Quantum moment map and obstructions to the existence of closed Fedosov star products
QQUANTUM MOMENT MAP AND OBSTRUCTIONS TO THE EXISTENCEOF CLOSED FEDOSOV STAR PRODUCTS
AKITO FUTAKI AND LAURENT LA FUENTE-GRAVY
Abstract.
It is shown that the normalized trace of Fedosov star product for quantum momentmap depends only on the path component in the cohomology class of the symplectic formand the cohomology class of the closed formal 2-form required to define Fedosov connections(Theorem 1.3). As an application we obtain a family of obstructions to the existence of closedFedosov star products naturally attached to symplectic manifolds (Theorem 1.5) and K¨ahlermanifolds (Theorem 1.6). These obstructions are integral invariants depending only on thepath component of the cohomology class of the symplectic form. Restricted to compact K¨ahlermanifolds we re-discover an obstruction found earlier in [29]. Introduction A star product [1] on a Poisson manifold M of dimension n = 2 m is an associative product ∗ on the space C ∞ ( M )[[ ν ]] of formal power series in ν with coefficients in C ∞ ( M ) such that if wewrite f ∗ g := ∞ (cid:88) r =0 ν r C r ( f, g ) for f, g ∈ C ∞ ( M )then(1) the C r ’s are bidifferential ν -linear operators,(2) C ( f, g ) = f g and C ( f, g ) − C ( g, f ) = { f, g } ,(3) the constant function 1 is a unit for ∗ (i.e. f ∗ f = 1 ∗ f ).Recall that a symplectic form ω is a closed nondegenerate 2-form. It induces the Poissonbracket { f, g } := − ω ( X f , X g ) for f, g ∈ C ∞ ( M ) and vector field X f uniquely determined by ı ( X f ) ω = df . Any star product ∗ on a symplectic manifold ( M, ω ) admits a unique normalizedtrace Tr : C ∞ ( M )[[ ν ]] → R [ ν − , ν ]]satisfying Tr([ f, g ] ∗ ) = 0 . Here normalization means as follows. On a contractible Darboux chart U we have an equivalence B : ( C ∞ ( U )[[ ν ]] , ∗ ) → ( C ∞ ( U )[[ ν ]] , ∗ Moyal ) of ∗| C ∞ ( U )[[ ν ]] with the Moyal star product ∗ Moyal satisfying Bf ∗ Moyal Bg = B ( f ∗ g ) . The normalization condition is(1) Tr( f ) = 1(2 πν ) m (cid:90) M Bf ω m m ! . It is known that the trace of a star product can always be written as an L -pairing with anessentially unique formal function ρ ∈ C ∞ ( M )[ ν − , ν ]], called the trace density. A star productis said to be (strongly) closed if the integration functional is a trace, c.f. [7]. Equivalently, itmeans that the trace density is a formal constant, i.e. ρ ∈ R [ ν − , ν ]]. If such a closed starproduct exists, it is possible to define its character [7], a cyclic cocycle in cyclic cohomology. See[13], [32], [25] for more on the trace and the trace density. a r X i v : . [ m a t h . S G ] J u l here are known constructions of star products [10], [12], [33], [27]. In this paper we considerFedosov star product constructed in [12] on symplectic manifolds. The Fedosov star product isdefined given a symplectic connection ∇ and a closed formal 2-form Ω ∈ ν Ω ( M )[[ ν ]], and thuswe denote it by ∗ ∇ , Ω . Here, a symplectic connection means a torsion free connection making ω parallel. It is known ([32], [8], [2]) that any star product on a symplectic manifold is equivalentto a Fedosov star product.In this paper, we study closedness of Fedosov star products naturally attached to symplecticor K¨ahler manifolds. On a compact symplectic manifold, we fix the de Rham class [ ω ] of thesymplectic form. We study the following problem: Problem 1.1 (Symplectic version) . Can one find a pair ( ω, ∇ ) consisting of a symplectic form ω ∈ [ ω ] and a symplectic connection ∇ with respect to ω such that ∗ ∇ , is closed? This problem is motivated by the study of moment map geometry of the space of symplecticconnections. As noticed in [28], since the trace density of ∗ ∇ , is given by(2 πν ) m ρ ∇ , := 1 − ν µ ( ∇ ) + O ( ν )where µ ( ∇ ) is the Cahen-Gutt momentum [6] of the symplectic connection ∇ , an affirmativeanswer to the above problem for ∗ ∇ , implies the constancy of the Cahen-Gutt momentum µ ( ∇ ).Recall that µ ( ∇ ) is given by µ ( ∇ ) := ( ∇ p,q ) Ric ∇ ) pq −
12 Ric ∇ pq Ric ∇ pq + 14 R ∇ pqrs R ∇ pqrs , where R ∇ is the curvature of ∇ and Ric ∇ ( · , · ) := tr[ V (cid:55)→ R ∇ ( V, · ) · ] is the Ricci tensor.On a closed K¨ahler manifold ( M, ω , J ), one consider the space M [ ω ] of K¨ahler forms inthe cohomology class of ω , the complex structure being fixed. To ω ∈ M [ ω ] , one can attacha natural family of Fedosov star products ∗ ∇ , Ω k ( ω ) described as follows. For k ∈ R , considerclosed 2-form Ω k ( ω ) := ν k Ric( ω ) , with Ric( ω ) := Ric ∇ ( J · , · ) being the Ricci form of the K¨ahler manifold ( M, ω, J ). Problem 1.2 (K¨ahler version) . For a fixed real number k , can one find ω ∈ M [ ω ] with Levi-Civita connection ∇ and Ricci form Ric( ω ) such that ∗ ∇ , Ω k ( ω ) is closed? A trace density for ∗ ∇ , Ω k ( ω ) is given by(2 πν ) m ρ ∇ , Ω k ( ω ) = 1 − ν k ω + O ( ν ) , with S ω being the scalar curvature (see Remark 5.2). So a necessary condition for ∗ ∇ , Ω k ( ω ) , with k (cid:54) = 0, to be closed is the existence of a constant scalar curvature K¨ahler metric.Our obstructions come from the presence of symmetries of the symplectic manifolds. Whena compact Lie group G acts on ( M, ω ), it is natural to restrict the above problem on G -invariant symplectic forms in [ ω ] and to consider G -invariant Fedosov star products (built with G -invariant ∇ and Ω). An important feature in this context is the notion of quantum momentmap [30, 35, 24, 31] which leads to phase space reduction in deformation quantization [3, 14].Let G be a compact Lie group acting effectively on a compact symplectic manifold M preservingthe symplectic form ω , a closed formal 2-form Ω ∈ ν Ω ( M )[[ ν ]] and a symplectic connection ∇ so that the Fedosov star product ∗ ∇ , Ω is G-invariant. We identify a Lie algebra element X ∈ g with a vector field on M by the action of G . In [30, 35, 24], a map µ · : g → C ∞ ( M )[[ ν ]] iscalled a quantum moment map if µ · is a Lie algebra morphism with respect to the commutator ν [ · , · ] ∗ ∇ , Ω on C ∞ ( M )[[ ν ]] satisfying(2) X ( u ) = 1 ν ad ∗ ∇ , Ω µ X ( u )for any u ∈ C ∞ ( M )[[ ν ]]. It follows from Theorem 8.2 in [24] or Deduction 4.4 in [30] that (2) isequivalent to(3) dµ X = i ( X )( ω − Ω) . If a formal function f ∈ C ∞ ( M )[[ ν ]] satisfies i ( X f )( ω − Ω) = df for some vector field X f wecall X f the quantum Hamiltonian vector field of f , and also call f the quantum Hamiltonianfunction of X f . In this paper we adopt (3) as the definition of quantum moment map withoutmentioning ∇ , and we say that, given a symplectic form ω and a closed 2-form Ω, a G -equivariantmap µ : M → g ∗ [[ ν ]] is a quantum moment map if µ X := (cid:104) µ, X (cid:105) ∈ C ∞ ( M )[[ ν ]] is a quantumHamiltonian function of X ∈ g . If there is a quantum moment map we say that G -actionon ( M, ω,
Ω) is quantum-Hamiltonian. Naturally if (
M, ω,
Ω) is quantum-Hamiltonian G -spacethen ω and Ω are G -invariant if G is connected. Given a G -invariant symplectic connection ∇ and Fedosov star product ∗ ∇ , Ω , the quantum moment map µ : M → g ∗ [[ ν ]] in our senseinduces µ · : g → C ∞ ( M )[[ ν ]] satisfying (2) by Theorem 8.2 in [24] or Deduction 4.4 in [30] asquoted above. Further, by the G -equivariance we required for µ : M → g ∗ [[ ν ]], the induced map µ · : g → C ∞ ( M )[[ ν ]] is a Lie algebra morphism and thus a quantum moment map in the senseof [30, 35, 24].Quantum moment maps are not unique, and any two of them differ by a map b : g → R [[ ν ]]that vanishes on Lie bracket. As a consequence, we can assume the quantum moment map isnormalized so that(4) (cid:90) M µ X ( ω − Ω) m = 0 . Given a quantum-Hamiltonian G -space ( M, ω , Ω ), we denote by C G ([ ω ] , [Ω ]) the spaceconsisting of all triples ( ω, Ω , ∇ ) such that(a) ( M, ω,
Ω) is a quantum-Hamiltonian G -space,(b) ω is cohomologous to ω and there is a smooth path { ω s } ≤ s ≤ consisting of G -invariantsymplectic forms joining ω and ω in the cohomology class [ ω ],(c) Ω is cohomologous to Ω , and(d) ∇ is a G -invariant symplectic connection with respect to ω .For each triple ( ω, Ω , ∇ ) in C G ([ ω ] , [Ω ]) we have the Fedosov star product ∗ ∇ , Ω . Theorem 1.3.
Let ( M, ω , Ω ) be a quantum-Hamiltonian G -space and consider a triple ( ω, Ω , ∇ ) in C G ([ ω ] , [Ω ]) . For X ∈ g , let µ X be the quantum Hamiltonian function of X with respect to ω − Ω with normalization (4) . Then the trace Tr ∗ ∇ , Ω ( µ X ) of the Fedosov star product ∗ ∇ , Ω isindependent of the choice of ( ω, Ω , ∇ ) in C G ([ ω ] , [Ω ]) . Hence, one can define a symplectic invariant :
Definition 1.4.
We define a character Tr [ ω ] , [Ω ] : g → R [[ ν ]] by Tr [ ω ] , [Ω ] ( X ) := Tr ∗ ∇ , Ω ( µ X ) where the right hand side is given by Theorem 1.3 with normalization as in (4) . In the particular case Ω = 0, we obtain an obstruction to the existence of closed Fedosov starproducts, answering to Problem 1.1.
Theorem 1.5.
Let ( M, ω ) be a compact symplectic manifold. If there exists a closed Fedosovstar product ∗ ∇ , for ( ω, , ∇ ) in C G ([ ω ] , then Tr [ ω ] , vanishes. xpanding Tr [ ω ] , ( X ) in terms of power series in ν we obtain a series of integral invariantsobstructing the existence of closed Fedosov star products. The ν − m -term is exactly the invariantfound in [29]. See also [21] for a different derivation of this invariant using Donaldson-Fujikitype picture. This is one of the obstructions to asymptotic Chow semi-stability found by thefirst author in [17]. As discussed in [23], the trace density is considered to play the same roleas the Bergman function for the Berezin-Toeplitz star product [34], [5]. See also [11], [18], [20],[22], [19], [9], [36], [26] for related topics.On a compact K¨ahler manifold ( M, ω , J ) admitting an effective action of a compact Lie group G preserving ω and J , it is natural to study M G [ ω ] the space of G -invariant K¨ahler forms inthe cohomology class of ω . For ω ∈ M G [ ω ] and k ∈ R , the closed 2-form Ω k ( ω ) is G -invariant.Thus, for ω ∈ M G [ ω ] and k ∈ R , we may consider the G -invariant Fedosov star product ∗ ∇ , Ω k ( ω ) ,where ∇ is the Levi-Civita connection of the K¨ahler form ω .Assume ω ∈ M G [ ω ] makes ( M, ω,
0) a quantum-Hamiltonian G -space with quantum momentmap µ · normalized by (4). Then, we will show that the triple ( ω, Ω k ( ω ) , ∇ ) is in C G ([ ω ] , [Ω k ( ω )])with some quantum moment map, which we denote by µ k · , normalized by (4), i.e. in this case (cid:90) M µ kX ( ω − Ω k ( ω )) m = 0for any X ∈ g . Another natural normalization for quantum moment maps is given by theintegral. We define ˜ µ k · to be the quantum moment map of the quantum-Hamiltonian G -space( M, ω, Ω k ( ω )) normalized by(5) (cid:90) M ˜ µ k · ω m = 0 . In Proposition 4.3, we show ˜ µ k · differs from µ k · by a K¨ahler invariant, i.e. a constant dependingonly on the K¨ahler class. Applying Theorem 1.3, we obtain a K¨ahler invariant obstructing theclosedness of the Fedosov star product ∗ ∇ , Ω k ( ω ) . Theorem 1.6.
Let ( M, ω , J ) be a compact K¨ahler manifold with ( ω, , ∇ ) ∈ C G ([ ω ] , . Thenfor all k ∈ R , Tr M G [ ω ,k ( X ) := Tr ∗ ∇ , Ω k ( ω ) (˜ µ kX ) is independent of the choice of ω ∈ M G [ ω ] . Moreover, if there exists a closed Fedosov star product ∗ ∇ , Ω k ( ω ) for ω ∈ M G [ ω ] , then Tr M G [ ω ,k ( X ) vanishes. The plan after this introduction is as follows. In section 2, we review Fedosov’s constructionof star product, particularly Fedosov connection on Weyl algebra bundle. The description of flatsections in Darboux charts is given in section 2. In section 3, the variation formula of the traceis given. In section 4, we apply the variation formula to the quantum moment map, discuss onthe two normalizations (4) and (5) and prove Theorem 1.3, 1.5 and 1.6. In section 5, we giveexplicit formulas of the invariants up to terms in ν .2. Prelimaries
In this section we describe the equivalence B in (1) when ∗ is the Fedosov star product ∗ ∇ , Ω (c.f. [12], [13]). We mainly follow Fedosov’s paper [15] incorporating non-zero Ω. We firstrecall the construction of Fedosov star product. Let T x M be the tangent space at x ∈ M ofthe symplectic manifold M with symplectic form ω . We choose a basis ( e , · · · , e n ) of T x M and write ω ij = ω ( e i , e j ) and express a tangent vector y as y = y e + · · · y n e n . Typically, wemay take e i = ∂/∂x i for a choice of local coordinates ( x , · · · x n ). The formal Weyl algebra W x orresponding to the symplectic space T x M is an associative algebra consisting of the formalseries(6) a ( y, ν ) = (cid:88) k,(cid:96) ≥ ν k a k,i , ··· ,i (cid:96) y i · · · y i (cid:96) where ν is a formal parameter and a k,i , ··· ,i (cid:96) are real coefficients. The product ◦ of the elements a, b ∈ W x is defined by the Weyl rule ( a ◦ b )( y, ν ) = (cid:20) exp (cid:18) ν ij ∂∂y i ∂∂z j (cid:19) a ( y, ν ) b ( z, ν ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) y = z where (Λ ij ) is the inverse matrix of the symplectic form ( ω ij ). Note that this description of ◦ isindependent of the choice of a basis of T x M . We prescribe the degree, called the Weyl degree ,of the variables by deg y i = 1 and deg ν = 2. Then each term of (6) has Weyl degree 2 k + (cid:96) .Taking a union of T x M over all x ∈ M we obtain a bundle W of the formal Weyl algebras.The local sections of W are of the form a ( x, y, ν ) = (cid:88) k + (cid:96) ≥ ν k a ( x ) k,i , ··· ,i (cid:96) y i · · · y i (cid:96) where the Weyl degree is used in the summation expression. These can be regarded as sectionsof (cid:80) r ≥ ν r (cid:80) S (cid:96) T ∗ M . The product ◦ extends to an algebra structure on the space Γ( W ) of thesections the Weyl algebra bundle W by( a ◦ b )( x, y, ν ) = (cid:20) exp (cid:18) ν ij ∂∂y i ∂∂z j (cid:19) a ( x, y, ν ) b ( x, z, ν ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) y = z We set Γ( W ) ⊗ ∧ ( M ) to be the set of the Weyl algebra bundle valued differential forms whichare expressed locally as (cid:88) k + (cid:96) ≥ ν k a ( x ) k i ··· i (cid:96) j ··· j p y i · · · y i (cid:96) dx j ∧ · · · ∧ dx j p . We extend ◦ to Γ( W ) ⊗ ∧ ( M ) by a ⊗ α ◦ b ⊗ β = a ◦ b ⊗ α ∧ β where a, b ∈ Γ( W ) and α, β ∈ ∧ ( M ). Then the commutator is naturally described as[ s, s (cid:48) ] = s ◦ s (cid:48) − ( − q q s (cid:48) ◦ s for s ∈ Γ( W ) ⊗∧ q ( M ) and s (cid:48) ∈ Γ( W ) ⊗∧ q ( M ). Note that the center of the algebra Γ( W ) ⊗∧ ( M )consists of the elements of the form (cid:80) ∞ k =0 ν k α k with α k differential forms in ∧ ( M ). We call theseelements the central elements or central forms .It is well-known that there is a torsion-free connection making ω parallel, called a symplecticconnection . In terms of the Christoffel symbols the condition for symplectic connection is that ω i(cid:96) Γ (cid:96)jk is symmetric in i, j, k . It is not unique, and for any two symplectic connections withChristoffel symbols Γ ijk and Γ (cid:48) ijk , ω i(cid:96) (Γ (cid:96)jk − Γ (cid:48) (cid:96)jk ) is symmetric in i, j, k . Conversely, given asymplectic connection and a symmetric covariant 3-tensor one can construct another symplecticconnection in this way. Thus the space of symplectic connections on ( M, ω ) is an affine spacemodeled on the vector space of all symmetric covariant tensors of degree 3.Let ∇ be a symplectic connection on ( M, ω ), and Γ kij be its Christoffel symbols. Let Γ( W ) ⊗∧ ( M ) be the space of W -valued differential forms on M . Then the induced exterior covariantderivative ∂ on Γ( W ) ⊗ ∧ ( M ) is described as ∂a := da + 1 ν [Γ , a ] here Γ = 12 ω (cid:96)k Γ kij y (cid:96) y j dx i . Its curvature is described as ∂ a = 1 ν [ R, a ]where R = 14 ω ir R rjk(cid:96) y i y j dx k ∧ dx (cid:96) . For a W -valued 1-form γ ∈ Γ( W ) ⊗ ∧ ( M ) we consider a more general connection D = ∂ + 1 ν [ γ, · ] . The connection D is determined up to a central term of γ . For the uniqueness of γ we require γ := γ | y =0 = 0 . This condition is called the
Weyl normalization . The curvature Θ of D is given byΘ = R + ∂γ + 1 ν γ ◦ γ. Following [12] we call Θ the
Weyl curvature when D satisfies the Weyl normalization.We wish to obtain a flattening D = ∂ + ν [ γ, · ] of ∂ in the form γ = ω ij y i dx j + r for some r ∈ Γ( W ) ⊗ ∧ ( M ). Since δ = dx (cid:96) ∧ ∂∂y (cid:96) = − ν [ ω ij y i dx j , · ]we may put(7) D := ∂ − δ + 1 ν [ r, · ]and seek r such that D = 0. In [12] such D is called an Abelian connection . Under the Weylnormalization condition r | y =0 = 0, one can see using δ = 0 and δ∂ + ∂δ = 0 that the Weylcurvature Θ is given by Θ = − ω ij dx i ∧ dx j + R − δr + ∂r + 1 ν r ◦ r. Since D = 1 ν [Θ , · ] D is Abelian if R + ∂r − δr + ν r ◦ r is a central 2-form, that is, a 2-form in ν Ω ( M )[[ ν ]]. Introducean operator δ − by δ − ( a pq ) = (cid:26) p + q y k i ( ∂∂x k ) a pq ( p + q (cid:54) = 0)0 ( p + q = 0)for a pq ∈ Γ( W ) ⊗ ∧ q ( M ) with degree p symmetric term in y . By the Hodge decomposition (seee.g. (5.1.7) in [12]), we have for a ∈ Γ( W ) ⊗ ∧ ( M )(8) δ − δa + δδ − a = a − a . Then we have now the standard theorem by Fedosov: heorem 2.1 (Fedosov [12]) . For any Ω ∈ ν Ω ( M )[[ ν ]] there exists a unique r ∈ Γ( W ) ⊗∧ ( M ) with δ − r = 0 and W -degree larger than 2 such that R + ∂r − δr + 1 ν r ◦ r = Ω , that is D is Abelian. Note that the condition δ − r = 0 implies the Weyl normalization condition r | y =0 = 0 and theWeyl curvature is Θ = − ω + Ωwhere we put(9) ω = 12 ω ij dx i ∧ dx j . The proof is given by showing that r is obtained recursively in terms of its degrees by using r = δ − ( R − Ω) + δ − ( ∂r + 1 ν r ◦ r ) . Thus D is uniquely determined under the conditions δ − r = 0 and W -degree of r largerthan 2 once we are given a symplectic connection ∇ and the central formal 2-form Ω. Wecall this connection D the Fedosov connection . Consider the space Γ( W ) D of flat (or parallel)sections with respect to D , i.e. the sections a with Da = 0. Then it is shown [12] that σ : Γ( W ) D → C ∞ ( M )[[ ν ]] sending a ∈ Γ( W ) D to a ∈ C ∞ ( M )[[ ν ]] is a bijection. Its inverse,denoted by Q , sending a to a := Qa can be constructed by solving recursively a = a + δ − ( ∂a + 1 ν [ r, a ])since δ − increases the Weyl degree at least by 1. Q is explicitly expressed as(10) Qa = (cid:88) k ≥ ( δ − ( ∂ + 1 ν [ r, · ])) k a . Since D is a derivation of ◦ , i.e. D ( a ◦ b ) = Da ◦ b + a ◦ Db, Γ( W ) D is closed under the product ◦ . Then the product ◦ on Γ( W ) D induces through Q a ∗ -product ∗ ∇ , Ω on C ∞ ( M )[[ ν ]] which we call the Fedosov star product .In the same way we can prove the following lemma.
Lemma 2.2.
Suppose b ∈ Γ( W ) ⊗ ∧ ( M ) satisfy Db = 0 . Then the equation Da = b admits aunique solution a ∈ Γ( W ) , denoted by a = D − b , such that a | y =0 = 0 .Proof. We use the Hodge decomposition (8) for our a . Since a is a 0-form we have δ − a = 0,and also have a = a | y =0 = 0. Thus we have a = δ − δa. From this and the equation Da = ∂a − δa + 1 ν [ r, a ] = b we need to solve a = − δ − b + δ − ( ∂a + 1 ν [ r, a ]) . his can be solved recursively since δ − raises degree by 1, and the solution is given explicitlyusing the same expression as Q (Equation (10)) in the form(11) a = (cid:88) k ≥ ( δ − ( ∂ + 1 ν [ r, · ])) k ( − δ − b ) . (cid:3) Later we will often use the relation(12) D − = − Q ◦ δ − . Next we recall the following proposition due to Fedosov, see Proposition 5.5.5 and 5.5.6 in [13].Since the characterization of V is used in later arguments we re-produce its proof in this paper. Proposition 2.3 ([13]) . On contractible Darboux chart U we have an equivalence A : Γ( W ) D | U → Γ( W | U ) D flat between the Fedosov connection D and D flat = d − δ and this equivalence A is expressed as Aa = V ◦ a ◦ V − for some V ∈ Γ( W | U ) .Proof. We look for A s and D s interpolating between D and D flat . By (7) we have D = D flat + 1 ν [ r, · ]on U . We write the symplectic connection we chose as d + Γ and the flat connection ∇ flat = d ,and join them by ∇ s = d + (1 − s )Γ. Since ∇ and ∇ flat are both symplectic connections for thesymplectic form in the Darboux chart U the affine line ∇ s are symplectic connections for all s .We also set Ω s := (1 − s )Ω, and build the Fedosov connection for ∇ s and Ω s so that the Weylcurvature of D s is − ω + Ω s = − ω + (1 − s )Ω. We define r s ∈ Γ( W ) ⊗ ∧ ( M ) by D s = D − ν [ r s , · ] . Note that r = 0. Since D = D s + ν [ r s , · ] and D = D s + 1 ν [ D s r s , · ] + 1 ν [ r s ◦ r s , · ]we have(13) Ω = (1 − s )Ω + D s r s + 1 ν r s ◦ r s . Taking the derivative of (13) with respect to s we obtain(14) − Ω + D s ˙ r s = 0since ˙ D s r s = − ν [ ˙ r s , r s ] = − ν ( ˙ r s ◦ r s + r s ◦ ˙ r s ) . On the contractible Darboux chart we can write the closed 2-form Ω asΩ = dα for some 1-form α . Then by using (14) we have D s ( − α + ˙ r s ) = − dα + D s ˙ r s = − Ω + D s ˙ r s = 0 . y Lemma 2.2 there is a unique solution H ( s ) ∈ Γ( W | U ) such that H ( s ) | y =0 = 0 and(15) D s H ( s ) = − α + ˙ r s . Note that, from the construction of the Fedosov connection in Theorem 2.1 and (11), the degree H ( s ) is at least 3 as δ − raises the Weyl degree by 1. We then solve dds V s = 1 ν H ( s ) ◦ V s of V s ∈ Γ( W | U ) with V = 1. This can be solved solving the integral equation V s = 1 + (cid:90) s ν H ( σ ) ◦ V σ dσ recursively using the Weyl degree. The iterations are completed since the integral operator onthe right hand side raises the Weyl degree by 1. Then we have dds ( V − s ◦ D s V s − ν r s )= − ν V − s ◦ H ( s ) ◦ D s V s − V − s ◦ ( 1 ν [ ˙ r s , V s ]) + V − s ◦ D s ( 1 ν H ( s ) ◦ V s ) − ν ˙ r s = 1 ν V − s ◦ ( D s H ( s ) − ˙ r s ) ◦ V s = − ν V − s ◦ α ◦ V s = − ν α where we have used (15) and the fact that α is central. This shows(16) V − s ◦ D s V s − ν r s = − sν α. Finally we define A s : Γ( W ) D | U → (Γ( W ) | U ) D s by a (cid:55)→ V s ◦ a ◦ V − s . Then we see for a ∈ Γ( W ) D | U that D s A s a = V s ◦ [ V − s ◦ D s V s − ν r s , a ] ◦ V − s = V s ◦ [ − ν α, a ] ◦ V − s = 0since α is central. V := V is the one we desired. This completes the proof of Proposition2.3. (cid:3) Note in particular, (16) shows(17) V − ◦ D flat V − ν r = − ν α. Now, since B = ev y =0 ◦ A ◦ Q , the normalized trace (1) can be expressed asTr ∗ ∇ , Ω ( F ) = (2 πν ) − m (cid:90) M ( AQ ( F )) | y =0 ω m m != (2 πν ) − m (cid:90) M ( V ◦ Q ( F ) ◦ V − ) | y =0 ω m m ! . (18) . Variation Formula
Let ∇ t be a family of symplectic connections, and Ω t a family of closed formal 2-forms in thesame cohomology class. We write Ω t − Ω = dβ t . For the pair of ∇ t and Ω t we have the Fedosov connection D t and the Fedosov star product ∗ ∇ t , Ω t . We denote respectively by Tr ∗ ∇ t, Ω t and ρ ∇ t , Ω t the trace and its trace density with respectto ∗ ∇ t , Ω t . We also use the notations D − t and Q t for D − in Lemma 2.2 and Q in (10) withrespect to t . Let us write the variation formula for the trace, our proof follows Fedosov’s paper[15] incorporating variations of Ω. Theorem 3.1.
With the notations being as above we have for any formal function F ∈ C ∞ ( M )[[ ν ]] ddt Tr ∗ ∇ t, Ω t ( F ) = (2 πν ) − m (cid:90) M ν [ D − t ( ˙Γ − ˙ β ) , Q t ( F )] | y =0 ρ ∇ t , Ω t ω m m ! . Proof.
Write locally on the contractible Darboux Chart U as(19) D t = D flat + 1 ν [ r t , · ] . (Note that this r t is different from r s in the previous section.) For each t , we built in the previoussection A t : (Γ( W ) D t ) | U → Γ( W | U ) D flat such that A t ( a ) = V t ◦ a ◦ V − t and(20) V − t ◦ D flat V t − ν r t = − ν α t for some V t ∈ Γ( W ) (see (17)) where Ω t = dα t on U . By (18) the normalized trace can beexpressed as(21) Tr ∗ ∇ t, Ω t ( F ) = (2 πν ) − m (cid:90) M (cid:0) V t ◦ Q t ( F ) ◦ V − t (cid:1) | y =0 ω m m ! . Hereafter we omit the notation ◦ . To compute the derivative of (21) with respect to t we see(22) ddt ( V t Q t ( F ) V − t ) = V t ([ V − t ˙ V t , Q t ( F )] + ˙ Q t ( F )) V − t . We first treat [ V − t ˙ V t , Q t ( F )]. Taking the derivative of (20) with respect to t we obtain(23) [ V − t D flat V t , V − t ˙ V t ] + D flat ( V − t ˙ V t ) = 1 ν ( ˙ r t − ˙ α t ) . Using (20), the fact that α t is central and (19) we obtain from (23)(24) D t ( V − t ˙ V t ) = 1 ν ( ˙ r t − ˙ α t ) . On the other hand, from D t = ( D flat + 1 ν [ r t , · ]) we have Ω t = D flat r t + 1 ν r t ◦ r t and thus D t ( ˙ r t − ˙ α t ) = D t ˙ r t − ˙Ω t = D flat ˙ r t + 1 ν [ r t , ˙ r t ] − ˙Ω t = 0(25) hus by Lemma 2.2, (24) and (25) it follows that V − t ˙ V t = D − t ( ˙ r t − ˙ α t ) + b for some b ∈ Γ( W ) D t . Note that this b is necessary because for uniqueness we have to impose V − t ˙ V t | y =0 = 0. Hence (22) has become(26) ddt ( V t Q t ( F ) V − t ) = V t (cid:18) ν [ D − t ( ˙ r t − ˙ α t ) + b, Q t ( F )] + ˙ Q t ( F ) (cid:19) V − t . Now we treat ˙ Q t ( F ). Taking the derivative of D t Q t ( F ) = 0 we obtain D t ˙ Q t ( F ) = − ν [ ˙ r t − ˙ α t , Q t ( F )]since ˙ α t is central. Using (25) again we have D t [ ˙ r t − ˙ α t , Q t ( F )] = 0 . Since Q t ( F ) | y =0 = F we also have the uniqueness condition ˙ Q t ( F ) | y =0 = 0. Thus by Lemma2.2 ˙ Q t ( F ) = − ν D − t [ ˙ r t − ˙ α t , Q t ( F )] . Now (26) has become ddt ( V t Q t ( F ) V − t ) = V t (cid:18) ν [ D − t ( ˙ r t − ˙ α t ) + b, Q t ( F )] − ν D − t [ ˙ r t − ˙ α t , Q t ( F )] (cid:19) V − t . Thus we obtain ddt Tr ∗ ∇ t, Ω t ( F ) = (cid:90) M V t (cid:18) ν [ D − t ( ˙ r t − ˙ α t ) + b, Q t ( F )] − ν D − t [ ˙ r t − ˙ α t , Q t ( F )] (cid:19) V − t | y =0 ω m m != Tr ∗ ∇ t, Ω t (cid:18) ν [ D − t ( ˙ r t − ˙ α t ) + b, Q t ( F )] − ν D − t [ ˙ r t − ˙ α t , Q t ( F )] (cid:19) | y =0 ) . Recall by (11) that D − t | y =0 = − Q ( δ − · ) | y =0 = 0since δ − increases y -degree by 1. Thus(27) ddt Tr ∗ ∇ t, Ω t ( F ) = Tr ∗ ∇ t, Ω t (cid:18) ν [ D − t ( ˙ r t − ˙ α t ) + b, Q t ( F )] | y =0 (cid:19) . Also b = Q t ( b ) for some b ∈ C ∞ ( M ), and by the property of the trace we haveTr ∗ ∇ t, Ω t ([ b, Q t ( F )] | y =0 ) = Tr ∗ ∇ t, Ω t ([ b , F ] ∗ ∇ t, Ω t ) = 0 . Thus (27) becomes ddt Tr ∗ ∇ t, Ω t ( F ) = Tr ∗ ∇ t, Ω t (cid:18) ν [ D − t ( ˙ r t − ˙ α t ) , Q t ( F )] | y =0 (cid:19) . Recall also (12) so that D − t ˙ r t = − Q ( δ − ˙ r t ) . Our ˙ r t comes from the variation ˙Γ of the symplectic connection and the variation of the r -termin Fedosov’s construction in Theorem 2.1. But the r -term in Fedosov’s construction is required δ − r = 0. Hence we have D − t ˙ r t = D − t ˙Γ . The Theorem 3.1 follows by noting ˙ α = ˙ β . This completes the proof. (cid:3) . Quantum moment map
The formula in the next proposition can be found in page 135 in [24], but we will re-produceits proof as we wish to make clear how the assumptions are used.
Proposition 4.1 ([24]) . For any triple ( ω, Ω , ∇ ) ∈ C G ([ ω ] , [Ω ]) and X ∈ g we have the identity L X = D ◦ i ( X ) + i ( X ) ◦ D + 1 ν ad ∗ ( Q ( µ X )) . Proof.
We start with the general formula in page 135, [24], for the Fedosov star product and asymplectic vector field X i.e. di ( X ) ω = 0(28) L X = D ◦ i ( X ) + i ( X ) ◦ D + 1 ν ad ∗ ( T ( X ))where T ( X ) = − i ( X ) r + ω ij X i y j + 12 ( ∇ i ( i ( X ) ω ) j y i y j ) , but note that the sign of r in [24] is opposite from ours. To prove the proposition it is sufficientto show(29) D ( µ X + T ( X )) = 0 . First of all, since L X ∇ = 0 and L X ω = L X Ω = 0 we have L X r = 0. Thus by (28) we have(30) − Di ( X ) r = i ( X ) Dr + 1 ν [ T ( X ) , r ] . From Theorem 2.1 and (7) we see(31) Dr = − R + Ω + 12 ν [ r, r ] . From (30) and (31) we obtain(32) − Di ( X ) r = − i ( X ) R + i ( X )Ω − ν [ ω ij X i y j + 12 ( ∇ i ( i ( X ) ω ) j y i y j ) , r ] . Secondly, using (7) we obtain(33) D ( ω ij X i y j ) = − i ( X ) ω + ∂ ( ω ij X i y j ) + 1 ν [ ω ij X i y j , r ] . Thirdly, using (7) again we have D ( 12 ( ∇ i ( i ( X ) ω ) j y i y j ))(34) = −∇ i ( i ( X ) ω ) j dx i y j + ∂ ( 12 ( ∇ i ( i ( X ) ω ) j y i y j )) + 1 ν [ 12 ( ∇ i ( i ( X ) ω ) j y i y j ) , r ] . The condition that ∇ is G -invariant implies L X ∇ = 0, which is equivalent to say( ∇ X )( Y, Z ) = ( −∇ X ∇ Y + ∇ Y ∇ X + ∇ [ X,Y ] ) Z, or equivalently ∇ k ∇ i X p = − R pi(cid:96)k X (cid:96) where (cid:96), k are regarded as indices of the form part and p, i are regarded as the indices of theendomorphism part. rom this and (9) we obtain ∂ ( 12 ( ∇ i ( i ( X ) ω ) j y i y j )) = 14 ∇ k ∇ i X p ω pj y i y j dx k = −
14 R pi(cid:96)k X (cid:96) ω pj y i y j dx k = 14 ω jp R pi(cid:96)k y i y j X (cid:96) dx k = i ( X ) R. Thus (34) becomes D ( 12 ( ∇ i ( i ( X ) ω ) j y i y j ))(35) = −∇ i ( i ( X ) ω ) j dx i y j + i ( X ) R + 1 ν [ 12 ( ∇ i ( i ( X ) ω ) j y i y j ) , r ] . Adding (32), (33) and (35) we obtain DT = i ( X )( − ω + Ω)= − dµ X = − Dµ X . This shows (29) completing the proof of Proposition 4.1. (cid:3)
Proposition 4.2.
The following two hold about normalization. (a)
For a quantum Hamiltonian vector field X , the quantum Hamiltonian function is deter-mined uniquely under the normalization condition (4) . (b) Let ( M, ω t , Ω t ) be quantum Hamiltonian G -spaces, for t ∈ I such that (36) ω t − Ω t = ω − Ω + dτ t for a smooth family of G -invariant formal -form τ t . Then the normalized quantumHamiltonian functions u X,t for ω t − Ω t with normalization condition (4) are related by u X,t = u X, − τ t ( X ) . Proof.
The statement of item (a) is obvious because if we have two quantum Hamiltonian func-tions of the same vector field X then the difference of the two is a formal constant.To show (b) one can see i ( X )( ω t − Ω t ) = du X,t and that u X,t is independent of the choice of the G -invariant 1-form τ t satisfying (36) since another τ (cid:48) t satisfying (36) is of the form τ (cid:48) t = τ t + dh t for a G -invariant smooth function h t . One further sees ddt (cid:90) M u X,t ( ω t − Ω t ) m = − (cid:90) M ˙ τ ( X )( ω t − Ω t ) m + (cid:90) M mu X,t d ˙ τ ∧ ( ω t − Ω t ) m − = − (cid:90) M ˙ τ ∧ i ( X )( ω t − Ω t ) m − (cid:90) M mdu X,t ∧ ˙ τ ∧ ( ω t − Ω t ) m − = 0 . Thus if u X, satisfies normalization (4) then so does u X,t for all t . This proves (b). (cid:3) Proof of Theorem 1.3.
As Step 1, we consider the case when we have ( ω , Ω , ∇ ) and ( ω , Ω , ∇ ) ∈C G ([ ω ] , [Ω ]). We take a family ( ω , Ω t , ∇ t ) in C G ([ ω ] , [Ω ]) joining ( ω , Ω , ∇ ) and ( ω , Ω , ∇ ).We put Ω t = Ω + dβ t for a G -invariant formal 1-form β t . Then by Proposition 4.2, the quantum Hamiltonian function µ X,t for ω − Ω t with normalization (4) is given by µ X,t = µ X + β t ( X ) . y Theorem 3.1 we have(37) ddt Tr ∗ ∇ t, Ω t ( µ X,t ) = (2 πν ) − m (cid:90) M (cid:18) ν [ D − t ( ˙Γ − ˙ β ) , Q t ( µ X,t )] | y =0 + ˙ β ( X ) (cid:19) ρ ∇ t , Ω t ω m m ! . By Proposition 4.1 we have1 ν [ D − t ( ˙Γ − ˙ β ) , Q t ( µ X,t )] | y =0 = ( − L X + D t ◦ i ( X ) + i ( X ) D t ) D − t ( ˙Γ − ˙ β ) | y =0 . But D − t ( ˙Γ − ˙ β ) is a 0-form, i.e. a function, so that i ( X ) D − t ( ˙Γ − ˙ β ) = 0 . Further, recall D − t = − Q t ◦ δ − by (12) and δ − increases y -degree by 1 so that L X D − t ( ˙Γ − ˙ β ) | y =0 = 0 . The remaining term becomes i ( X ) D t D − t ( ˙Γ − ˙ β ) | y =0 = i ( X )( ˙Γ − ˙ β ) | y =0 = − ˙ β ( X )since ˙Γ has y -degree 2. Thus the right hand side of (37) vanishes. ThusTr ∗ ∇ , Ω ( µ X ) = Tr ∗ ∇ , Ω0 ( µ X, )for fixed ω = ω . This completes the proof of Theorem 1.3 in the case when ω is fixed to be ω .As Step 2, we consider the case when we have ( ω , Ω , ∇ ) and ( ω, Ω , ∇ ) ∈ C G ([ ω ] , [Ω ]).Then there is a smooth path { ω s } ≤ s ≤ consisting of G -invariant symplectic forms joining ω and ω = ω in the cohomology class [ ω ] Then we have G -equivariant diffeomorphisms f s : M → M such that f ∗ s ω s = ω by Moser’s theorem. We put f := f for notational convenience. Then wehave f ∗ ( ω, Ω , ∇ ) = ( ω , f ∗ Ω , f ∗ ∇ )with f ∗ Ω cohomologous to Ω and f ∗ ∇ being a symplectic connection for f ∗ ω = ω . Then weare in a position where the same arguments as in Step 1 apply for the pair ( ω , Ω , ∇ ) and( ω , f ∗ Ω , f ∗ ∇ ). We obtain from Step 1Tr ∗ ω , ∇ , Ω0 ( µ X, ) = Tr ∗ ω ,f ∗∇ ,f ∗ Ω ( f ∗ µ X )where µ X, indicates the quantum moment map for ω − Ω and where we indicated the symplec-tic forms with respect to which the star products are considered. But since f is a G -equivariantsymplectomorphism the right hand side is equal toTr ∗ ω ,f ∗∇ ,f ∗ Ω ( f ∗ µ X ) = Tr ∗ ω, ∇ , Ω ( µ X ) . Thus Tr ∗ ω, ∇ , Ω ( µ X ) is independent of ( ω, ∇ , Ω) ∈ C G ([ ω ] , [Ω ]) with the normalization condition(4) of µ X . This completes the proof of Theorem 1.3. (cid:3) We now apply Theorem 1.3 to the K¨ahler situation. Consider a K¨ahler manifold (
M, ω , J )with G -invariant ω and J . For ω ∈ M G [ ω ] we take the Levi-Civita connection ∇ . Then ( ω, , ∇ )is in C G ([ ω ] ,
0) and the quantum Hamiltonian G -space ( M, ω,
0) has quantum moment map µ · normalized by (4), which means that for X ∈ g the normalization gives (cid:90) M µ X ω m = 0since we are taking Ω = 0. Proposition 4.3.
Under the above situation the following two hold. µ X − ν k ∆ ( ω ) µ X is a quantum-Hamiltonian with respect to the star product ∗ ∇ , Ω k ( ω ) ,with ∆ ( ω ) being the Laplacian with respect to ( ω, J ) . In particular, ( ω, Ω k ( ω ) , ∇ ) ∈C G ([ ω ] , [Ω k ( ω )]) . (2) The integral (38) (cid:90) M (cid:18) µ X − ν k ( ω ) µ X (cid:19) ( ω − Ω k ( ω )) m is independent of the choice of ω ∈ M G [ ω ] . Before going to the proof, let us recall a particular case of the construction from [17]. Ona compact K¨ahler manifold (
M, ω, J ), consider the holomorphic bundle T (1 , M consisting oftangent vectors of type (1 , , ∇ on T (1 , M with curvature R ∇ .For Z in h the reduced Lie algebra of holomorphic vector fields, define L ( Z (1 , ) := ∇ Z (1 , −L Z (1 , , it is a 0-form with values in End( T (1 , M ). Let q be a Gl( m, C )-invariant polynomialon gl ( m, C ) of degree p , the first author defined in [17], the map F q : h → C by(39) F q ( Z ) := (cid:90) M − ( m − p + 1) u Z q ( R ∇ ) ∧ ω ( m − p ) + q ( L ( Z (1 , ) + R ∇ ) ∧ ω ( m − p +1) , where u Z = f + ih ∈ C ∞ ( M, C ) for Z = X f + J X h ∈ h . Remark that as L ( Z (1 , ) + R ∇ isa form of mixed degree, the form q ( L ( Z (1 , ) + R ∇ ) in the second term of F q is also of mixeddegree but only the component of degree 2( p −
1) will contribute to the integral.One shows F q depends neither on the choice of the (1 , M [ ω ] , see [17]. Lemma 4.4.
For the polynomial q := ( c ) p , ∇ = ∇ the Levi-Civita connection and Z = X f ∈ h with f ∈ C ∞ ( M ) , the invariant F q writes as: F ( c ) p ( Z ) = (cid:18) π (cid:19) p (cid:90) M − ( m − p + 1) f Ric( ω ) p ∧ ω ( m − p ) − p ( ω ) f Ric( ω ) ( p − ∧ ω ( m − p +1) Proof. As c ( · ) := i π tr C ( · ) then c ( R ∇ ) = π Ric( ω ) and as u Z = f , the first term of thestatement comes from the first term of the general formula (39). For the second term, we have (cid:90) M ( c ) p ( L ( Z (1 , ) + R ∇ ) ∧ ω ( m − p +1) = (cid:90) M p c ( L ( Z (1 , ))( c ) p − ( R ∇ ) ∧ ω ( m − p +1) . Now, c ( L ( Z (1 , )) = π tr C (cid:16) Y (1 , (cid:55)→ ∇ Y (1 , X (1 , f (cid:17) = − π ∆ ω f . (cid:3) Now, we can prove proposition 4.3.
Proof of Proposition 4.3. (1) It comes from i X Ric( ω ) = d ( ∆ f ) for i X ω = df and L X J = 0.(2) We compute the terms of order ν p in the integral (38): • at p = 0, we have (cid:82) M µ X ω m = 0. Notice that this is the normalization (4) sinceΩ = 0 for our quantum-Hamiltonian G -space ( M, ω, • at p = 1, we have − nk (cid:82) M µ X Ric( ω ) ∧ ω ( m − , which is the original Futaki invariant(the Laplacian does not contribute to the integral). • at p >
1, we have( − p k p m − p + 1 (cid:18) mp (cid:19) (cid:90) M ( m − p + 1) µ X Ric( ω ) p ∧ ω ( m − p ) + p ( ω ) µ X Ric( ω ) ( p − ∧ ω ( m − p +1) , which is a K¨ahler invariant by Lemma 4.4 (cid:3) e set µ kX to be the quantum moment map with respect to ∗ ∇ , Ω k ( ω ) normalized by (4), i.e. (cid:90) M µ kX ( ω − Ω k ( ω )) m = 0 , and ˜ µ kX := µ X − ν k ∆ ( ω ) µ X which is normalized by (5), i.e. (cid:90) M ˜ µ kX ω m = 0 . Proof of Theorem 1.6.
Let us compute for X ∈ g :(40) Tr ∗ ∇ , Ω k ( ω ) (˜ µ kX ) = Tr ∗ ∇ , Ω k ( ω ) ( µ kX ) + Tr ∗ ∇ , Ω k ( ω ) (˜ µ kX − µ kX ) . The first term of the right hand side is an invariant by Theorem 1.3. About the second term,note that ˜ µ kX − µ kX = 1 (cid:82) M ( ω − Ω k ( ω )) m (cid:90) M (cid:18) µ X − ν k ( ω ) µ X (cid:19) ( ω − Ω k ( ω )) m which is an invariant by Proposition 4.3. Hence,Tr ∗ ∇ , Ω k ( ω ) (˜ µ kX − µ kX ) = (˜ µ kX − µ kX ) Tr ∗ ∇ , Ω k ( ω ) (1)is an invariant since Tr ∗ ∇ , Ω k ( ω ) (1) is a topological invariant by the index theorem [32]. Thisproves the first statement of Theorem 1.6. Moreover, if ∗ ∇ , Ω k ( ω ) is closed, then the trace densityis 1 and Tr ∗ ∇ , Ω k ( ω ) (˜ µ kX ) = (2 πν ) − m (cid:90) M ˜ µ kX ω m = 0 . This completes the proof of Theorem 1.6. (cid:3) Computations by hand up to ν We are able to compute by hand the first order terms of the invariants Tr [ ω ] , [Ω] and Tr M G [ ω ] ,k .To this end we compute the trace density up to order ν . Proposition 5.1.
For the symplectic connection ∇ and the formal -form Ω := να + ν α + O ( ν ) , denote by ρ ∇ , Ω := πν ) m (cid:0) νρ + ν ρ + O ( ν ) (cid:1) the trace density of the Fedosov starproduct ∗ ∇ , Ω . We have: ρ := − m α ∧ ω m − ω m ρ := − µ ( ∇ ) − m α ∧ ω m − ω m + 12 m ( m − α ∧ α ∧ ω m − ω m for µ ( ∇ ) being the Cahen-Gutt momentum of ∇ .Proof. Performing the Fedosov construction with symplectic connection ∇ and the formal 2-formΩ := να + ν α + O ( ν ), one obtains [4]: f ∗ ∇ , Ω g = f.g + ν { f, g } + ν C ( f, g ) + ν C ( f, g ) + O ( ν )with : C ( f, g ) = 18 Λ i j Λ i j ∇ i i f ∇ j j g − α ( X f , X g ) C ( f, g ) = 148 S ∇ ( f, g ) + 12 ( ı X f α ) i Λ ik ( ı X g α ) k − α ( X f , X g ) + B ∇ [ α ]( f, g )where(41) S ∇ ( f, g ) = Λ i j Λ i j Λ i j L X f ∇ i i i L X g ∇ j j j , or L X f ∇ i i i being the component of the Lie derivative of ∇ seen as a symmetric 3-tensor on M , and B ∇ [ α ]( f, g ) := 132 (cid:16) Λ ta ( α ) au Λ ui Λ kj + Λ ta ( α ) au Λ uj Λ ki (cid:17) (cid:0) ∇ tk f ∇ ij g + ∇ tk g ∇ ij f (cid:1) + 148 (cid:16) Λ ui Λ kj + Λ uj Λ ki (cid:17) (cid:0) ( ı X f ∇ k α ) u ∇ ij g + ∇ ij f ( ı X g ∇ k α ) u (cid:1) . Note that in [4] the conventions are slightly different from here: the formal parameter is rescaledby a factor 2 as well as the formal 2-form Ω.As B ∇ [ α ]( f, g ) and Λ i j Λ i j ∇ i i f ∇ j j g are symmetric in f, g and the other terms of C ( f, g ) are anti-symmetric in f, g , we have:[ f, g ] ∗ ∇ , Ω = ν { f, g } − ν α ( X f , X g ) + ν C − ( f, g ) + O ( ν ) , where C − ( f, g ) = 124 S ∇ ( f, g ) + ( ı X f α ) i Λ ik ( ı X g α ) k − α ( X f , X g ) . The fact that ρ and ρ are the first terms of the trace density is summurised in the followingequations: For ρ we have − (cid:90) α ( X f , X g ) ω m m ! = (cid:90) M { f, g } α ∧ ω m − ( m − . For ρ , first by the moment map property [6] of µ we have (cid:90) S ∇ ( f, g ) ω m m ! = (cid:90) { f, g } µ ( ∇ ) ω m m ! , also, − (cid:90) α ( X f , X g ) ω m m ! = (cid:90) M { f, g } α ∧ ω m − ( m − , and finally,12 (cid:90) M { f, g } α ∧ α ∧ ω m − ( m − (cid:90) α ( X f , X g ) ρ ω m m ! − (cid:90) ( ı X f α ) i Λ ik ( ı X g α ) k ω m m ! . (cid:3) Remark 5.2.
On a K¨ahler manifold ( M, ω, J ) , applying Proposition 5.1 to Ω = Ω k ( ω ) yields ρ ∇ , Ω k ( ω ) = 1(2 πν ) m (cid:18) − ν k ω + O ( ν ) (cid:19) We can now compute the first terms of the invariants.
Proposition 5.3.
The invariant Tr [ ω ] , [Ω] ( X ) = − π ) m ν m − (cid:90) µ X µ ( ∇ ) ω m m ! + O ( ν m − ) , for µ X := µ X + νµ X + ν µ X + O ( ν ) , the quantum moment map normalised by (4) . roof. We compute the trace(2 πν ) m Tr [ ω ] , [Ω] ( X ) = (2 πν ) m Tr ∗ ∇ , Ω ( µ X )= (cid:90) µ X ω m m ! + ν (cid:18) − (cid:90) µ X α ∧ ω m − ( m − (cid:90) µ X ω m m ! (cid:19) + ν (cid:18) − (cid:90) µ X µ ( ∇ ) ω m m ! − (cid:90) µ X α ∧ ω m − ( m − (cid:90) µ X α ∧ α ∧ ω m − ( m − − (cid:90) µ X α ∧ ω m − ( m − (cid:90) µ X ω m m ! (cid:19) + O ( ν )But the moment map is normalised so that (cid:82) µ X ( ω − Ω) n = 0, hence the above becomes:(2 πν ) m Tr [ ω ] , [Ω] ( X ) = − ν (cid:90) µ X µ ( ∇ ) ω m m ! + O ( ν ) . (cid:3) Proposition 5.4.
The invariant Tr M G [ ω ] ,k ( X ) = 1(2 πν ) m (cid:18) πνm ! F kc ( X ) + (2 πν ) ( m − F c − k c ( X ) + O ( ν ) (cid:19) . Proof.
We consider the moment map ˜ µ kX := µ X − νk ∆ ( ω ) µ X given in Proposition 4.3, normalisedby the integral. Using Proposition 5.1 with Ω = Ω k ( ω ), the trace is(2 πν ) m Tr M G [ ω ] ,k ( X ) = − kν (cid:90) µ X Ric( ω ) ∧ ω m − ( m − ν (cid:18) − (cid:90) µ X µ ( ∇ ) ω m m !+ k (cid:90) µ X Ric( ω ) ∧ Ric( ω ) ∧ ω m − ( m − ( ω ) µ X Ric( ω ) ∧ ω m − ( m − (cid:19) + O ( ν )The term in ν is visibly πm ! F kc ( X ). From Lemma 4.4, one sees that − F k c ( X ) = k (cid:18)(cid:90) µ X Ric( ω ) ∧ Ric( ω ) ∧ ω m − ( m − ( ω ) µ X Ric( ω ) ∧ ω m − ( m − (cid:19) . Finally, the remaining term in ν involving the Cahen-Gutt moment map was identified in [29]to be (2 π ) ( m − F c − c ( X ). (cid:3) References [1] F. Bayen, M. Flato, C. Fronsdal, A. Lichn´erowicz, D. Sternheimer : Deformation theory and quantization,Annals of Physics , part I : 61–110, part II : 111–151 (1978).[2] M. Bertelson, M. Cahen, S. Gutt : Equivalence of star products, Class. Quan. Grav. , A93–A107 (1997).[3] Bordemann, M., Herbig, H.-C., Waldmann, S. : BRST Cohomology and Phase Space Reduction in Defor-mation Quantization, Commun. Math. Phys. , 107–144 (2000).[4] M. Bordemann : (Bi)Modules, morphisms, and reduction of star-products: the symplectic case, foliations,and obstructions, Trav. Math. , 9–40 (2005).[5] M. Bordemann, E. Meinrenken, M. Schlichenmaier : Toeplitz quantization of K¨ahler manifolds and gl n , n → + ∞ limits, Comm. Math. Phys. , 281–296 (1994).[6] M. Cahen, S. Gutt : Moment map for the space of symplectic connections, Liber Amicorum Delanghe, F.Brackx and H. De Schepper eds., Gent Academia Press, 2005, 27–36.[7] A. Connes, M. Flato, D. Sternheimer : Closed star products and cyclic cohomology, Lett. in Math. Phys ,1–12 (1992).[8] P. Deligne : D´eformation de l’alg`ebre des fonctions d’une vari´et´e symplectique : comparaison entre Fedosovet De Wilde, Lecomte., Selecta Math. , 667–697 (1995).[9] A. Della Vedova and F. Zuddas : Scalar curvature and asymptotic Chow stability of projective bundles andblowups. Trans. Amer. Math. Soc. , no. 12, 6495–6511 (2012).
10] M. De Wilde, P.B.A. Lecomte : Existence of star-products and of formal deformations of the Poisson LieAlgebra of arbitrary symplectic manifolds., Lett. Math. Phys. , 487–496 (1983).[11] S.K. Donaldson : Scalar curvature and projective embeddings, I, J. Differential Geometry , 479–522 (2001).[12] B.V. Fedosov : A simple geometrical construction of deformation quantization, Journal of Differential Ge-ometry , 213-238 (1994).[13] B.V. Fedosov : Deformation quantization and index theory , Mathematical Topics vol. 9, Akademie Verlag,Berlin, 1996.[14] Fedosov, B. V. : Non-Abelian reduction in deformation quantization,
Lett. Math. Phys. , 137–154. (1998)[15] B.V. Fedosov : On the trace density in deformation quantization, in Deformation quantization (Strasbourg,2001), vol. 1 of IRMA Lect. Math. Theor. Phys., de Gruyter, Berlin, 67–83 (2002).[16] B.V. Fedosov : Quantization and The Index, Dokl. Akad. Nauk. SSSR , 82–86 (1986).[17] A. Futaki : Asymptotic Chow semi-stability and integral invariants, Internat. Journ. of Math., (9), 967–979(2004).[18] A. Futaki : Stability, integral invariants and canonical K¨ahler metrics, Proc. 9-th Internat. Conf. on Dif-ferential Geometry and its Applications, 2004 Prague, (eds. J. Bures et al), 2005, 45-58, MATFYZPRESS,Prague.[19] A. Futaki : Asymptotic Chow polystability in K¨ahler geometry, Fifth International Congress of ChineseMathematicians. Part 1, 2, 139–153, AMS/IP Stud. Adv. Math., 51, pt. 1, 2, Amer. Math. Soc., Providence,RI, 2012.[20] A. Futaki, H. Ono : Einstein metrics and GIT stability, Sugaku Expositions, 24(2011), 93-122.arXiv:0811.0067.[21] A. Futaki, H. Ono : Cahen-Gutt moment map, closed Fedosov star product and structure of the automor-phism group, J. Symplectic Geom. 18(2020), 123-145.[22] A. Futaki, H. Ono and Y. Sano : Hilbert series and obstructions to asymptotic semistability, Advances inMath., 226 (2011), 254–284. DOI: 10.1016/j.aim.2010.06.018[23] A. Futaki, L. La Fuente-Gravy : Deformation quantization and K¨ahler geometry with moment map, toappear in Proc. of ICCM 2018. arXiv1904.11749.[24] S. Gutt, J. Rawnsley : Natural star products on symplectic manifolds and quantum moment maps, Lett. inMath. Phys. , 123–139 (2003).[25] S. Gutt, J. Rawnsley : Traces for star products on symplectic manifolds, Journ. of Geom. and Phys. ,12–18 (2002).[26] L. Ioos : Anticanonicallly balanced metrics on Fano manifolds. Preprint arXiv:2006.05989. June 2020.[27] M. Kontsevitch : Deformation Quantization of Poisson Manifolds. Letters in Math. Phys. 66(2003), 157–216.[28] L. La Fuente-Gravy : Infinite dimensional moment map geometry and closed Fedosov’s star products, Ann.of Glob. Anal. and Geom. (1), 1–22 (2015).[29] L. La Fuente-Gravy, Futaki invariant for Fedosov’s star products. J. Symplectic Geom., 17(2019), 1317-1330.[30] M. M¨uller, N. Neumaier : Some Remarks on g-invariant Fedosov Star Products and Quantum MomentumMappings, J. Geom. Phys. , no. 1-4, 257–272. (2004)[31] M. M¨uller, N. Neumaier : Invariant star products of Wick type: classification and quantum momentummappings, Lett. Math. Phys. , no. 1, 1–15. (2004)[32] R. Nest, B. Tsygan : Algebraic index theorem for families, Advances in Math. , 151–205 (1995).[33] H. Omori, Y. Maeda, A. Yoshioka : Weyl manifolds and deformation quantization, Adv. in Math. , 224–255, 1991.[34] M. Schlichenmaier : Berezin-Toeplitz quantization of compact K¨ahler manifolds, in Quantization, Coher-ent States and Poisson Structures, Proceedings of the 14th Workshop on Geometric Methods in Physics(Bialowieza, Poland, July 1995).[35] P. Xu : Fedosov ∗ -products and quantum momentum maps, Comm. Math. Phys. 197(1998), 167-197.[36] M. Yamashita : A new construction of strict deformation quantization for Lagrangian fiber bundles. Preprint,arXiv:2003.06732. March 2020. Yau Mathematical Sciences Center, Tsinghua University, Haidian district, Beijing 100084, China
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