Geometric quantization for proper moment maps

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Differential Geometry

Geometric quantization for proper moment maps

Xiaonan Ma

^a

, Weiping Zhang

^b

aUniversité Denis-Diderot – Paris 7, UFR de mathématiques, case 7012, site Chevaleret, 75205 Paris cedex 13, France bChern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, PR China

Received 2 December 2008; accepted 28 January 2009 Available online 6 March 2009

Presented by Jean-Michel Bismut

Abstract

We establish a geometric quantization formula for Hamiltonian actions of a compact Lie group acting on a non-compact symplectic manifold such that the associated moment map is proper. In particular, we give a solution to a conjecture of Michèle Vergne.

To cite this article: X. Ma, W. Zhang, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Résumé

Quantification géométrique pour les applications moment propres. Nous établissons une formule de quantification géo- métrique pour les actions hamiltoniennes d’un groupe de Lie compact agissant sur une variété symplectique non-compacte dont l’application moment est propre. En particulier, nous résolvons une conjecture formulée par Michèle Vergne dans son exposé à l’ICM 2006.Pour citer cet article : X. Ma, W. Zhang, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Version française abrégée

SoitGun groupe de Lie compact connexe agissant sur une variété symplectique non-compacte(M, ω)par une action hamiltonienne. Soit(L,∇^L)un fibré en droites hermitien muni d’une connexion hermitienne etG-invariante, et tel que(∇^L)²=^√^2π₋₁ω. SoitJ une structure presque complexeG-invariante surT M, telle queg^{T M}(u, v)=ω(u, J v) est une métrique riemannienne surT M. Soitgl’algèbre de Lie deG. On munitg^∗d’une métrique AdG-invariante.

On suppose que l’application moment associéeμ:M→g^∗est propre.

SoitH= |μ|². SoitX^H= −J (dH)^∗le champ de vecteurs hamiltonien associé àH.

Poura >0, on poseUa:= {x∈M: H(x)a}. Pour une valeur régulièreadeH, d’après Atiyah [1], le symbole

√−1c(·+X^H)⊗IdL(c(·)est l’action de Clifford) définit un symbole transversalement elliptique surU_a, et son indice transversalQ(L)_aest une distribution surG. Tian et Zhang [13] ont introduit l’opérateur de Dirac correspondant dans leur approche de la quantification géométrique quandMest compacte. Le symbole associé a été utilisé par Paradan [10,11].

E-mail addresses:[email protected] (X. Ma), [email protected] (W. Zhang).

doi:10.1016/j.crma.2009.02.003

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Soit Λ^∗₊ l’ensemble des poids dominants. Pourγ ∈Λ^∗₊, soitV_γ^G la représentationG-irréductible de plus haut poidsγ, soitQ(L)^γ_a ∈Zla multiplicité de la représentationV_γ^GdansQ(L)_a. Pourγ∈Λ^∗₊, on noteQ(L_γ)l’indice de l’opérateur de Dirac Spin^csur la réduction symplectique deMenγ.

Théorème 0.1. Pourγ ∈Λ^∗₊, il existea_γ >0 telle queQ(L)^γ_a ∈Zne dépend pas deaa_γ, poura une valeur régulière deH. On le note commeQ(L)^γ.

Théorème 0.2. Pourγ∈Λ^∗₊, on aQ(L)^γ =Q(L_γ).

Si {X^H=0}est compact, le Théorème 0.2 a été conjecturé par Michèle Vergne dans [15, §4.3], et des cas par- ticuliers de cette conjecture ont été démontrés par Paradan [11,12], dans le cadre des séries discrètes. Les résultats annoncés dans cette note sont démontrés dans [8].

1. Transversal index and the APS index

In this Note, we present a solution of an extended version of the conjecture of Michèle Vergne in her ICM 2006 lecture. Details will appear in [8].

LetMbe a compact manifold with non-empty boundary∂M. LetJ be an almost complex structure onT M. Let g^{T M} be aJ-invariant Riemannian metric on the tangent vector bundleπ:T M →M. Let(E, h^E)be a Hermitian vector bundle overMwith Hermitian connection∇^E.

LetGbe a compact connected Lie group with Lie algebrag. We assume thatGacts onMand that this action lifts to an action onE, and preservesJ,g^{T M},h^Eand∇^E.

For any W ∈T M such that W =w+w∈ T^(1,0)M⊕T^(0,1)M=T M ⊗_R C, the Clifford action c(W ) on Λ(T^∗^(0,1)M) is defined by c(W )=√

2w^∗∧ −√

2iw, where w^∗∈T^∗^(0,1)M is the metric dual of w. Recall that the spin^cstructure associated to the Clifford moduleΛ(T^∗^(0,1)M)is induced by the line bundle det(T^(1,0)M).

Let∇^T^(1,0)^Mbe the Hermitian connection onT^(1,0)Minduced by projection by the Levi-Civita connection∇^{T M}on (T M, g^{T M}). Let∇^detbe the Hermitian connection on det(T^(1,0)M)induced by∇^T^(1,0)^M. Let∇^Λ^0,^•^⊗^Ebe the Clifford connection onΛ(T^∗^(0,1)M)⊗Einduced by∇^{T M},∇^det, and∇^E. Let{e_i}be an orthonormal basis ofT M.

Then one can construct canonically the Spin^c-Dirac operator (twisted by E), D^E =_dimM

i=1 c(e_i)∇_e^Λ_i^0,^•^⊗^E: Ω^0,^•(M, E), withΩ^0,^•(M, E)the space of smooth sections of(T^∗^(0,1)M)⊗EonM.

Lete_n be the inward unit normal vector field perpendicular to∂M. Lete₁, . . . , e_dimM₋₁be an orthonormal basis of T ∂M. Letπ_ij= ∇_e^{T M}_i e_j, e_nbe the second fundamental form of the isometric embeddingı_∂M :∂M →M. Let D_∂M^E :Ω^0,^•(M, E)|∂M→Ω^0,^•(M, E)|∂M be the induced (byD^E) Dirac operator on∂Mdefined by (cf. [6, p. 142])

D_∂M^E = −

dimM−1 i=1

c(e_n)c(e_i)∇e^Λi^0,^•^⊗^E+1 2

dimM−1 i=1

π_ii. (1)

LetΨ :M→gbe aG-equivariant map with AdG-action ong. LetΨ^M denote the vector field overMsuch that Ψ^M(x)equals to the value atx of the vector field generated byΨ (x)∈goverM.

We suppose thatΨ^M|∂M∈T ∂Mis nowhere zero on∂M.

Following [6, Lemma 2.2] and [14, §1c)], set forT ∈R, D_T^E=D^E+

√−1T 2 c

Ψ^M

, D^E_±_,T=D_T^E

Ω^0,êvenôdd(M,E), D_∂M,TÊ =D_∂MÊ −

√−1T 2 c(e_n)c

Ψ^M

, D_∂M,^E _±_,T =D^E_∂M,T|

Ω^0,êvenôdd(M,E)_∂M. (2) Then DÊ_∂M,T preserves the Z2-grading on Ω^0,^•(M, E)|∂M. For any λ∈Spec{D_∂M,Ê _±_,T}, let E_λ,_±_,T be the corresponding eigenspace. Let P_0,_±_,T (resp. P_>0,_±_,T) be the orthogonal projections from the L²-completions of Ω^0,êvenôdd(M, E)|∂M onto

λ0E_λ,_±_,T (resp.

λ>0E_λ,_±_,T).

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For anyT ∈R, let(DÊ₊_,T, P_0,₊_,T)(resp.(D₋Ê_,T, P_>0,₋_,T)) denote the corresponding Atiyah–Patodi–Singer type boundary valued problem [2], [6, Theorem 2.3]. Then both(DÊ₊_,T, P_0,₊_,T)and(D₋Ê_,T, P_>0,₋_,T)are elliptic and G-equivariant.

LetT be a maximal torus ofG, and lettbe its Lie algebra andt^∗its dual. Then the set of the finite dimensional G-irreducible representations is parameterized by the set of dominant weightsΛ^∗₊⊂t^∗. Forγ ∈Λ^∗₊, we denote by V_γ^G the irreducible G-representation with highest weight γ. ThenV_γ^G,γ ∈Λ^∗₊ is aZ-basis of the representation ringR(G).

Recall thatg=t⊕rwithr= [t,g], and sog^∗=t^∗⊕r^∗. Thus we identify naturallyΛ^∗₊as a subset ofg^∗. LetQ^M_APS,T(E, Ψ^M)^γ ∈Z,γ∈Λ^∗₊, be defined by

γ∈Λ^∗₊

Q^M_APS,T

E, Ψ^Mγ

·V_γ^G:=Ker

D₊^E_,T, P_0,₊_,T

−Ker

D^E₋_,T, P_>0,₋_,T

. (3)

Proposition 1.1.For anyγ∈Λ^∗₊, there existsTγ0such thatQ^M_APS,T(E, Ψ^M)^γ does not depend onT Tγ. Denote byQ^M_APS(E, Ψ^M)^γ the quantization number Q^M_APS,T(E, Ψ^M)^γ forT T_γ. ThenQ^M_APS(E, Ψ^M)^γ does not depend ong^{T M},h^E,∇^Eand depends only on the homotopy classes ofJ,Ψ^M.

Let M = M \ ∂M be the interior of M. One identifies T M and T^∗M via g^{T M}. Let σ^M

E,Ψ^M ∈ Hom(π^∗(Λ^even(T^∗^(0,1)M) ⊗E), π^∗(Λ^odd(T^∗^(0,1)M) ⊗ E)) denote the symbol defined by σ^M

E,Ψ^M(x, v(x)) =

√−1π^∗(c(v+Ψ^M)⊗IdE)_(x,v(x)).¹ The symbol σ^M

E,Ψ^M defines a G-transversally elliptic symbol onT_GM in the sense of Atiyah [1, §3], which in turn determines a transversal index (cf. also [10, §3], [11, §3])

Ind σ^M

E,Ψ^M

=

γ∈Λ^∗₊

Indγ

σ^M

E,Ψ^M

·V_γ^G, with each Indγ

σ^M

E,Ψ^M

∈Z. (4)

The following result can be proved by using the main theorem of Braverman in [4]:

Theorem 1.2.For anyγ∈Λ^∗₊, one hasIndγ(σ^M

E,Ψ^M)=Q^M_APS(E, Ψ^M)^γ. 2. Geometric quantization for proper moment maps

Let(M, ω)be a non-compact symplectic manifold with symplectic formω. We assume that there exists a Hermitian line bundle(L, h^L)(called a prequantized line bundle) carrying a Hermitian connection∇^Lsuch that(∇^L)²=^√^2π₋₁ω.

LetJ be an almost complex structure on T M such that g^{T M}(u, v)=ω(u, J v) defines a J-invariant Riemannian metric onT M.

We assume that the compact connected Lie groupGacts onMand this action can be lifted to an action onL. We assume also thatGpreservesg^{T M},J,h^Land∇^L. Then the moment mapμ:M→g^∗is defined by 2π√

−1μ(K):=

∇_K^LM −LK, for anyK∈g, which verifiesdμ(K)=i_KMω.

Basic Assumption. The corresponding moment mapμ:M→g^∗is proper.

Take anyγ ∈Λ^∗₊. Ifγ is a regular value of the moment mapμ, then one can construct the Marsden–Weinstein symplectic reduction(M_γ, ω_γ), whereM_γ =μ⁻¹(G·γ )/Gis acompactorbifold. Moreover,L(resp.J) induces a prequantized line bundle L_γ (resp. an almost complex structureJ_γ) over (M_γ, ω_γ). One can then construct the associated Spin^c-Dirac operator (twisted byL_γ) onM_γ whose indexQ(L_γ)is well-defined. If γ ∈Λ^∗₊ is not a regular value ofμ, then by proceeding as in [9], one still gets a well-defined quantization numberQ(L_γ)extending the above definition.

1 This kind of deformation (byΨ^M) has been used by Tian–Zhang [13,14] and Paradan [10,11] in their approaches to the Guillemin–Sternberg geometric quantization conjecture [7].

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We equip gwith a AdG-invariant metric, and we identifyg withg^∗ by the metric. Set H= |μ|². Let X^H=

−J (dH)^∗=2μ^M be the Hamiltonian vector field associated toH.

For anya >0,U_a:= {x∈M: H(x)a}is a compact subset ofM. For any regular valuea >0 ofH,X^H is nowhere zero on∂U_a=H⁻¹(a).

Theorem 2.1.For anyγ∈Λ^∗₊, there existsa_γ >0such thatQ^U_APS^a (L, μ^M)^γ∈Zdoes not depend onaa_γ, witha a regular value ofH.

For anyγ∈Λ^∗₊, we denote byQ(L)^γ the well-defined integerQ^U_APS^a (L, μ^M)^γ not depending on the regular value a0.

Let(N, ω^N)be a compact symplectic manifold and(F, h^F,∇^F)the prequantized line bundle overN, andGacts onN, F and preservesJ^N, h^F,∇^F. Letη:N→g^∗be the corresponding moment map.

We will use the same notation for the natural extension of the objects onM,Nto(M×N, ω⊕ω^N). In particular, L⊗F is the prequantized line bundle overM×N obtained by the tensor product of the natural liftings ofLandF toM×N. Then the induced moment mapθ:M×N→g^∗is given byθ (x, y)=μ(x)+η(y).

Theorem 2.2.For the induced action ofGon(M×N, ω⊕ω^N)andL⊗F, we have, Q

(L⊗F )_γ₌₀

=

γ∈Λ^∗₊

Q(L)^γQ(F )⁻^γ. (5)

The proof of Theorem 2.2 is deferred to Section 4.

By takingNin (5) to be the orbits of the co-adjoint action ofGong^∗, we get:

Theorem 2.3.For anyγ∈Λ^∗₊, one hasQ(L)^γ=Q(L_γ).

Remark 2.4.(i) IfMis compact, then Theorem 2.1 holds tautologically and Theorem 2.3 is the Guillemin–Sternberg geometric quantization conjecture [7] proved in [9].

(ii) In view of Theorem 1.2, Theorem 2.3 was conjectured by Michèle Vergne in [15, §4.3] in the case when the zero set ofX^His compact. Special cases of this conjecture, related to the discrete series of semi-simple Lie groups, have been proved by Paradan [11,12].

Theorem 2.3 provides a proof of this conjecture even when the zero set ofX^His non-compact.

3. A vanishing result

Our proof of Theorem 2.2 relies essentially on a vanishing result, Theorem 3.1, which is established in this section.

ForA >0 large enough which is a regular value of the functions|μ|² and ¹₂|θ|² onM×N, we define M= {(x, y)∈M×N; |μ|²A, |θ|²2A},M1= {(x, y)∈M×N; |μ|²=A},M_A= {x∈M; |μ|²A}andM2= {(x, y)∈M×N; |θ|²=2A}. LetMA= {(x, y)∈M×N,|μ(x)|²A},M=M∪MA= {|θ|²2A}.

ThenMis a smooth manifold with boundary∂Mand∂M=M1∪M2,M1=∂M_A×N. Setα,˜ φ˜∈C^∞(R)verify the following conditions,

˜ α(t )=

t², fort¹₃,

1, fort²₃, φ(t )˜ =

1−t³, fort¹₃, 2(1−t ), fort²₃,

˜

α(t )+ ˜φ(t )29

27, φ˜(t ) <0, for 1

3t2

3. (6)

ForA >0, setα(t )= ˜α(_A^t −1),φ(t)= ˜φ(_A^t −1).

We defineβ∈C^∞(M×N ),ρ:M×N→g^∗gbyβ= |μ|²+α(|μ|²)(|θ|²− |μ|²),ρ=θ−φ(β)η.

LetV₁, . . . , V_dimG be an orthonormal basis ofg. Denote by V_i^M, V_i^N the Killing vector fields onM, N induced byV_i. For any functionQwith values ing, we will denoteQ_i itsi-component with respect to the basis{V_i}, and when a subscript index appears two times in a formula, we sum up with this index. Set

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γ_j=2 1+α

|μ|²

|θ|²− |μ|²

μ_j+2α

|μ|² η_j,

ψ_j=2ρj−2φ(β)ρ_iη_iγ_j. (7)

Letψ:=ψjVj :M→gbe the inducedG-equivariant map and letY be the vector field onMinduced byψ, then Y :=ψ^M=ψ_j(V_j^M+V_j^N). We also haveψ|_M₁=2μ, andψ|_M₂=(2+_A⁸ρ_iη_i)θ.

Theorem 3.1.WhenA >0is large enough, we haveQ^M_APS(L⊗F, Y )^γ⁼⁰=0.

Outline of the proof of Theorem 3.1.ForT ∈R, letD_T^M be the operator defined byD_T^M=D^L^⊗^F +^√⁻₂^1Tc(Y ): Ω^0,^•(M, L⊗F )→Ω^0,^•(M, L⊗F ). For anyT ∈R, letF_T^M:Ω^0,^•(M, L⊗F )→Ω^0,^•(M, L⊗F )be defined byF_T^M=D_T^M^,2+√

−1T ψ_jL_V_j.

Let{e_k}^dimM_k₌₁ (resp.{f_i}^dimN_i₌₁ ) be an orthonormal frame ofT M (resp.T N). Let∇^T^(1,0)^M,∇^T^(1,0)^N be the connections onT^(1,0)M,T^(1,0)Ninduced by the Levi-Civita connections. Set

I₁=1 4c

d^Mψ_j_∗ c

V_j^M+2V_j^N +1

2c

d^Nψ_j_∗ c

V_j^M+V_j^N , I₂=1

4

1+ J

√−1

V_j^M,

d^Mψ_j_∗ ,

Rj=

√−1 4

dimN

i=1

c(fi)c

∇f^{T N}_i V_j^N

−

√−1 2 Tr

∇^T^(1,0)^NV_j^N

T^(1,0)N

. (8)

Proposition 3.2.The following Bochner type identity holds onM, F_T^M=D^L^⊗^F,2+T

2π ψjθ_j+ψ_jR_j+√

−1(I1+I₂) +T²

4 |Y|² +

√−1T 4

dimM

k=1

c(e_k)c

∇e^{T M}_k

ψ_jV_j^M

−

√−1T

2 Tr

∇^T^(1,0)^M

ψ_jV_j^M

T^(1,0)M

. (9)

Proposition 3.3.There existsA₀>0such that for anyA > A₀,z∈M\∂Mthere exists an open neighborhoodU_z ofz;C_z, C_1,z>0such that for anyT 1ands∈Ω^0,^•(M, L⊗F )withsupp(s)⊂U_z, one has

Re

F_T^Ms, s

C_zD^L^⊗^Fs²

0+(T−C_1,z)s²₀

. (10)

Proof. IfY (z)=0, then by (9), we get easily (10). IfY (z)=0, from our choice ofψ_j, we have the following crucial estimate: there existC >0,A₀>0 such that for anyA > A₀and(x, y)∈ {Y =0} ⊂M, we have

√−1(I1+I₂)_(x,y)−CId_Λ(T∗(0,1)(M×N ))⊗L⊗F. (11)

Moreover,ψ_jθ_j= |μ|²+O(A^1/2),|R_j|’s are bounded, andψ_jV_j^M= −J (d^M|ρ|²)^∗. We then apply the arguments in [13, §2.4] to get (10). 2

From Proposition 3.2, asY is nowhere zero on∂M, by proceeding as in the proof of [14, Proposition 2.4], we get an estimate similar to [14, Proposition 2.4] forD_T^Mon an open neighborhood of∂M. Now from Proposition 3.3, the estimate near∂MforD_T^M, and the gluing arguments in [3, p. 115–116], we get finally:

Theorem 3.4.ForAas in Proposition3.3, there existC, C₁>0such that for anyT 1ands∈Ω^0,^•(M, L⊗F )^G withP_0,_±_,T(s|∂M)=0, one has

D^M_T s²

0CD^L^⊗^Fs²

0+(T−C1)s²0

. (12) In particular, forT >0large enough, Theorem3.1holds.

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Remark 3.5.There are other choices ofαandφfor which Theorem 3.1 is still valid. A possible simpler choice for Y would be to useα≡0 orα≡1 in (7), and then choose a suitableφ. However, we could not get the corresponding estimate (11) for these choices, thus we could not eliminate the potential contributions caused by the possible zero set ofY. This explains why we introduce the non-linear deformationβ in Theorem 3.1.

4. Proof of Theorem 2.2

Let ψ^M be a smoothG-invariant vector field onM induced by a G-equivariant mapψ:M→g, such that ψ^M=Y onM. Then forA >0 large enough, we have

Q^M_APS

L⊗F, θ^Mγ=0

=Q^M_APS

L⊗F, ψ^Mγ=0

=Q^M_APS^A

L⊗F, ψ^Mγ=0

+Q^M_APS(L⊗F, Y )^γ⁼⁰

=Q^M_APS^A

L⊗F, ψ^Mγ=0

, (13)

where in the first equation, we use the deformationt θ^M+(1−t )ψ^M, 0t1, which is nowhere zero onM2; while in the second equation, we make use of the splitting property of the APS type index (cf. [5, Theorem 1.1] for the product metrics case, and one deduces the general case by a deformation argument); while in the third equation, we use Theorem 3.1. We then use the deformationt Y+(1−t )μ^M, 0t1, which isμ^MalongM_Aand thus nowhere zero onM1, to complete the proof of Theorem 2.2.

Acknowledgements

We thank the referee for very helpful suggestions. The work of the second author was partially supported by MOEC and NSFC. Part of the Note was written while the second author was visiting the School of Mathematics of Fudan University during November and December of 2008. He would like to thank Professor Jiaxing Hong and other members of the School for hospitality.

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