ∗ ∇ = −∇ in the notations of Eq. (22). After some algebra we obtain the quantitiesappearing in the canonical form (22) of the Laplacian¯ ∇ X ϕ = ∇ X ϕ + g ( X, v ) ϕ (36) v = ı2 2 n X j,k =1 T ( e j , e k ) J e j · e k · + J T (37)¯ E = − div LC v − g ( v, v ) − ı n X jk J e j · e k · R Q ( e j , e k ) (38)In the last formula the divergence corresponds to the Levi-Civita connection. In local terms, div LC v = g µν ∇ LC µ v ν .To calculate the traces we have to define the u ( n ) representations corresponding to the objects ap-pearing the formulas above. Just looking at the expression (18) for the curvature and at the relations (6),(7) to the Yang-Mills field strength one may conjecture that the combinations e j · J e k · are generators of u ( n ). Let us show that this is indeed the case.Let A ∈ u ( n ) be given by a matrix A jk in the 2 n -dimensional real defining representation. This meansthat A jk is a real antisymmetric 2 n × n matrix which satisfies A jk = A j + n,k + n and A j,k + n = − A j + n,k for j, k ≤ n . Then r Q ( A ) = ı2 2 n X j,k =1 A jk e k · J e j · (39)is a representation of A on the symplectic spinors. Indeed,[ r Q ( A ) , r Q ( B )] ϕ = (cid:0) ı2 (cid:1) [ A jk e k · J e j , B lm e m · J e l ] · ϕ = ı2 [ A, B ] jk e k · J e j · ϕ = r Q ([ A, B ]) ϕ . (40)This representation is, of course, reducible. The action is fiberwise, so that it is enough to understand r Q on each fiber. Lemma 4.1.
On the fibers of Q Jl the representation r Q is equivalent to an su ( n ) representation with theDynkin indices ( l, , . . . , , while the u (1) charge is q l .Proof. In the 2 n -dimensional real defining representation non-zero matrix elements of the Cartan gener-ators of u ( n ) are ( K j ) j,j + n = − ( K j ) j + n,j = 1, j = 1 , . . . , n . Therefore, r Q ( K j ) ϕ = ı2 (ˆ e j · ˆ e j + ˆ f j · ˆ f j ) · ϕ (41)6ocally, a fiber of Q Jl can be viewed as a linear space spanned by products h α ( x ) · · · h α n ( x n ) of theHermite functions h α ( x ) with α + · · · + α n = l and with Clifford multiplications by ˆ e j and ˆ f j representedby ix j and ∂ j , respectively [9]. Hence, r Q ( K j ) h α j ( x j ) = − ı( α j + ) h α j ( x j ). Let us take the Cartangenerators corresponding to ordered positive simple roots of su ( n ) as K − K , K − K , ..., K n − − K n . Bycalculating the eigenvalues of this generators on h l ( x ) h ( x ) · · · h ( x n ), we conclude that this monomialis the highest weight vector of the representation ( l, , . . . , Q Jl is justthis representation and nothing else, it is enough to compare the dimensions (cf [9]),dim ( l, , . . . ,
0) = ( l + n − l !( n − Q Jl . (42)The u (1) charge is, up to the imaginary unit, the eigenvalue of the u (1) generator n X j =1 r Q ( K j ) = i H J , (43)which is iq l .Let us define a projector Π on the u (1) generatorΠ r Q ( A ) = ı H J n n X j =1 A j,j + n (44)and a pull-back of Π to the 2 n -dimensional real representation, Π r Q ( A ) = r Q (Π A ). Corollary 4.2. tr l (cid:0) r Q ( A ) (cid:1) = ı q l rank Q Jl n n X j =1 A j,j + n (45)tr l (cid:0) Π r Q ( A )Π r Q ( B ) (cid:1) = − q l rank Q Jl n n X j,k =1 A j,j + n B k,k + n (46)tr l (cid:0) (1 − Π) r Q ( A )(1 − Π) r Q ( B ) (cid:1) = ( l + n )!4( l − n + 1)! tr (cid:0) (1 − Π) A (1 − Π) B (cid:1) (47) Proof.
The first two lines above immediately follow from the fact that on each Q Jl the operator H is anidentity matrix times q l . To get the last line one uses that the trace forms in irreducible representationsof a simple Lie algebra are proportional between themselves. I.e., for A and B being matrices in the 2 n dimensional real representation of su ( n )tr l (cid:0) r Q ( A ) r Q ( B ) (cid:1) = c ( n, l ) tr (cid:0) AB (cid:1) . (48)Next, we take the quadratic Casimir element C and calculate its’ trace once by using the relation above,and then by using the fact that C in any irreducible representation is a unit matrix times the eigenvalues C ( n, l ), tr l C = µ n ( n − c ( n, l ) = rank Q Jl C ( n, l ) , (49)where n − su ( n ) and µ n is a normalization coefficient. Hence, c ( n, l ) = c ( n,
1) rank Q Jl C ( n, l )rank Q J C ( n, . (50)The value c ( n,
1) = can be easily recovered by considering the realization on products of Hermitefunctions, C ( n, l ) = l ( n + l )( n − /n is to be found in any textbook, see [1] for instance, and dimensionsof relevant representations have been given above. By collecting everything together, one arrives at(47). 7xplicitly, tr (cid:0) (1 − Π) A (1 − Π) B (cid:1) = n X j,k =1 A jk B kj + 2 n n X i,l =1 A i,i + n B l,l + n . (51)With these formulas,we calculate traces of the terms appearing in Eq. (38) for ¯ E tr l g ( v, v ) = rank Q Jl (cid:18) − q l n + ( l + n ) l n ( n + 1 (cid:19) g ( T , T ) − ( l + n )!4( l − n + 1)! n X j,k =1 g ( T ( e j , e k ) , T ( e j , e k )) , (52)tr l (cid:0) − ı n X jk =1 J e j · e k · R Q ( e j , e k ) (cid:1) = (cid:18) − q l rank Q Jl n + ( l + n )! n ( l − n + 1)! (cid:19) n X i,l =1 g ( R ( e i , e i + n ) e l , e l + n ) − ( l + n )!2( l − n + 1)! n X j,k =1 g ( R ( e j , e k ) e j , e k ) . Let us introduce a short hand notation α ( n, l ) ≡ l ( l + n ) n ( n + 1)Then the first two heat trace coefficients read a ( P l ) = (4 π ) − n rank Q Jl vol ( M ) (53) a ( P l ) = (4 π ) − n rank Q Jl Z M h(cid:16)
16 + α ( n, l )2 (cid:17) ¯ ρ + (cid:16) α ( n, l ) − q l n (cid:17) n X i,j =1 g ( R (ˆ e i , ˆ f i )ˆ e j , ˆ f j )+ α ( n, l ) n X i,j =1 (cid:16) g ( T ( T ( e i , e j ) , e i ) , e j ) + 38 g ( T ( e i , e j ) , T ( e i , e j )) (cid:17) + (cid:16) −
14 + q l n − α ( n, l )2 n (cid:17) g ( T , T ) − α ( n, l ) n X j =1 τ ( e j ) i (54) Remarks.
1. The coefficients (53) and (54) carry no dependence on the metaplectic structure. Thestructure of invariants (36) - (38) tells us that higher heat kernel coefficients have the same property. Inthe language of Kac [12] this means that through the heat trace expansion of P l one can hear the shapeof symplectic almost hermitian manifolds, but not of the metaplectic structures.2. In general, the operator P is not self-adjoint. However, the coefficients (53) and (54) are real. We donot expect this property to hold for higher terms in the heat trace expansion.3. One may define a family of spectral actions S l = Tr (cid:0) χ ( P l / Λ (cid:1) similarly to [2] with a cut-off function χ and a scale parameter Λ. The large Λ expansion of S l is given by the heat trace asymptotics thosestructure differs considerably for the standard case of spin Dirac operators with torsion, cf. [10, 11].Note that in four dimensions the spectral action for the spin Dirac operator is restricted by the chiralsymmetry [11], that is not present in the symplectic case. C P M is a K¨ahler manifold, so that the torsion vanishes. This reduces considerably8he combinatorial complexity of the heat trace asymptotic expansion and will allow to compute more heattrace coefficients. We also like to mention compact expressions [16] for the heat trace asymptotics of thescalar Laplace operator on K¨ahler manifolds.In two dimensions, the expression for curvature simplifies R Q ( X, Y ) ϕ = − ı( R ( X, Y )ˆ e , ˆ f ) H J ϕ , (55)where ˆ e ≡ ˆ e , ˆ f ≡ ˆ f . For vanishing torsion P ϕ = ∇ ∗ ∇ ϕ + ı X j,k =1 J e j · e k · R Q ( e j , e k ) ϕ = −∇ ϕ − ρ (cid:0) H J (cid:1) ϕ , (56)where ρ = ¯ ρ is the scalar curvature.Let us consider the operator P l . We remind that H J | Q Jl = − ( l + ) ≡ q l . The heat trace asymptoticsare characterized by the following Proposition 5.1.
The operator P l has the form (22) with ¯ ∇ = ∇ , E = ρq l , Ω ij = ı2 q l ρω ij . (57) The heat kernel coefficients read a ( P l ) = (4 π ) − vol M , (58) a ( P l ) = (4 π ) − (1 + 6 q l ) Z M ρ , (59) a ( P l ) = (4 π ) − (2 + 15 q l + 60 q l ) Z M ρ . (60) Proof.
First, we recall that in two dimensions there is only one indepedent component of the Riemanntensor, so that if ( R (ˆ e , ˆ f )ˆ e , ˆ f ) = r , then Ric (ˆ e , ˆ e ) = Ric (ˆ f , ˆ f ) = − r and ¯ ρ = ρ = − r . Then, ¯ ∇ = ∇ byinspection, so that the corresponding curvature is just R Q . By Eq. (55), we haveΩ(ˆ e , ˆ f ) ϕ = ı2 ρ H J ω (ˆ e , ˆ f ) ϕ , that yields the 2nd equation in (57). The first equation there follows from (56). Substitutions in (25) –(27) lead to the desired result.Let us restrict our attention further by taking M = CP . Then the eigenvalues λ l,j of P l and theirdegeneracies m l,j read (see [9, Proposition 6.3.5]) λ l,j = 4( l + j + 1) − l + 1) − , m l,j = 2( l + j + 1) , j = 0 , , , . . . (61)To calculate the heat trace asymptotics one has to evaluate the asymptotic expansion for K ( P l , t ) = ∞ X j =0 m l,j e − tλ l,j = e t (3(2 l +1) +1) ∞ X k = l +1 k e − tk . (62)To this end one may use the Euler-Maclaurin formula q X k = m f ( k ) = Z qm f ( x ) dx + 12 ( f ( q ) + f ( m )) + ∞ X i =1 B i (2 i )! (cid:0) f (2 i − ( q ) − f (2 i − ( m ) (cid:1) (63)with f ( x ) = 2 xe − tx . Since f (2 i − = O ( t i − ), only a finite number of terms on the right hand sideof (63) contribute to any given order of the expansion. Here B = , B = − , . . . are the Bernoullinumbers. We have ∞ X k = m k e − tk ≃ e − m t (cid:16) t + (cid:0) m − (cid:1) + t (cid:0) m − (cid:1) + O ( t ) (cid:17) . (64)9onsequently, K ( P l , t ) ≃ t + (cid:16) − m + 2 m (cid:17) + t (cid:16) − m + 14 m − m + 8 m (cid:17) + O ( t ) , (65)where m = l + 1. This expansion gives the values of a , a and a for M = CP .The CP with Fubini-Study metric is isometric to S with the radius 1 /
2. Consequently,vol CP = π, Z CP ρ = 8 π , Z CP ρ = 64 π . (66)Therefore, (58) - (60) are consistent with (65).The eigenvalues of P l on other complex projective spaces can be found in [19]. Acknowledgments
The author is grateful to Rold˜ao da Rocha for many fruitful discussions. This work was supported inpart by CNPq, projects 306208/2013-0 and 456698/2014-0, and FAPESP, project 2012/00333-7.
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