Some naturally defined star products for Kaehler manifolds

SOME NATURALLY DEFINED STAR PRODUCTS

FOR

KÄHLER MANIFOLDS

Martin Schlichenmaier Mathematics Research Unit

University of Luxembourg

Bayrischzell Workshop 2016 April 29- May 3, 2016

I One mathematical aspect of quantization is thepassage from thecommutative worldto thenon-commutative world.

I one waya deformation quantization(also calledstar product)

I can only be done on the level offormal power seriesover the algebra of functions

OUTLINE

I give an overview of somenaturally defined star productsin the case that our “phase-space manifold” is a (compact)

Kähler manifold

I here we have additionalcomplex structureand search for star productsrespectingit

I yield star products ofseparation of variables type

(Karabegov) resp. Wick or anti-Wick type(Bordemann and Waldmann)

I both constructions are quite different, but there is a1:1 correspondence(Neumaier)

I single outcertainnaturallygiven ones.

I restrict toquantizableKähler manifolds

I Berezin-Toeplitzstar product,Berezintransform,Berezin starproduct

I a side result: star product ofgeometric quantization I all of the above areequivalent star product, but not the

same

I giveDeligne-Fedosov classandKarabegov forms

GEOMETRIC SET-UP

I (M, ω)apseudo-Kählermanifold.

M a complex manifold, and ω, a non-degenerate closed (1, 1)-form

I if ω is apositive formthen (M, ω) is a honestKähler manifold

I C∞(M)the algebra of complex-valued differentiable functions with associative product given bypoint-wise multiplication

I define thePoisson bracket

{ f , g } := ω(Xf,Xg) ω(Xf, ·) =df (·)

STAR PRODUCT

star product for M is anassociative product?on A := C∞(M)[[ν]], such 1. f ? g = f · g mod ν, 2. (f ? g − g ? f ) /ν = −i{f , g} mod ν. Also f ? g = ∞ X k =0 νkCk(f , g), Ck(f , g) ∈ C∞(M),

differential(orlocal) ifCk( , )are bidifferential operators.

Equivalence of star products

?and ?0 (the same Poisson structure) are equivalent means there exists

a formal series of linear operators

B = ∞ X i=0 B_iνi, B_i :C∞(M) → C∞(M), with B0=id and B(f ) ?0B(g) = B(f ? g).

to every equivalence class of a differential star product one assigns itsDeligne-Fedosov class

cl(?) ∈ 1 i( 1 ν[ω] + H 2 dR(M, C)[[ν]]). gives a1:1 correspondence

SEPARATION OF

V

I pseudo-Kählercase: we look for star products adapted to the complex structure

I separation of variables type(Karabegov)

I Wick and anti-Wick type(Bordemann - Waldmann)

I Karabegov convention: of separation of variables type if in Ck(., .)for k ≥ 1 the first argument differentiated in

anti-holomorphic and the second argument in holomorphic directions.

I we call this conventionseparation of variables (anti-Wick) typeand call a star product ofseparation of variables (Wick) typeif the role of the variables is switched

KARABEGOV CONSTRUCTION

(SKETCH OF A SKETCH

)

I (M, ω₋₁)thepseudo-Kählermanifold

I aformal deformationof the form (1/ν)ω₋₁is a formal form

b

ω = (1/ν)ω₋₁+ ω0+ ν ω1+ . . .

ωr, r ≥ 0, closed (1,1)-forms on M.

I Karabegov: toeverysuch_bωthere exists a star product ? of anti-Wick type

I andvice-versa

I Karabegov formof the star product ? is kf (?) :=_bω,

I Formal Berezin transform

I for localantiholomorphicfunctions a andholomorphic

functions b on U ⊂ M we have the relation a ? b =I?(b ? a) =I?(b · a), I can be written as I? = ∞ X i=0 Ii νi, Ii :C∞(M) → C∞(M), I0=id , I1= ∆.

I Start with?separation of variables type(anti-Wick) (M, ω−1)

I opposite of the dual

f ?0g = I−1(I(f ) ? I(g)).

on (M, ω−1), is ofWick type

I theformal Berezin transform I? establishes anequivalence

of the star products

C

?star product of anti-Wick type with Karabegov formkf (?) =ω_b Deligne-Fedosov classcalculates as

cl(?) = 1 i([ω] −b

δ 2).

[..]denotes the de-Rham class of the forms and δ is the

canonical classof the manifold i.e. δ := c1(KM).

standardstar product ?K (withKarabegov formω = (1/ν)ωb −1)

I For the Karabegov form to be in 1:1 correspondence, we need to fix a convention: Wickoranti-Wickfor reference

I here we refer to theanti-Wicktype product

I it ? is of Wick type we set

kf (?) := kf (?op),

I where

f ?opg = g ? f

OTHER GENERAL CONSTRUCTIONS

I Bordemann and Waldmann: modification of Fedosov’s geometric existence proof.

I fibre-wise Wick product.

I by amodified Fedosov connectiona star product?BW of

Wick type is obtained.

I Karabegov formis −(1/ν)ω

I Deligne class class

Neumaier:byadding a formal closed (1, 1) formas parameter each star product of separation of variables type can be obtained by the Bordemann-Waldmann construction Reshetikhin and Takhtajan:

formal Laplace expansions of formal integralsrelated to the star product.

coefficientsof the star product can be expressed (roughly) by

B

STAR PRODUCT

I compactandquantizableKähler manifold (M, ω),

I quantum line bundle(L, h, ∇),Lis a holomorphic line bundle over M,ha hermitian metric on L,∇a compatible connection

I recall (M, ω) is quantizable, if there exists such(L, h, ∇), with

curv_(L,∇)= −i ω

scalar product hϕ, ψi := Z M h(m)(ϕ, ψ) Ω, Ω := 1 n!ω ∧ ω · · · ∧ ω| {z } n Π(m) :L2(M, Lm) −→ Γhol(M, Lm)

Takef ∈ C∞(M), ands ∈ Γhol(M, Lm)

s 7→ T_f(m)(s) := Π(m)(f · s) defines

Berezin-Toeplitz operator quantization f 7→T_f(m)

m∈N0

.

has the correctsemi-classical behavior

Theorem (Bordemann, Meinrenken, and Schl.)

C

C

I Statement of the previous theoremcorresponds to the fact that we have a continuous field of C∗-algebras (with additionallyDirac conditionon commutators).

I over I ={0} ∪ {_m1 ∈ N},

I over {0} we set the algebraC∞(M), over _m1 the algebra

End(Γhol(M, Lm)),

I sectionis given by f ∈ C∞(M)

I

Theorem (BMS, Schl., Karabegov and Schl.) ∃ aunique differential star product

f ?BT g = X νkCk(f , g) such that T_f(m)T_g(m) ∼ ∞ X k =0  1 m k T_C(m) k(f ,g)

Further properties: is ofseparation of variables type (Wick type)

classifyingDeligne-Fedosov class 1_i(1_ν[ω] −δ₂)andKarabegov form −1_ν ω + ωcan

GEOMETRIC QUANTIZATION

Further result:The Toeplitz map of level m T(m):C∞(M) → End (Γhol(M, Lm))

issurjective

implies that the operator Q_f(m)ofgeometric quantization(with holomorphic polarization) can be written asToeplitz operatorof a function fm(maybe different for every m)

indeedTuynman relation:

I star product of geometric quantization I set B(f ) := (id − ν∆₂)f

f ?_GQg = B−1(B(f ) ?BT B(g))

defines anequivalent star product

I can also be given by theasymptotic expansionof product of geometric quantisation operators

I it isnotof separation of variable type

Where is the Berezin star product ??

I It is an important star product: Berezin, Cahen-Gutt-Rawnsley, etc.

I The original definition islimitedin applicability.

I We will give a definition forquantizable Kähler manifold.

I Clue: define it as theopposite of the dualof ?BT.

I f ?Bg := I(I−1(f ) ?BT I−1(g)) I Problem: Howto determineI?

COHERENT STATES

- BEREZIN SYMBOLS

I assume the bundle L isvery ample(i.e. has enough global sections)

I pass to itsdual (U, k ) := (L∗,h−1) with dual metric k

I inside of the total space U, consider thecircle bundle

Q := {λ ∈ U | k (λ, λ) = 1},

APPLICATIONS

I Bergman projectors Π(m),Bergman kernels, ....

I Covariant Berezin symbol σ(m)(A) (of level m)of an operator A ∈ End(Γhol(M, L(m)))

I

I Construction of theBerezin star product,only for limited classes of manifolds(see Berezin, Cahen-Gutt-Rawnsley)

I A(m) _{≤ C}∞_{(M),of level m covariant symbols.} I symbol map isinjective(follows from Toeplitz map

surjective)

I for σ(m)(A) and σ(m)(B) the operators A and B are uniquely fixed

I

σ(m)(A) ?_(m)σ(m)(B) := σ(m)(A · B)

I ?(m)on A(m)is an associative and noncommutative product

B

I(m) :C∞(M) → C∞(M), f 7→I(m)(f ):= σ(m)(T_f(m))

Theorem:(Karabegov - Schl.)

I(m)(f ) has a completeasymptotic expansionas m → ∞

I(m)(f )(x ) ∼ ∞ X i=0 Ii(f )(x ) 1 mi, Ii :C∞(M) → C∞(M),I0(f ) = f , I1(f ) = ∆f.

B

I from asymptotic expansion of the Berezin transform get

formal expression I = ∞ X i=0 I_i νi, I_i :C∞(M) → C∞(M) I setf ?Bg := I(I−1(f ) ?BT I−1(g)) I ?B is called theBerezin star product

I I gives theequivalenceto?BT (I0=id ). Hence, the same Deligne-Fedosov classes

I separation of variables type (but now ofanti-Wick type).

I Karabegov formis 1_νω + F(i ∂∂ log um)

I um is theBergman kernelBm(α, β) = heα(m),e(m)_β i evaluated along the diagonal

I _{F means: take asymptotic expansion in 1/m as}formal seriesin ν

I I = I?B, thegeometric Berezintransform equals theformal

Berezin transformof Karabegov for ?B

I both star products?B and?BT aredual and oppositeto

SUMMARY OF NATURALLY DEFINED STAR PRODUCT

name _{Karabegov form Deligne}

Fedosov class ?BT Berezin-Toeplitz −1 ν ω + ωcan(Wick) 1 i( 1 ν[ω] − δ 2). ?B Berezin 1 νω + F(i ∂∂ log um) (anti-Wick) 1 i( 1 ν[ω] − δ 2). ?_{GQ geometric} quantization (—) 1 i( 1 ν[ω] − δ 2).

?K standard product (1/ν) ω(anti-Wick) 1

I Berezin transformis not only the equivalence relating ?BT with ?B

I also it (resp. the Karabegov form) can be used tocalculate the coefficientsof these naturally defined star products,

I eitherdirectly