Kapranov's L ∞ structures, Fedosov's star products, and one-loop exact BV quantizations on Kähler manifolds
KKAPRANOV’S L ∞ STRUCTURES, FEDOSOV’S STAR PRODUCTS, ANDONE-LOOP EXACT BV QUANTIZATIONS ON K ¨AHLER MANIFOLDS
KWOKWAI CHAN, NAICHUNG CONAN LEUNG, AND QIN LIA
BSTRACT . We study quantization schemes on a K¨ahler manifold and relate several in-teresting structures. We first construct Fedosov’s star products on a K¨ahler manifold X as quantizations of Kapranov’s L ∞ -algebra structure. Then we investigate the Batalin-Vilkovisky (BV) quantizations associated to these star products. A remarkable feature isthat they are all one-loop exact, meaning that the Feynman weights associated to graphswith two or more loops all vanish. This leads to a succinct cochain level formula for thealgebraic index.
1. I
NTRODUCTION
K¨ahler manifolds possess very rich structures because they lie at the crossroad of com-plex geometry and symplectic geometry. On the other hand, as symplectic manifoldswhich admit natural polarizations (namely, the complex polarization), K ¨ahler manifoldsprovide a natural ground for constructing quantum theories. In particular, there havebeen extensive studies on deformation quantization on K¨ahler manifolds. A notable ex-ample is the Berezin-Toeplitz quantization, which is closely related to geometric quanti-zation in the complex polarization [3, 4, 17–20, 22, 23].This paper is another attempt to understand the relation between the K ¨ahlerian condi-tion and properties of quantum theories. Our starting point is Kapranov’s famous con-struction of an L ∞ -algebra structure for a K ¨ahler manifold in [16], which was motivatedby the study of Rozansky-Witten theory [24] (and also [21]) and has since been playingimportant roles in many different subjects.Let X be a K ¨ahler manifold. Using the Atiyah class of the holomorphic tangent bun-dle TX , Kapranov constructed a natural L ∞ -algebra structure on the Dolbeault complex A •− X ( TX ) , enabling us to view TX [ − ] as a Lie algebra object in the derived category ofcoherent sheaves on X . In Section 2.2, we reformulate this structure as a flat connection D K (where the subscript “K” stands for “Kapranov”) on the holomorphic Weyl bundle W X over X . On the other hand, Fedosov abelian connections , which give rise to deformationquantizations or star products on X , are connections of the form D = ∇ − δ + h [ I , − ] (cid:63) on thecomplexified Weyl bundle W X , C satisfying D =
0; here ∇ is the Levi-Civita connection, Mathematics Subject Classification.
Key words and phrases. L ∞ structure, deformation quantization, star product, BV quantization, algebraicindex theorem, K¨ahler manifold. a r X i v : . [ m a t h . QA ] A ug CHAN, LEUNG, AND LI I ∈ A ( X , W X , C ) is a 1-form valued section of W X , C , and (cid:63) is the fiberwise Wick producton W X , C . The flatness condition D = Fedosov equation (1.1) ∇ I − δ I + h I (cid:63) I + R ∇ = α .Our first main result says that Kapranov’s L ∞ structure can naturally be quantized to pro-duce Fedosov abelian connections D F , α (where the subscript “F” stands for “Fedosov”): Theorem 1.1 (=Theorem 2.17) . For a representative α of any given formal cohomology class [ α ] ∈ ¯ hH dR ( X )[[ ¯ h ]] of type (
1, 1 ) , there exists a Fedosov abelian connection D F , α = ∇ − δ + h [ I α , − ] (cid:63) such that D F , α | W X = D K . There are some interesting features of the resulting star products (cid:63) α on X . First of all,they are of so-called Wick type , which roughly means that the corresponding bidiffer-ential operators respect the complex polarization (Proposition 2.20). By computing theKarabegov form associated to our star products, we can also show that every Wick typestar product arises from our construction (Corollary 2.26). More importantly, our solu-tions of the Fedosov equation (1.1) satisfy a gaugue condition (Proposition 2.21) which isdifferent from all previous constructions of Fedosov quantization on K¨ahler manifolds.Because of this, our construction is more consistent with the Berezin-Toeplitz quantiza-tion and also the local picture that z acts as the creation operator (which is classical) while ¯ z acts as the annihilation operator ¯ h ∂∂ z (which is quantum). Furthremore, our Fedosov quan-tization is, in a certain sense, polarized because only half of the functions, namely, theanti-holomorphic ones, receive quantum corrections. See Section 2.3 for more details.Next we go from deformation quantization to quantum field theory (QFT). From theQFT viewpoint, the Fedosov quantization describes the local data of a quantum mechan-ical system, namely, the cochain complex ( A • X ( W X , C ) , D F , α ) gives the cochain complex of local quantum observables of a sigma model from S to the targetmanifold X . To obtain global quantum observables and also define the correlation functions ,we construct the Batalin-Vilkovisky (BV) quantization [2] of this quantum mechanical sys-tem, which comes with a map from local to global quantum observables called the factor-ization map . We will mainly follow Costello’s approach to the BV formalism [7] and relyon Costello-Gwilliam’s foundational work on factorization algebras in QFT [9, 10].To construct a BV quantization, one wants to produce solutions of the quantum masterequation (QME) (see Lemma 3.3, Definition 3.4 and the equation (3.1)):(1.2) Q BV ( e r /¯ h ) = Q BV : = ∇ + ¯ h ∆ + h d TX R ∇ is the so-called BV differential , by running the homotopygroup flow which is defined by choosing a suitable propagator. For general symplectic Combining with the results in [5], we can show that these Fedosov star products actually coincide withthe Berezin-Toeplitz star products [6].
APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 3 manifolds, it was shown in the joint work [15] of the third author with Grady and Si Lithat canonical solutions of the QME can be constructed by applying the homotopy groupflow operator to solutions of the Fedosov equation (1.1).In the K¨ahler setting, it is desirable to choose the propagator differently to make it morecompatible with the complex polarization. This leads to what we call the polarized prop-agator (see Definition 3.8) and hence a slightly different form of the canonical solutionsto the QME. More precisely, in Theorem 3.15, we will construct the canonical solution e ˜ R ∇ /2¯ h · e γ ∞ /¯ h of the QME (1.2) from a solution γ of the Fedosov equation (2.8) (which isequivalent to (1.1)); here the curvature term ˜ R ∇ appears precisely because of the differ-ent choice of the propagator. The second main result of this paper is that the resultingBV quantization is one-loop exact , meaning that the above canonical solution of the QMEadmits a Feynman graph expansion that involves only trees and one-loop graphs : Theorem 1.2 (=Theorem 3.26) . Let γ be a solution of the Fedosov equation (2.8) . Then theFeynman weight associated to a graph G with two or more loops vanishes, i.e.,W G ( P , d TX γ ) = whenever b ( G ) ≥ Hence, the graph expansion of the canonical solution of the QME associated to the Fedosov con-nection D F , α by Theorem 3.15 involves only trees and one-loop graphs: γ ∞ = ∑ G : connected , b ( G )= ¯ h g ( G ) | Aut ( G ) | W G ( P , d TX γ ) .This is in sharp contrast with the general symplectic case studied in [15] where the BVquantization involves quantum corrections from graphs with any number of loops. Thesame kind of one-loop exactness has been observed in a few cases before, including theholomorphic Chern-Simons theory studied by Costello [8] and a sigma model from S tothe target T ∗ Y (cotangent bundle of a smooth manifold Y ) studied by Gwilliam-Grady[14]. Theorem 1.2 shows that K¨ahler manifolds provide a natural geometric ground forproducing such one-loop exact QFTs. Remark . If the Feynman weights associated to graphs of higher genera ( ≥
2) giverise to exact differential forms and thereby contributing trivially to the computation ofcorrelation functions, we may call the quantization cohomologically one-loop exact . Thisis much more commonly found in the mathematical physics literature and should bedistinguished from our notion of one-loop exactness here.As in the symplectic case [15], from the canonical QME solution e ˜ R ∇ /2¯ h · e γ ∞ /¯ h , we ob-tain the local-to-global factorization map, which can be used to define the correlationfunction (cid:104) f (cid:105) of a smooth function f ∈ C ∞ ( X )[[ ¯ h ]] (see Definition 3.18 and Proposition3.20). The association Tr : f (cid:55)→ (cid:104) f (cid:105) then gives a trace of the star product (cid:63) α associated tothe Fedosov connection D F , α (Corollary 3.21). CHAN, LEUNG, AND LI
As an application of the one-loop exactness of our BV quantization in Theorem 1.2,we deduce a novel cochain level formula for the for the algebraic index Tr ( ) , which is thecorrelation function of the constant function 1, and thus a particularly neat presentationof the algebraic index theorem : Theorem 1.3 (=Theorem 3.39 & Corollary 3.40) . We have σ (cid:16) e ¯ h ι Π ( e ˜ R ∇ /2¯ h e γ ∞ /¯ h ) (cid:17) = ˆ A ( X ) · e − ω ¯ h ¯ h + Tr ( R + ) = Td ( X ) · e − ω ¯ h ¯ h + Tr ( R + ) , where Td ( X ) is the Todd class of X and R + is the curvature of the holomorphic tangent bundledefined in (2.1) . In particular, we obtain the algebraic index theorem , namely, the trace of thefunction is given by Tr ( ) = (cid:90) X ˆ A ( X ) · e − ω ¯ h ¯ h + Tr ( R + ) = (cid:90) X Td ( X ) · e − ω ¯ h ¯ h + Tr ( R + ) .This theorem can be regarded as a cochain level enhancement of the result in [15]. Inthe forthcoming work [6], by combining with the results in [5], we will prove that thestar product (cid:63) α associated to α = ¯ h · Tr ( R + ) is precisely the Berezin-Toeplitz star product on a prequantizable K¨ahler manifold studied in [3, 4, 20]. In this case, the algebraic indextheorem is of the simple form: Tr ( ) = (cid:90) X Td ( X ) · e ω /¯ h . Acknowledgement.
We thank Si Li and Siye Wu for useful discussions. The first named author thanks Mar-tin Schlichenmaier and Siye Wu for inviting him to attend the conference GEOQUANT2019 held in September 2019 in Taiwan, in which he had stimulating and very helpfuldiscussions with both of them as well as Jrgen Ellegaard Andersen, Motohico Mulase,Georgiy Sharygin and Steve Zelditch.K. Chan was supported by a grant of the Hong Kong Research Grants Council (ProjectNo. CUHK14303019) and direct grant (No. 4053395) from CUHK. N. C. Leung was sup-ported by grants of the Hong Kong Research Grants Council (Project No. CUHK14301117& CUHK14303518) and direct grant (No. 4053400) from CUHK. Q. Li was supported by agrant from National Natural Science Foundation of China for young scholars (Project No.11501537). 2. F
ROM K APRANOV TO F EDOSOV
There have been extensive studies on the Fedosov quantization of K¨ahler manifolds. Inthis section, we give a new construction of the Fedosov quantization (i.e., solutions of theFedosov equation for abelian connections on the Weyl bundle) as a natural quantizationof Kapranov’s L ∞ -algebra structure on a K¨ahler manifold. APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 5
The organization of this section is as follows: In Section 2.1, we review some basicK¨ahler geometry, including the geometry of Weyl bundles on a K ¨ahler manifold X . InSection 2.2, we recall the L ∞ -algebra structure introduced by Kapranov [16], reformu-lated using the geometry of Weyl bundles on X . In Section 2.3, we construct Fedosov’sflat connections as quantizations of Kapranov’s L ∞ -algebra structure, which produce starproducts on X of Wick type. We also compute the Karabegov forms associated to suchstar products and prove that every Wick type star product arises from our construction.2.1. Preliminaries in K¨ahler geometry.
Some basic identities.
We first collect some basic identities in K ¨ahler geometry, which are needed in latercomputations.First of all, writing the K¨ahler form as ω = ω α ¯ β dz α ∧ d ¯ z β = √− g α ¯ β dz α ∧ d ¯ z β ,where we adopt the convention that ω ¯ γα ω α ¯ β = δ ¯ γ ¯ β , then a simple computation shows that g α ¯ β = −√− ω α ¯ β .In local coordinates, the curvature of the Levi-Civita connection is given by ∇ ( ∂ x k ) = R lijk dx i ∧ dx j ⊗ ∂ x l ,or in complex coordinates: ∇ ( ∂ z k ) = R li ¯ jk dz i ∧ d ¯ z j ⊗ ∂ z l ,where the coefficients can be written as the derivatives of Christoffel symbols: R li ¯ jk = − ∂ ¯ z j ( Γ lik ) .We define R i ¯ jk ¯ l by R i ¯ jk ¯ l : = g ( R ( ∂ z i , ∂ ¯ z j ) ∂ z k , ∂ ¯ z l ) = g ( R mi ¯ jk ∂ z m , ∂ ¯ z l ) = R mi ¯ jk g m ¯ l .We can also compute the curvature on the cotangent bundle: R ( ∂ z i , ∂ ¯ z j )( y k ) = −R ( ∂ ¯ z j , ∂ z i )( y k ) = − ∂ ¯ z j ( − Γ kil dz i ⊗ y l )= ∂ ¯ z j ( Γ kij ) d ¯ z j ∧ dz i ⊗ y l = R li ¯ jk dz i ∧ d ¯ z j ⊗ y l .The following computation shows that the curvature operator R can be expressed as abracket: √− h [ R i ¯ jk ¯ l dz i ∧ d ¯ z j ⊗ y k ¯ y l , y m ] (cid:63) = − √− h R i ¯ jk ¯ l dz i ∧ d ¯ z j ⊗ (cid:16) y m (cid:63) y k ¯ y l − y m y k ¯ y l (cid:17) = −√− R i ¯ jk ¯ l ω m ¯ l dz i ∧ d ¯ z j ⊗ y k = −√− R ni ¯ jk g n ¯ l ω m ¯ l dz i ∧ d ¯ z j ⊗ y k CHAN, LEUNG, AND LI = −√− R ni ¯ jk ( −√− ω n ¯ l ) ω m ¯ l dz i ∧ d ¯ z j ⊗ y k = R mi ¯ jk dz i ∧ d ¯ z j ⊗ y k = ∇ ( y m ) .For later computations, we also use the notation R + to denote the curvature of theholomorphic tangent bundle:(2.1) R + : = R mi ¯ jk dz i ∧ d ¯ z j ⊗ ( ∂ z m ⊗ y k ) .In particular, we have an explicit formula for its trace:(2.2) Tr ( R + ) = R ki ¯ jk = R i ¯ jk ¯ l g k ¯ l = −√− R i ¯ jk ¯ l ω k ¯ l .2.1.2. Weyl bundles on K¨ahler manifolds.
Here we recall the definitions and basic properties of various types of Weyl bundles onsymplectic and K¨ahler manifolds.
Definition 2.1.
For a symplectic manifold ( M , ω ) , its (real) Weyl bundle is defined as W M , R : = (cid:100) Sym ( T ∗ M R )[[ ¯ h ]] ,where (cid:100) Sym ( T ∗ M R ) is the completed symmetric power of the cotangent bundle T ∗ M R of M and ¯ h is a formal variable. A (smooth) section a of this infinite rank bundle is givenlocally by a formal series a ( x , y ) = ∑ k , l ≥ ∑ i ,..., i l ≥ ¯ h k a k , i ··· i l ( x ) y i · · · y i l ,where the a k , i ··· i l ( x ) (cid:48) s are smooth functions on M . Remark . We use the following notation for the symmetric tensor power: y i · · · y i l : = ∑ τ ∈ S l y i τ ( ) ⊗ · · · ⊗ y i τ ( l ) .Here the product on the tensor algebra is given by ( y i ⊗ · · · ⊗ y i k ) · ( y i k + ⊗ · · · ⊗ y i k + l ) : = ∑ τ ∈ Sh ( k , l ) y i τ ( ) ⊗ · · · ⊗ y i τ ( k + l ) ,where Sh ( k , l ) denotes the set of all ( k , l ) -shuffles.There is a canonical (classical) fiberwise multiplication, denoted as · , which makes W M , R an (infinite rank) algebra bundle over M . We will also consider differential formswith values in W M , R , i.e., A • M ( W M , R ) . Remark . In this subsection, we are only concerned with the classical geometry of Weylbundles; the formal variable ¯ h is included in Definition 2.1 for discussing their quantumgeometry in later sections. APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 7
Definition 2.2.
There are two important operators on A • M ( W M , R ) which are A • M -linear: δ a = dx k ∧ ∂ a ∂ y k , δ ∗ a = y k · ι ∂ xk a .Here ι ∂ xk denotes the contraction of differential forms by the vector field ∂∂ x k . We can nor-malize the operator δ ∗ by letting δ − : = l + m δ ∗ when acting on the monomial y i · · · y i l dx j ∧· · · ∧ dx j m . Then for any form a ∈ A • M ( W M , R ) , we have a = δδ − a + δ − δ a + a ,where a denotes the constant term, i.e., the term without any dx i ’s or y j ’s in a . Definition 2.3.
For a K ¨ahler manifold X , we define the holomorphic and anti-holomorphicWeyl bundles respectively by W X : = (cid:100) Sym ( T ∗ X )[[ ¯ h ]] , W X : = (cid:100) Sym ( T ∗ X )[[ ¯ h ]] ,where T ∗ X and T ∗ X are the holomorphic and anti-holomorphic cotangent bundles of X respectively. With respect to a local holomorphic coordinate system { z , . . . , z n } , we let y i ’s and ¯ y j ’s denote the local frames of T ∗ X and T ∗ X respectively. A local section of thecomplexification W X , C : = W X , R ⊗ R C of the real Weyl bundle is then of the form: ∑ k , m ≥ ∑ i ,..., i m ≥ ∑ j ,..., j l ≥ a k , i , ··· , i m , j , ··· , j l ¯ h k y i · · · y i m ¯ y j · · · ¯ y j l ,from which we see that W X , C = W X ⊗ C ∞ X W X . The symbol map σ : A • X ⊗ W X , C → A • X [[ ¯ h ]] is defined by setting y i ’s and ¯ y j ’s to be 0. Notation 2.4.
We will use the notation W p , q to denote the component of W X , C of type ( p , q ) . Also we will often abuse the names “Weyl bundle” and “symbol map” when thereis no ambiguity.We introduce several operators on A • X ( W X , C ) , similar to those in Definition 2.2. Definition 2.5.
There are 4 natural operators acting as derivations on A • X ( W X , C ) : δ a = dz i ∧ ∂ a ∂ y i , δ a = d ¯ z j ∧ ∂ a ∂ ¯ y j ,as well as ( δ ) ∗ a = y k · ι ∂ zk a , ( δ ) ∗ a = ¯ y j · ι ∂ ¯ zj a .We define the operators ( δ ) − and ( δ ) − by normalizing ( δ ) ∗ and ( δ ) ∗ respec-tively: ( δ ) − : = p + p ( δ ) ∗ on A p , q X ( W p , q ) , The Einstein summation rule will be used throughout this paper.
CHAN, LEUNG, AND LI ( δ ) − : = q + q ( δ ) ∗ on A p , q X ( W p , q ) . Remark . We use the same notation for the operator ( δ ) − as in Fedosov’s originalpaper [11], although it could be confusing since it is actually not inverse to δ . Lemma 2.6.
We have the following identities: δ = δ + δ , id − π ∗ = δ ◦ ( δ ) − + ( δ ) − ◦ δ , id − π ∗ ,0 = δ ◦ ( δ ) − + ( δ ) − ◦ δ , where π ∗ and π ∗ ,0 denote the natural projections from A • X ( W X , C ) to A • X ( W X ) and A • ,0 X ( W X ) respectively. A reformulation of Kapranov’s L ∞ structure on a K¨ahler manifold. In this subsection, we reformulate Kapranov’s L ∞ -algebra structure [16] on a K¨ahler man-ifold X in terms of the holomorphic Weyl bundle W X on X . Let us start with Kapranov’soriginal theorem: Theorem 2.7 (Theorem 2.6 and Reformulation 2.8.1 in [16]) . Let X be a K¨ahler manifold.Then there exist R ∗ n ∈ A X ( Hom ( T ∗ X , Sym n ( T ∗ X ))) , n ≥ such that their extensions ˜ R ∗ n to the holomorphic Weyl bundle W X by derivation satisfy (cid:32) ¯ ∂ + ∑ n ≥ ˜ R ∗ n (cid:33) = or equivalently, (2.3) ¯ ∂ ˜ R ∗ n + ∑ j + k = n + ˜ R ∗ j ◦ ˜ R ∗ k = for any n ≥ . The R ∗ n ’s are defined as partial transposes of the higher covariant derivatives of thecurvature tensor: R ∗ = R mi ¯ jk d ¯ z j ⊗ ( y i y k ⊗ ∂ z m ) , R ∗ n = ( δ ) − ◦ ∇ ( R ∗ n − ) ,where, by abuse of notations, we use ∇ to denote the Levi-Civita connection on the(anti)holomorphic (co)tangent bundle of X , as well as their tensor products includingthe Weyl bundles. We can write these R ∗ n ’s locally in a more consistent way as:(2.4) R ∗ n = R ji ··· i n ,¯ l d ¯ z l ⊗ ( y i · · · y i n ⊗ ∂ z j ) .Readers are referred to [16, Section 2.5] for a detailed exposition. APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 9
The L ∞ relations (2.3) can be reformulated as the flatness of a natural connection D K (where the subscript “K” stands for “Kapranov”) on the holomorphic Weyl bundle, whose (
0, 1 ) -part is exactly the differential operator in Theorem 2.7: Proposition 2.8.
The operator D K = ∇ − δ + ∑ n ≥ ˜ R ∗ n defines a flat connection on the holomorphic Weyl bundle W X , which is compatible with the clas-sical (commutative) product.Proof. To show the vanishing of the (
2, 0 ) , (
0, 2 ) and (
1, 1 ) parts of D K , we let D K and D K denote the (
1, 0 ) and (
0, 1 ) parts of the connection D K respectively, and note thatthe Levi-Civita connection ∇ on W X has the type decomposition ∇ = ∇ + ¯ ∂ . First, itis clear that D K is exactly Kapranov’s differential in Theorem 2.7. Thus the vanishingof the (
0, 2 ) part of D K follows from Theorem 2.7. Next, the vanishing of the square of D K = ∇ − δ follows from the following computation: ( ∇ − δ ) = ( ∇ ) + ( δ ) − ( ∇ ◦ δ + δ ◦ ∇ )= − ( ∇ ◦ δ + δ ◦ ∇ ) = ∇ . Finally, for the (
1, 1 ) part,we have D K ◦ D K + D K ◦ D K = ( ∇ − δ ) (cid:32) ∑ n ≥ R ∗ n (cid:33) + ∇ .Also, δ ( R ∗ ) = ∇ . So we only need to show that ∇ ( R ∗ n ) = δ ( R ∗ n + ) for n ≥ R ∗ n is inductively defined by R ∗ n + = ( δ ) − ◦ ∇ ( R ∗ n ) for n ≥
2. It followsthat δ ( R ∗ n + ) = δ ◦ ( δ ) − (cid:16) ∇ ( R ∗ n ) (cid:17) = ∇ ( R ∗ n ) − ( δ ) − ◦ δ (cid:16) ∇ ( R ∗ n ) (cid:17) = ∇ ( R ∗ n ) ,as desired; here the last equality follows from the fact that ∇ ( R ∗ n ) is symmetric in alllower subscripts, which was shown in [16]. (cid:3) Now if α is a local flat section of W X under D K , it is easy to see that σ ( α ) must be aholomorphic function. Actually, the symbol map defines an isomorphism: Proposition 2.9.
The space of (local) flat sections of the holomorphic Weyl bundle with respect tothe connection D K is isomorphic to the space of holomorphic functions. More precisely, the symbolmap σ : Γ flat ( U , W X ) → O X ( U )[[ ¯ h ]] is an isomorphism for any open subset U ⊂ X. To prove this proposition, we mimic Fedosov’s arguments in [11]. First we need alemma:
Lemma 2.10.
Let α ∈ A qX be a smooth differential form on X of type ( q ) . Define a sequence { α k } by α k : = (( δ ) − ◦ ∇ ) k ( α ) ∈ A qX ( W k ,0 ) for k ≥ . Then we have ( δ ◦ ∇ )( α k ) = for all k ≥ , and hence D K (cid:0) ∑ k ≥ α k (cid:1) = .Proof. Without loss of generality we can assume that q =
0, i.e., α is a function on X . Letus write α k as α k = a i ··· i k y i ⊗ · · · ⊗ y i k , where the coefficients a i ··· i k are totally symmetricwith respect to all indices. We also write ∇ ( α k ) = b i ··· i k + dz i ⊗ ( y i ⊗ · · · ⊗ y i k + ) . Wewill show by induction that the coefficients b i ··· i k + are totally symmetric with respect toall indices i , · · · , i k + . This is clearly true for k =
1. For general k ≥
1, it is clear that ∇ ( α k ) = b i ··· i k + dz i ⊗ ( y i ⊗ · · · ⊗ y i k + ) is symmetric in i , · · · , i k + , so we only needto show that it is symmetric in the first two indices i and i . We know from constructionthat ∇ ( α k − ) = a i ··· i k dz i ⊗ y i · · · ⊗ y i k . Since ( ∇ ) =
0, we have(2.5) − dz i ∧ ∇ ( a i ··· i k y i ⊗ · · · ⊗ y i k ) = ∇ ( a i ··· i k dz i ⊗ y i · · · ⊗ y i k ) = ∇ ( α k ) = ∇ ( a i ··· i k y i ⊗ · · · ⊗ y i k )= y i ⊗ ∇ ( a i ··· i k y i ⊗ · · · ⊗ y i k ) + ∇ ( y i ) ⊗ ( a i ··· i k y i ⊗ · · · ⊗ y i k ) .Since the Levi-Civita connection is torsion-free, the second term in the last expression issymmetric in the first two indices. Now the first term has the same symmetric propertyby (2.5). Hence we have ( δ ◦ ∇ )( α k ) = ∇ ( α k ) = ( δ ◦ ( δ ) − + ( δ ) − ◦ δ )( ∇ ( α k ))= ( δ ◦ ( δ ) − )( ∇ ( α k )) = δ ( α k + ) .Therefore D K (cid:32) ∑ k ≥ α k (cid:33) = ( ∇ − δ ) ( α + α + α + · · · )= − δ ( α ) + ( ∇ ( α ) − δ ( α )) + ( ∇ ( α ) − δ ( α )) + · · · = (cid:3) Proof of Proposition 2.9.
The injectivity follows by observing that any nonzero section s of W X with zero constant term cannot be flat since δ ( s ) , where s is the leading term (i.e.of least weight) in s , is of smaller weight and nonzero. APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 11
For the surjectivity, we claim that given a holomorphic function f , the section(2.6) J f : = ∑ k ≥ (( δ ) − ◦ ∇ ) k ( f ) ,with leading term f is flat with respect to D K . From Lemma 2.10, we know that D K ( J f ) =
0. It follows that D K ◦ D K ( J f ) = − D K ◦ D K ( J f ) =
0. Observe that σ ( D K ( J f )) = ¯ ∂ f =
0. Thus, by the same reason as in the proof of the injectivity of σ , we must have D K ( J f ) = (cid:3) Similar arguments give the following proposition, which will be useful later:
Proposition 2.11.
Let α be a ¯ ∂ -closed ( q ) form on X. Then the section α + ∑ k ≥ (( δ ) − ◦ ∇ ) k ( α ) is flat with respect to the connection D K . Fedosov quantization from Kapranov’s L ∞ structure. In [11], Fedosov gave an elegant geometric construction of deformation quantization ona general symplectic manifold (see also [12, 13]). The goal of this section is to show thatFedosov’s quantization arises naturally as a quantization of Kapranov’s L ∞ structure. Thekey is to adapt Fedosov’s construction in the K¨ahler setting by incorporating the complexstructure, or complex polarization, on X .We begin by considering the complexified Weyl bundle W X , C , equipped with the fiber-wise Wick product induced by the K¨ahler form ω : if α , β are sections of W X , C , the Wickproduct is explicitly defined by: α (cid:63) β : = ∑ k ≥ ¯ h k k ! · ω i ¯ j · · · ω i k ¯ j k · ∂ k α∂ y i · · · ∂ y i k ∂ k β∂ ¯ y j · · · ∂ ¯ y j k .With respect to this product, the operator δ on W X , C can be expressed as: δ = h (cid:104) ω i ¯ j ( dz i ⊗ ¯ y j − d ¯ z j ⊗ y i ) , − (cid:105) (cid:63) ,where [ − , − ] (cid:63) denotes the bracket associated to the Wick product. Definition 2.12.
A connection on W X , C of the form D = ∇ − δ + h [ I , − ] (cid:63) is called a Fedosov abelian connection if D =
0. Here ∇ is the Levi-Civita connection, and I ∈ A ( X , W X , C ) is a 1-form valued section of W X , C .Recall the following calculation in [4]: Lemma 2.13 (Proposition 4.1 in [4]) . The curvature of the Levi-Civita connection on the Weylbundle is given by ∇ = h [ R ∇ , − ] (cid:63) , where R ∇ : = √− · R i ¯ jk ¯ l dz i ∧ d ¯ z j ⊗ y k ¯ y l . A simple computation together with Lemma 2.13 shows that the flatness of D is equiv-alent to the Fedosov equation :(2.7) ∇ I − δ I + h I (cid:63) I + R ∇ = α ,where α = ∑ k ≥ ¯ h k ω k ∈ A X [[ ¯ h ]] is a closed formal 2-form on X . Letting γ : = I + ω i ¯ j ( dz i ⊗ ¯ y j − d ¯ z j ⊗ y i ) and ω ¯ h : = − ω + α , then equation (2.7) is equivalent to(2.8) ∇ γ + h γ (cid:63) γ + R ∇ = ω ¯ h .We will show that the flat connection D K on W X gives rise to a Fedosov abelian con-nection D F (where the subscript “F” stands for “Fedosov”), which is a quantization (or quantum extension ) because the Wick product (cid:63) is non-commutative. First recall that D K = ∇ − δ + ∑ n ≥ ˜ R ∗ n ,where each ˜ R ∗ n is an A • X -linear derivation on W X , C via R ∗ n ∈ A X ( Sym n ( T ∗ X ) ⊗ TX ) (seeequation (2.4)). Consider the A • X -linear operator L : A • X (cid:16) (cid:100) Sym ( T ∗ X ) ⊗ TX (cid:17) → A • X (cid:16) (cid:100) Sym ( T ∗ X ) ⊗ T ∗ X (cid:17) given by “lifting the last subscript” using the K¨ahler form ω . Then we can define I n : = L ( R ∗ n ) = R ji ··· i n ,¯ l ω j ¯ k d ¯ z l ⊗ ( y i · · · y i n ¯ y k ) ∈ A X ( W X , C ) . Lemma 2.14.
We have (2.9) ˜ R ∗ n = h [ I n , − ] (cid:63) | W X ;(2.10) ∇ ◦ L = L ◦ ∇ ;(2.11) L ◦ δ = δ ◦ L ;(2.12) δ ◦ L ( R ∗ ) = R ∇ ;(2.13) L ([ R ∗ m , R ∗ n ]) = [ L ( R ∗ m ) , L ( R ∗ n )] (cid:63) . APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 13
Proof.
The construction of I n ’s implies equation (2.9). Equation (2.10) follows from the factthat ∇ ( ω ) =
0. Equation (2.11) is obvious. Equation (2.12) follows from a straightforwardcomputation: δ ◦ L ( R ∗ ) = L ◦ δ (cid:18) R mi ¯ jk d ¯ z j ⊗ ( y i y k ⊗ ∂ z m ) (cid:19) = L (cid:16) R mi ¯ jk dz i ∧ d ¯ z j ⊗ ( y k ⊗ ∂ z m ) (cid:17) = R mi ¯ jk ω m ¯ l dz i ∧ d ¯ z j ⊗ y k y l = R mi ¯ jk √− g m ¯ l dz i ∧ d ¯ z j ⊗ y k y l = R ∇ .To show equation (2.13), notice that L ([ R ∗ m , R ∗ n ]) = L (cid:0) ˜ R ∗ m ( R ∗ n ) + ˜ R ∗ n ( R ∗ m ) (cid:1) . We then havethe explicit computation: L (cid:0) ˜ R ∗ m ( R ∗ n ) (cid:1) = L (cid:16) R ji ··· i m ,¯ l d ¯ z l ⊗ ( y i · · · y i m ⊗ ∂ z j )( R j (cid:48) i (cid:48) ··· i (cid:48) n ,¯ l (cid:48) d ¯ z l (cid:48) ⊗ ( y i (cid:48) · · · y i (cid:48) n ⊗ ∂ z j (cid:48) )) (cid:17) = L (cid:32) ∑ ≤ α ≤ n R ji ··· i m ,¯ l R j (cid:48) i (cid:48) ··· i (cid:48) n ,¯ l (cid:48) d ¯ z l ∧ d ¯ z l (cid:48) ⊗ ( y i · · · y i m y i (cid:48) · · · (cid:99) y i (cid:48) α · · · y i (cid:48) n ⊗ ∂ z j (cid:48) ) (cid:33) = ∑ ≤ α ≤ n ω j (cid:48) ¯ k R i (cid:48) α i ··· i m ,¯ l R j (cid:48) i (cid:48) ··· i (cid:48) n ,¯ l (cid:48) d ¯ z l ∧ d ¯ z l (cid:48) ⊗ ( y i · · · y i m y i (cid:48) · · · (cid:99) y i (cid:48) α · · · y i (cid:48) n ¯ y k ) .On the other hand, we have L ( R ∗ n ) (cid:63) L ( R ∗ m )= (cid:16) R j (cid:48) i (cid:48) ··· i (cid:48) n ,¯ l (cid:48) ω j (cid:48) ¯ k (cid:48) d ¯ z l (cid:48) ⊗ ( y i (cid:48) · · · y i (cid:48) n y ¯ k (cid:48) ) (cid:17) (cid:63) (cid:16) R ji ··· i m ,¯ l ω j ¯ k d ¯ z l ⊗ ( y i · · · y i m y ¯ k ) (cid:17) = L ( R ∗ n ) · L ( R ∗ m )+ ∑ ≤ α ≤ n ω j ¯ k ω j (cid:48) ¯ k (cid:48) ω i (cid:48) α ¯ k R ji ··· i m ,¯ l R j (cid:48) i (cid:48) ··· i (cid:48) n ,¯ l (cid:48) d ¯ z l (cid:48) ∧ d ¯ z l ⊗ ( y i · · · y i m y i (cid:48) · · · (cid:99) y i (cid:48) α · · · y i (cid:48) n ¯ y k (cid:48) )= − L ( R ∗ m ) · L ( R ∗ n )+ ∑ ≤ α ≤ n ( − ) ω j (cid:48) ¯ k (cid:48) R i (cid:48) α i ··· i m ,¯ l R j (cid:48) i (cid:48) ··· i (cid:48) n ,¯ l (cid:48) d ¯ z l (cid:48) ∧ d ¯ z l ⊗ ( y i · · · y i m y i (cid:48) · · · (cid:99) y i (cid:48) α · · · y i (cid:48) n ⊗ ∂ z j (cid:48) )= − L ( R ∗ m ) · L ( R ∗ n ) + L (cid:0) ˜ R ∗ m ( R ∗ n ) (cid:1) . (cid:3) Let I : = ∑ n ≥ I n . Then D F : = ∇ − δ + h [ I , − ] (cid:63) defines a connection on W X , C . Equation (2.9) says that D F is an extension of the D K ,namely, D F | W X = D K . Lemma 2.15.
We have δ I = .Proof. Recall that I = √− R i ¯ jk ¯ l d ¯ z l ⊗ y i y k ¯ y j , where R i ¯ jk ¯ l is symmetric in ¯ j and ¯ l , fromwhich we know that δ I =
0. The statement for the general I n ’s follows from the itera-tive equation I n = ( δ ) − ◦ ∇ ( I n − ) and the commutativity relations [ δ , ( δ ) − ] =[ δ , ∇ ] = (cid:3) Here comes the first main result of this paper:
Theorem 2.16.
The connection D F is flat.Proof. Notice that I = ∑ n ≥ I n is a (
0, 1 ) -form valued in W X , C . We only need to show thevanishing of the (
2, 0 ) , (
0, 2 ) and (
1, 1 ) parts of D F . The vanishing of the (
2, 0 ) part of D F follows from that of D K . The (
0, 2 ) part of D F is given by1¯ h (cid:20) ∇ I − δ I + h [ I , I ] (cid:63) , − (cid:21) (cid:63) .By Lemma 2.14 and the flatness of D K , we have ∇ I − δ I + h [ I , I ] (cid:63) = ∇ I + h [ I , I ] (cid:63) = L (cid:32) ∇ ( ∑ k ≥ ˜ R ∗ k ) + (cid:34) ∑ k ≥ ˜ R ∗ k , ∑ k ≥ ˜ R ∗ k (cid:35)(cid:33) = (
1, 1 ) part of D F is given by ∇ + h (cid:104) ∇ I − δ I , − (cid:105) (cid:63) = h (cid:104) R ∇ + ∇ I − δ I , − (cid:105) (cid:63) = h (cid:34) R ∇ + ( ∇ − δ ) ◦ L (cid:32) ∑ k ≥ R ∗ k (cid:33) , − (cid:35) (cid:63) ( ∗ ) = h (cid:34) δ ◦ L ( R ∗ ) + ( ∇ − δ ) ◦ L (cid:32) ∑ k ≥ R ∗ k (cid:33) , − (cid:35) (cid:63) = h (cid:34) L ◦ δ ( R ∗ ) + L ◦ ( ∇ − δ ) (cid:32) ∑ k ≥ R ∗ k (cid:33) , − (cid:35) (cid:63) = h (cid:104) L (cid:16) δ ( R ∗ ) − δ ( R ∗ ) + ( ∇ R ∗ − δ R ∗ ) + · · · + ( ∇ R ∗ k − δ R ∗ k + ) + · · · (cid:17) , − (cid:105) (cid:63) = ( ∗ ) , and also the fact that L commuteswith both δ and ∇ . Hence D F is a Fedosov abelian connection. A simple computationshows that I actually satisfies the Fedosov equation (2.7) with α = ∇ I − δ I + h I (cid:63) I + R ∇ = APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 15 (cid:3)
Remark . It is worth pointing out that this flat connection D F only has a “classical” part,i.e., the sections I n ∈ A • X ( W X , C ) have no higher order terms in the ¯ h -power expansion.This is very different from Fedosov’s original solutions of his equation.More generally, given any closed formal 2-form α on X with [ α ] ∈ ¯ hH dR ( X )[[ ¯ h ]] of type (
1, 1 ) , the following theorem produces an explicit solution of the Fedosov equation: (2.7): Theorem 2.17.
Let α be a representative of a formal cohomology class in ¯ hH dR ( X )[[ ¯ h ]] of type (
1, 1 ) . Then there exists a solution of the Fedosov equation of the form I α = I + J α ∈ A X ( W X , C ) : ∇ I α − δ I α + h I α (cid:63) I α + R ∇ = α . We denote the corresponding Fedosov abelian connection by D F , α .Proof. The ∂ ¯ ∂ -lemma guarantees the local existence of a formal function g such that α = ¯ ∂∂ ( g ) . We define a section J α of A X ( W X ) by J α : = ∑ k ≥ (cid:16) ( δ ) − ◦ ∇ (cid:17) k ( ¯ ∂ g ) .Such a function g is unique up to a sum of purely holomorphic and purely anti-holomorphicfunctions. It follows that J α is independent of the choice of g . In particular, this impliesthat these local sections J α ’s patch together to give a global section over X .By Proposition 2.11, ¯ ∂ g + J α is closed under D K , so it is also closed under D F by the factthat D F | W X = D K . Now D F ( ¯ ∂ g + J α ) = ∂ ¯ ∂ g + ∇ J α − δ ( J α ) + h [ I , J α ] (cid:63) = − α + ∇ J α − δ J α + h [ I , J α ] (cid:63) ,so ∇ J α − δ J α + h [ I , J α ] (cid:63) = α . Together with Theorem 2.16 and the fact that J α has onlyholomorphic components in W X , C which implies that J α (cid:63) J α = ∇ I α − δ I α + h I α (cid:63) I α + R ∇ = ∇ J α − δ J α + h ([ I , J α ] (cid:63) + J α (cid:63) J α ) = α . (cid:3) We now briefly recall the construction of star products from Fedosov connections.
Proposition 2.18 (Theorem 3.3 in [11]) . Let D = ∇ − δ + h [ I , − ] (cid:63) be a Fedosov abelian con-nection. Then there is a one-to-one correspondence induced by the symbol map σ : σ : Γ f lat ( X , W X , C ) ∼ −→ C ∞ ( X )[[ ¯ h ]] . The flat section O f corresponding to f ∈ C ∞ ( X ) is the unique solution of the iterative equation: (2.14) O f = f + δ − (cid:18) ∇ O f + h [ I , O f ] (cid:63) (cid:19) . For the Fedosov connection D F , α defined in Theorem 2.17, the associated deformationquantization (star product) (cid:63) α on C ∞ ( X )[[ ¯ h ]] is given by(2.15) f (cid:63) α g : = σ ( O f (cid:63) O g ) . Definition 2.19.
We say that a deformation quantization ( C ∞ ( X )[[ ¯ h ]] , (cid:63) ) is of Wick type (or with separation of variables ) if we have f (cid:63) g = f · g whenever f is antiholomorphic or g isholomorphic. This is equivalent to requiring the corresponding bi-differential operators C i ( − , − ) to take holomorphic and anti-holomorphic derivatives of the first and secondarguments respectively. Proposition 2.20.
For every closed formal differential form α of type (
1, 1 ) , the star product (cid:63) α defined in (2.15) is of Wick type.Proof. To show that (cid:63) α is of Wick type, we only need to show that if both f and g are(anti-)holomorphic functions, then f (cid:63) α g = f · g .If f , g are holomorphic, then both O f and O g are sections of the holomorphic Weylbundle W X . Now D F , α = ∇ − δ + h [ I α , − ] (cid:63) where I α = I + J α and since J α ∈ A X ⊗ W X ,we have D F , α | W X = D F | W X = D K . It follows that O f = J f (where J f is defined in (2.6))and must be of the desired type. Thus we have σ ( O f (cid:63) O g ) = f · g by type reasons.If f , g are anti-holomorphic functions, then we are in the opposite situation. By a simpleinduction using equation (2.14), we can see that both O f and O g do not contain any termthat has only holomorphic components in W X , C . Hence it also follows that σ ( O f (cid:63) O g ) = f · g . (cid:3) Remark . The fact that O f = J f for any (local) holomorphic function f says that holo-morphic functions do not receive any quantum corrections in these Fedosov quantiza-tions.2.3.1. Gauge fixing conditions and comparison with previous works.
Fedosov’s original solutions [11] of his equation satisfy the gauge fixing condition that δ − ( I ) =
0. On the other hand, by a simple type reason argument, it is easy to see that oursolutions of the Fedosov equation satisfy instead the following gauge fixing condition: ( δ ) − ( I ) = Proposition 2.21.
The solution I of the Fedosov equation (2.7) satisfying the two conditions: (2.16) ( δ ) − ( I ) = π ∗ ( I ) = is unique. APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 17
Proof.
Equation (2.17) implies that ( δ ◦ ( δ ) − + ( δ ) − ◦ δ )( I ) = I . Together withthe gauge fixing condition (2.16), we have I = ( δ ) − ◦ δ ( I ) . By applying the operator ( δ ) − to equation (2.7), we see that I must satisfy the following iterative equation: I = ( δ ) − (cid:18) ∇ I + h [ I , I ] (cid:63) + R ∇ − ω ¯ h − δ I (cid:19) = ( δ ) − (cid:18) ∇ I + h [ I , I ] (cid:63) + R ∇ − ω ¯ h (cid:19) ,where we have used the fact that the two operators δ and ( δ ) − commute with eachother in the second equality. This iterative equation clearly has a unique solution. (cid:3) It is clear that our solution I = ∑ n ≥ I n of the Fedosov equation (2.7) is exactly theunique one satisfying the conditions (2.16) and (2.17).There were a number of works on the Fedosov construction of Wick type deformationquantizations on (pseudo-)K¨ahler manifolds [4, 18, 23]. Notice that, in all these works,the authors were using the same gauge condition, namely, Fedosov’s original condition δ − I =
0, when solving the Fedosov equation. For the purpose of deformation quan-tization, there is no essential difference between these two choices of gauge conditions.However, here are some interesting features of our construction which were not found inprevious ones:(1) As we have emphasized, our Fedosov connection is a quantization of Kapranov’s L ∞ -algebra structure on a K¨ahler manifold.(2) In our construction, the sections O f corresponding to holomorphic functions f donot receive any quantum corrections. This is consistent with the Berezin-Toeplitzquantization, and also the local picture where z acts as the creation operator whichis classical while ¯ z acts as the annihilation operator ¯ h ∂∂ z which is quantum.(3) Our quantization is in a certain sense “polarized”: only half of the functions, i.e.,the anti-holomorphic ones receive quantum corrections.2.3.2. The Karabegov form.
In [17], Karabegov gives a complete classification of deformation quantizations of Wicktype on a K¨ahler manifold: Theorem 2.22 (Theorem 2 in [17]) . Deformation quantizations of Wick type (or with separationof variables) on a K¨ahler manifold X are in one-to-one correspondence with formal deformations ofthe K¨ahler metric ω on X. To see how a formal deformation of ω is constructed from a Wick type star product, letus recall the following proposition in [17]: Proposition 2.23 (Proposition 1 in [17]) . Let X be a K¨ahler manifold with K¨ahler form ω and (cid:63) be a formal star product with separation of variables. Then, on each contractible coordinate Actually the roles of the holomorphic and anti-holomorphic variables in [17] were reversed, but thisdoes not affect the results. chart U ⊂ X with any holomorphic coordinate system ( z , · · · , z n ) , there exist formal functionsu , · · · , u m ∈ C ∞ ( U )[[ ¯ h ]] such that [ u k , z k (cid:48) ] (cid:63) = ¯ h δ kk (cid:48) . This proposition gives a locally defined formal differential form ¯ ∂ ( − u k dz k ) of type (
1, 1 ) on each chart U , which patch together to a globally defined closed formal differentialform, called the Karabegov form associated to (cid:63) . Moreover, this formal (
1, 1 ) -form is adeformation of ω , i.e., of the form ω + O ( ¯ h ) . Remark . In [17], the Karabegov form is defined as √− · ¯ ∂ ( − u k dz k ) . We are usinga different normalization since our star products satisfy the condition that C ( f , g ) − C ( g , f ) = { f , g } , instead of C ( f , g ) − C ( g , f ) = √− · { f , g } .Now let α = ∑ i ≥ ¯ h i ω i be a closed formal differential form of type (
1, 1 ) , and D F , α and (cid:63) α be respectively the associated Fedosov abelian connection and Wick type star productconstructed in Theorem 2.17. To calculate the Karabegov form associated to (cid:63) α , we beginwith a lemma. Lemma 2.24.
Let α = ∑ i ≥ ¯ h i ω i be a closed formal differential form of type (
1, 1 ) and − ∑ i ≥ ¯ h i ρ i be a potential of α (i.e., ¯ ∂∂ρ i = ω i ), and let ρ be a potential of ω . For the (locally defined) formalfunctions u k = ∂∂ z k (cid:32) ρ + ∑ i ≥ ¯ h i ρ i (cid:33) , the terms in O u k which live in W X (which we call “terms of purely anti-holomorphic type”) aregiven by u k + ω k ¯ m ¯ y m . Proof.
Recall that O u k is the unique solution of the iterative equation: O u k = u k + δ − ◦ (cid:18) ∇ + h [ I α , − ] (cid:63) (cid:19) ( O u k ) .Observe that if a monomial A does not live in A • X ( W X ) , then ∇ A + h [ I α , A ] (cid:63) does nothave terms living in A • X ( W X ) . So we can prove the theorem by an induction on theweights of “ terms of purely anti-holomorphic type” in O u k .The terms in O u k of weight 1 are given by ∂ ρ∂ ¯ z l ∂ z k ¯ y l = ω k ¯ l ¯ y l .We know from the iterative equation that the weight 2 terms are given by δ − ◦ ∇ ( ω k ¯ l ¯ y l ) ,which vanish since the Levi-Civita connection is compatible with both the symplecticform and the complex structure. The next terms are δ − (cid:18) ∇ (cid:18) ¯ h ∂ρ ∂ z k (cid:19) + h (cid:20) ¯ h ∂ ρ ∂ ¯ z n ∂ z m d ¯ z n ⊗ y m , ω k ¯ l ¯ y l (cid:21) (cid:63) (cid:19) APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 19 = δ − (cid:18) ¯ h ∂ ρ ∂ z k ∂ ¯ z l d ¯ z l + ¯ h ∂ ρ ∂ ¯ z n ∂ z m d ¯ z n ω k ¯ l ω m ¯ l (cid:19) = O u k of purely anti-holomorphic type vanish. This argumentcan be generalized to all such terms of higher weights. (cid:3) Theorem 2.25.
For every closed formal differential form α of type (
1, 1 ) , the star product (cid:63) α defined in (2.15) has Karabegov form ω − α .Proof. Let U be any contractible coordinate chart in X , with local holomorphic coordinates ( z , · · · , z k ) . We define the functions u k as in Lemma 2.24. Then the flat section O z k canbe explicitly written as O z k = z k + y k + · · · , where all the terms live in W X . From thedefinition of the fiberwise Wick product on W X , C and that of the symbol map, we onlyneed those terms in O u k which are of “purely anti-holomorphic type” for the followingcomputation: O u k (cid:63) O z k (cid:48) − O z k (cid:48) (cid:63) O u k = ω k ¯ m ¯ y m (cid:63) y k (cid:48) − y k (cid:48) (cid:63) ( ω k ¯ m ¯ y m ) = − y k (cid:48) (cid:63) ( ω k ¯ m ¯ y m ) = ¯ h δ kk (cid:48) .Thus we have shown that the functions u k ’s satisfy the condition in Proposition 2.23.According to its construction, the Karabegov form is then given by¯ ∂ ( − u k dz k ) = − ∂ u k ∂ ¯ z l d ¯ z l ∧ dz k = (cid:32) ∂ ρ∂ z k ∂ ¯ z l + ∑ i ≥ ¯ h i ∂ ρ i ∂ z k ∂ ¯ z l (cid:33) dz k ∧ d ¯ z l = ω − α . (cid:3) Combining Theorems 2.22 and 2.25, we see that any star product of Wick type on aK¨ahler manifold arises from our construction:
Corollary 2.26.
On a K¨ahler manifold X, any deformation quantization of Wick type is of theform (cid:63) α for some closed formal (
1, 1 ) form α .
3. F
ROM F EDOSOV TO B ATALIN -V ILKOVISKY
Let us first recall the definition of traces of a star product:
Definition 3.1.
Let ( C ∞ ( X )[[ ¯ h ]] , (cid:63) ) denote a deformation quantization of X . A trace of thestar product (cid:63) is a linear map Tr : C ∞ ( X )[[ ¯ h ]] → C [[ ¯ h ]] such that(1) Tr ( f (cid:63) g ) = Tr ( g (cid:63) f ) ;(2) Tr ( f ) = (cid:82) X f · ω n + O ( ¯ h ) .In particular, Tr ( ) is called the algebraic index of (cid:63) .From the point of view of quantum field theory (QFT), traces are defined by correlationfunctions of local quantum observables. The Fedosov quantization describes the localdata of a quantum mechanical system, namely, the cochain complex ( A • X ( W X , C ) , D F , α ) gives the cochain complex of local quantum observables of a sigma model from S to the target X . To get global quantum observables and define the correlation functions properly, westudy the Batalin-Vilkovisky (BV) quantization [2] of this quantum mechanical system,from which we would obtain a factorization map from local to global quantum observ-ables. For a detailed explanation of the physical background, we refer the readers to [15].Mathematically, a BV quantization can be formulated as a solution of the quantummaster equation (QME). Our main result in this section is that, the canonical solution ofthe QME associated to the Fedosov abelian connection D F , α is one-loop exact. This leadsto a very neat cochain level formula for the algebraic index.The organization of this section is as follows: Section 3.1 is a review of the constructionof BV quantization from Fedosov abelian connections. We mainly follow the treatment in[15, Sections 2.3-2.5] but there is one key difference, namely, the choice of the propagtor,in order to reflect the K ¨ahlerian condition. In Section 3.3, we prove the main result of thissection saying that our solutions of the QME are all one-loop exact, and we explain howthis can lead to the cochain level formula for the trace.3.1. BV quantization.
This subsection is largely a review of the construction of BV quantizations from Fe-dosov quantizations in [15, Sections 2.3-2.5]. The main difference lies in the choice of thepropagator – we will give a construction of the so-called polarized propagator , which ismore compatible with the K ¨ahlerian condition and leads to some special features of theresulting BV quantizations.3.1.1.
Geometry of BV bundles and the QME.
The cochain complex of global quantum observables can be described in a differentialgeometric way. We start with the BV bundle on X : Definition 3.2 (cf. Definition 2.19 in [15]) . The
BV bundle of a K¨ahler manifold X is definedto be (cid:98) Ω −• TX : = (cid:100) Sym ( T ∗ X C ) ⊗ ∧ −• ( T ∗ X C ) , ∧ −• ( T ∗ X C ) : = (cid:77) k ∧ k ( T ∗ X C )[ k ] ,where ∧ k ( T ∗ X C ) has cohomological degree − k .For any tensor power of the BV bundle, we have the canonically defined multiplication :Mult : ( (cid:98) Ω −• TX ) ⊗ k : = (cid:98) Ω −• TX ⊗ · · · ⊗ (cid:98) Ω −• TX → (cid:98) Ω −• TX ,which can be extended A • X -linearly to Mult : A • X ( (cid:98) Ω −• TX ) ⊗ k → A • X ( (cid:98) Ω −• TX ) . To describe thedifferential on the BV bundle, we consider the fiberwise de Rham operator d TX : (cid:98) Ω −• TX → (cid:98) Ω − ( • + ) TX , and the contraction ι Π : (cid:98) Ω −• TX → (cid:98) Ω − ( •− ) TX by the Poisson tensor. We also havesimilarly defined operators ∂ TX , ¯ ∂ TX . There is also the BV operator defined by ∆ : = [ d TX , ι Π ] . APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 21
The operators d TX , ι Π and ∆ all extend A • X -linearly to operators on A • X ( (cid:98) Ω −• TX ) ⊗ k . Thefailure of the BV operator ∆ being a derivation is known as the BV bracket : { A , B } ∆ : = ∆ ( A · B ) − ∆ ( A ) · B ± A · ∆ ( B ) .It is clear that the operators d TX , ι Π and ∆ all commute with the multiplication map Mult.We also have [ ∆ , ∇ ] = ∇ ( ω ) =
0, and ∆ = Lemma 3.3 (Lemma 2.21 in [15]) . The operatorQ BV : = ∇ + ¯ h ∆ + h d TX R ∇ is a differential on the BV bundle (i.e., Q BV = ), which we call the BV differential . Definition 3.4 (Definition 2.22 in [15]) . A section r of the BV bundle is said to satisfy thequantum master equation (QME) if(3.1) Q BV ( e r /¯ h ) = ∇ r + ¯ h ∆ r + { r , r } ∆ + d TX R ∇ = Q BV ( e r /¯ h ) = (cid:18) ∇ + ¯ h ∆ + h d TX R ∇ (cid:19) ( e r /¯ h ) = h ( ∇ r + ¯ h ∆ r + { r , r } ∆ + d TX R ∇ ) · e r /¯ h . Lemma 3.5 (Lemma 2.23 in [15]) . Given a solution γ ∞ of the QME (3.1) , the operator (3.2) ∇ + ¯ h ∆ + { γ ∞ , −} ∆ is a differential on the BV bundle. The cochain complex of global quantum observables isdefined as ( A • X (cid:0) ˆ Ω −• TX (cid:1) [[ ¯ h ]] , ∇ + ¯ h ∆ + { γ ∞ , −} ∆ ) . Lemma 3.6 (Lemma 2.24 in [15]) . The fiberwise Berezin integration , defined by taking thetop degree component in odd variables and setting the even variables to : (cid:90) Ber : A • X (cid:0) ˆ Ω −• TX (cid:1) → A • X , a (cid:55)→ n ! ( ι Π ) n ( a ) (cid:12)(cid:12)(cid:12)(cid:12) y i = ¯ y j = , is a cochain map, with respect to the BV differential Q BV on A • X (cid:0) ˆ Ω −• TX (cid:1) and the de Rham differ-ential on A • X . From this lemma we get a well-defined composition map on cohomology classes: H ∗ ( A • X (cid:0) ˆ Ω −• TX (cid:1) [[ ¯ h ]]) (cid:82) Ber −→ H ∗ dR ( X )[[ ¯ h ]] (cid:82) X −→ C [[ ¯ h ]] .We can thus define the correlation functions (or expectation values) of global quantumobservables: Definition 3.7.
Let γ ∞ be a solution of the QME (3.1) and let O be a global quantumobservable. The correlation function of O is defined as (cid:104) O (cid:105) : = (cid:90) X (cid:90) Ber O · e γ ∞ /¯ h . A polarized propagator and the homotopy group flow.
To construct solutions of the QME and define the local-to-global factorization map, weneed the homotopy group flow operator . Our formulation here follows that in [15, Section2.4] but with a significant modification of the definition of the propagator in order toadapt to the K¨ahler setting.First of all, for any positive integer k , let S [ k ] be the compactified configuration spaceof k ordered points on the circle S . Then the function P on S [ ] defined by(3.3) P ( θ , u ) : = u − propagator (see [15, Definition 2.30]) because it is the derivative of the Green’sfunction on S with respect to the standard flat metric, and thus represents the propagatorof topological quantum mechanics on S (see [15, Remarks 2.31 and B.6]). When restrictedto the open subset S × S \ ∆ ⊂ S [ ] , the propagator P is anti-symmetric in the twocopies of S .We can now define the polarized propagator in the K¨ahler setting as a combination of P and the K¨ahler form on X : Definition 3.8 (cf. Definition 2.32 in [15]) . We define the A • S [ k ] -linear operators ∂ P , D : A • S [ k ] ⊗ C A • X ( (cid:98) Ω −• TX ) ⊗ k → A • S [ k ] ⊗ C A • X ( (cid:98) Ω −• TX ) ⊗ k by (1) ∂ P ( a ⊗ · · · ⊗ a k ) : = ∑ ≤ α < β ≤ k π ∗ αβ ( P + ) ⊗ C ( ω i ¯ j ( x ) a ⊗ · · · ⊗ L ∂ zi a α ⊗ · · · ⊗ L ∂ ¯ zj a β ⊗ · · · ⊗ a k ) − ∑ ≤ α < β ≤ k π ∗ αβ ( P − ) ⊗ C ( ω i ¯ j ( x ) a ⊗ · · · ⊗ L ∂ ¯ zj a α ⊗ · · · ⊗ L ∂ zi a β ⊗ · · · ⊗ a k ) ;(2) D ( a ⊗ · · · ⊗ a k ) : = ∑ ≤ i ≤ k ± d θ i ⊗ C ( a ⊗ · · · d TX a i ⊗ · · · a k ) .Here a i ∈ A • X ( (cid:98) Ω −• TX ) , π αβ : S [ k ] → S [ ] is the forgetful map to the two points indexedby α , β , θ i ∈ [
0, 1 ) is the parameter on the S indexed by 1 ≤ i ≤ k and d θ i is a 1-form on S [ k ] via the pullback π i : S [ k ] → S , and finally ± are appropriate Koszul signs.We have the following decomposition of the operator ∂ P : Lemma 3.9.
The operator ∂ P in Definition 3.8 can be written as the sum ∂ P = ∂ P + ∂ P , where ∂ P ( a ⊗ · · · ⊗ a k ) = ∑ ≤ α < β ≤ k π ∗ αβ ( P ) ⊗ R ( ω i ¯ j · a ⊗ · · · ⊗ L ∂ zi a α ⊗ · · · ⊗ L ∂ ¯ zj a β ⊗ · · · ⊗ a k ) For the construction and basic facts of compactified configuration spaces, see [1].
APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 23 − ∑ ≤ α < β ≤ k π ∗ αβ ( P ) ⊗ R ( ω i ¯ j · a ⊗ · · · ⊗ L ∂ ¯ zj a α ⊗ · · · ⊗ L ∂ zi a β ⊗ · · · ⊗ a k ) , and ∂ P ( a ⊗ · · · ⊗ a k ) = ∑ ≤ α < β ≤ k ( ω i ¯ j · a ⊗ · · · ⊗ L ∂ zi a α ⊗ · · · ⊗ L ∂ ¯ zj a β ⊗ · · · ⊗ a k )+ ∑ ≤ α < β ≤ k ( ω i ¯ j · a ⊗ · · · ⊗ L ∂ ¯ zj a α ⊗ · · · ⊗ L ∂ zi a β ⊗ · · · ⊗ a k ) .Notice that both ∂ P and ∂ P are symmetric in the indices α , β , and that ∂ P coincideswith the propagator used in [15]. Lemma 3.10 (cf. Lemma 2.33 in [15]) . As operators on the BV bundle, we have [ d S , ∂ P ] = [ ∆ , D ] , D = [ ∂ P , D ] = h ∂ P and D commute, so we can formally define thefollowing operator on the BV bundle: e ¯ h ∂ P + D : = ∑ k ≥ k ! ( ¯ h ∂ P + D ) k .Here is the definition of the homotopy group flow operator on the BV bundle: Definition 3.11 (cf. Definition 2.34 in [15]) . Given any γ ∈ A • X ( W X , C ) , we define γ ∞ ∈A • X ( ˆ Ω −• TX )[[ ¯ h ]] by e γ ∞ /¯ h : = Mult (cid:90) S [ ∗ ] e ¯ h ∂ P + D e ⊗ γ /¯ h ,where e ⊗ γ /¯ h : = ∑ k ≥ k !¯ h k γ ⊗ k , γ ⊗ k ∈ A • X ( ˆ Ω −• TX ) ⊗ k ) , and (cid:82) S [ ∗ ] : A • S [ k ] ⊗ C A • X ( (cid:98) Ω −• TX ) ⊗ k →A • X ( (cid:98) Ω −• TX ) ⊗ k is the integration (cid:82) S [ k ] over the appropriate configuration space S [ k ] . Lemma 3.12 (cf. Lemma 2.37 in [15]) . For any γ ∈ A • X ( W X , C ) , we have ¯ h ∆ e γ ∞ /¯ h = Mult (cid:90) S [ ∗− ] e ¯ h ∂ P + D h ( γ (cid:63) γ ) ⊗ e ⊗ γ /¯ h . Proof.
The proof is very similar to that of [15, Lemma 2.37], except that here we use thepolarized propagator ∂ P instead of the propagator ∂ P in [15].Using the commutation relations in Lemma 3.10, we have¯ h ∆ e γ ∞ /¯ h = Mult (cid:90) S [ ∗ ] ¯ h ∆ e ¯ h ∂ P + D e ⊗ γ /¯ h = Mult (cid:90) S [ ∗ ] d S e ¯ h ∂ P + D e ⊗ γ /¯ h + Mult (cid:90) S [ ∗ ] ( ¯ h ∆ − d S ) e ¯ h ∂ P + D e ⊗ γ /¯ h = Mult (cid:90) ∂ S [ ∗ ] e ¯ h ∂ P + D e ⊗ γ /¯ h + Mult (cid:90) S [ ∗ ] e ¯ h ∂ P + D ( ¯ h ∆ − d S ) e ⊗ γ /¯ h = Mult (cid:90) ∂ S [ ∗ ] e ¯ h ∂ P + D e ⊗ γ /¯ h ,where in the last step we used the fact that ¯ h ∆ (cid:0) e ⊗ γ /¯ h (cid:1) = d S (cid:0) e ⊗ γ /¯ h (cid:1) = d S an-nihilates e ⊗ γ /¯ h because e ⊗ γ /¯ h does not depend on the configuration space S [ ∗ ] and ∆ annihilates e ⊗ γ /¯ h by type reasons).We now consider the term Mult (cid:82) ∂ S [ ∗ ] e ¯ h ∂ P + D e ⊗ γ /¯ h . Recall that there is an explicit de-scription of the boundary (i.e., codimension 1 strata) of the configuration space: ∂ S [ k ] = (cid:91) I ⊂{ ··· , k } , | I |≥ π − ( D I ) ,where D I ⊂ ( S ) k is the small diagonal where points in those S ’s indexed by I coincide(see e.g. [15, Appendix B] for more details). Similar to the real symplectic case, onlycomponents corresponding to those indices with | I | = I corresponds to a pair α < β ∈ { · · · , n } . Thecorresponding boundary strata has two components, each isomorphic to S [ ∗ − ] . So wehave Mult (cid:90) ∂ S [ ∗ ] e ¯ h ∂ P + D e ⊗ γ /¯ h = Mult (cid:90) S [ ∗− ] e ¯ h ∂ P + D h [ γ , γ ] (cid:63) ⊗ e ⊗ γ /¯ h = Mult (cid:90) S [ ∗− ] e ¯ h ∂ P + D h ( γ (cid:63) γ ) ⊗ e ⊗ γ /¯ h .We emphasize that the polarized propagator plays a key role here: the operators on thetwo connected components precisely correspond to A (cid:63) B and B (cid:63) A , and this explainswhy the bracket [ − , − ] (cid:63) shows up. (cid:3) Solutions of the QME from Fedosov abelian connections.
From the perspective of BV quantization, a solution of the QME (3.1) is the ∞ -scaleeffective interaction of the quantum mechanical system on X . Such a solution can beconstructed by running the homotopy group flow.We first consider the following section of the BV bundle:˜ R ∇ : = ( ∂ TX ◦ ¯ ∂ TX )( R ∇ ) = √− · R i ¯ jk ¯ l dz i ∧ d ¯ z j ⊗ dy k ∧ d ¯ y l . Lemma 3.13.
We have ∆ ( ˜ R ∇ ) = ∇ ( ˜ R ∇ ) = .Proof. The vanishing of the first term follows from the type of the BV operator ∆ , and thesecond one is the Bianchi identity. (cid:3) Lemma 3.14.
We have ∂ P ( d TX R ∇ ⊗ γ ) = − { ˜ R ∇ , γ } ∆ .Proof. This is a straightforward computation. For the LHS, we have ∂ P ( d TX R ∇ ⊗ γ ) APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 25 = ω p ¯ q (cid:16) L ∂ zp ( d TX R ∇ ) ⊗ L ∂ ¯ zq γ + L ∂ ¯ zq ( d TX R ∇ ) ⊗ L ∂ zp γ (cid:17) = √− ω p ¯ q (cid:16) L ∂ zp ( R i ¯ jk ¯ l dz i ∧ d ¯ z j ⊗ y k d ¯ y l ) ⊗ L ∂ ¯ zq γ + L ∂ ¯ zq ( R i ¯ jk ¯ l dz i ∧ d ¯ z j ⊗ ¯ y l dy k ) ⊗ L ∂ zp γ (cid:17) = √− ω p ¯ q (cid:16) ( R i ¯ jp ¯ l dz i ∧ d ¯ z j ⊗ d ¯ y l ) ⊗ L ∂ ¯ zq γ + ( R i ¯ jk ¯ q dz i ∧ d ¯ z j ⊗ dy k ) ⊗ L ∂ zp γ (cid:17) .On the other hand, recall that ∆ = ω p ¯ q (cid:16) L ∂ zp ι ∂ ¯ zq − L ∂ ¯ zq ι ∂ zp (cid:17) . Hence the RHS is given by { ˜ R ∇ , γ } ∆ = √− ω p ¯ q (cid:16) ι ∂ ¯ zq ( R i ¯ jk ¯ l dz i ∧ d ¯ z j ⊗ dy k ∧ d ¯ y l ) ⊗ L ∂ zp ( γ ) − ι ∂ zp ( R i ¯ jk ¯ l dz i ∧ d ¯ z j ⊗ dy k ∧ d ¯ y l ) ⊗ L ∂ ¯ zq ( γ ) (cid:17) = − √− ω p ¯ q (cid:16) ( R i ¯ jk ¯ q dz i ∧ d ¯ z j ⊗ dy k ) ⊗ L ∂ zp ( γ ) + ( R i ¯ jp ¯ l dz i ∧ d ¯ z j ⊗ d ¯ y l ) ⊗ L ∂ ¯ zq ( γ ) (cid:17) (cid:3) We are now ready to construct our solutions of the QME:
Theorem 3.15 (cf. Theorem 2.26 in [15]) . Suppose that γ is a solution of the Fedosov equation (2.8) and let γ ∞ be defined as in Definition 3.11. Then e ˜ R ∇ /2¯ h · e γ ∞ /¯ h is a solution of the QME (3.1) , i.e., Q BV (cid:16) e ˜ R ∇ /2¯ h · e γ ∞ /¯ h (cid:17) = .Proof. Recall that Q BV = ∇ + ¯ h ∆ + ¯ h − d TX R ∇ . We compute each term in Q BV ( e γ ∞ /¯ h ) :(1) The first term is given by: ∇ ( e γ ∞ /¯ h ) = ∇ (cid:18) Mult (cid:90) S [ ∗ ] e ¯ h ∂ P + D e ⊗ γ /¯ h (cid:19) = Mult (cid:90) S [ ∗ ] e ¯ h ∂ P + D ∇ ( e ⊗ γ /¯ h )= Mult (cid:90) S [ ∗ ] e ¯ h ∂ P + D ( ∇ γ ⊗ e γ /¯ h ) = ( ∇ γ ) · e γ ∞ /¯ h ,where we are using the relations [ ∇ , ¯ h ∂ P + D ] = (cid:104) ∇ , (cid:82) S [ ∗ ] (cid:105) = h ∆ ( e γ ∞ /¯ h ) = Mult (cid:90) S [ ∗− ] e ¯ h ∂ P + D h ( γ (cid:63) γ ) ⊗ e ⊗ γ /¯ h = h ( γ (cid:63) γ − ω ¯ h ) · e γ ∞ /¯ h .Note that the − ω ¯ h term appears because D ( − ω ¯ h ) = (cid:90) S [ ∗ ] e ¯ h ∂ P + D R ∇ ¯ h ⊗ e ⊗ γ /¯ h = Mult (cid:90) S [ ∗ ] e ¯ h ∂ P + D ( h d θ d TX R ∇ ) ⊗ e ⊗ γ /¯ h Notice that ¯ h ∂ P can be applied to d TX R ∇ once, which gives rise to the followingdifference:Mult (cid:90) S [ ∗ ] e ¯ h ∂ P + D ( h d θ d TX R ∇ ) ⊗ e ⊗ γ /¯ h − Mult (cid:90) S [ ∗ ] ( h d θ d TX R ∇ ) e ¯ h ∂ P + D (cid:16) e ⊗ γ /¯ h (cid:17) = Mult (cid:90) S [ ∗ ] e ¯ h ∂ P + D ( h d θ d TX R ∇ ) ⊗ e ⊗ γ /¯ h − Mult (cid:90) S [ ∗ ] ( h d TX R ∇ ) e ¯ h ∂ P + D ⊗ e ⊗ γ /¯ h = Mult (cid:90) S [ ∗ ] e ¯ h ∂ P + D d θ · h · ( − ) · { ˜ R ∇ , d TX γ } ∆ ⊗ e ⊗ γ /¯ h = − h { ˜ R ∇ , γ ∞ } ∆ · e γ ∞ /¯ h ,where we have used Lemma 3.14 and the relations (cid:2) { ˜ R ∇ , −} ∆ , ¯ h ∂ P (cid:3) = (cid:2) { ˜ R ∇ , −} ∆ , D (cid:3) = Q BV (cid:16) e ˜ R ∇ /2¯ h · e γ ∞ /¯ h (cid:17) = e ˜ R ∇ /2¯ h · Q BV ( e γ ∞ /¯ h ) + (cid:16) ( ∇ + ¯ h ∆ ) e ˜ R ∇ /2¯ h (cid:17) · e γ ∞ /¯ h + h { ˜ R ∇ , γ ∞ } ∆ · (cid:16) e ˜ R ∇ /2¯ h · e γ ∞ /¯ h (cid:17) = e ˜ R ∇ /2¯ h · Q BV ( e γ ∞ /¯ h ) + h { ˜ R ∇ , γ ∞ } ∆ · (cid:16) e ˜ R ∇ /2¯ h · e γ ∞ /¯ h (cid:17) = h (cid:18) ∇ γ + h γ (cid:63) γ − ω ¯ h + R ∇ − { ˜ R ∇ , γ ∞ } ∆ + { ˜ R ∇ , γ ∞ } ∆ (cid:19) · (cid:16) e ˜ R ∇ /2¯ h · e γ ∞ /¯ h (cid:17) = h (cid:18) ∇ γ + h γ (cid:63) γ − ω ¯ h + R ∇ (cid:19) · (cid:16) e ˜ R ∇ /2¯ h · e γ ∞ /¯ h (cid:17) = (cid:3) Remark . Compared with [15, Theorem 2.26], our QME solution has the additional fac-tor e ˜ R ∇ /2¯ h due to our different choice of the propagator. We will see in the next subsectionthat this leads to nontrivial contribution from the tadpole graph in computing the parti-tion function, which we do not see in the general symplectic case in [15].We are now ready to define the local-to-global factorization map of quantum observ-ables: Theorem 3.16 (cf. Theorem 2.40 in [15]) . The factorization map defined by [ − ] ∞ : A • X ( W X , C ) → A • X ( ˆ Ω −• TX )[[ ¯ h ]] O (cid:55)→ [ O ] ∞ : = e − γ ∞ /¯ h · (cid:18) Mult (cid:90) S [ ∗ ] e ¯ h ∂ P + D ( Od θ ⊗ e ⊗ γ /¯ h ) (cid:19) is a cochain map from the complex ( A • X ( W X , C ) , D F , α ) of local quantum observables to the complex ( A • X ( ˆ Ω −• TX )[[ ¯ h ]] , ∇ + { γ ∞ , −} ∆ + ¯ h ∆ ) of global quantum observables. Corollary 3.17.
Let f ∈ C ∞ ( X )[[ ¯ h ]] be any smooth function and O f be the corresponding flatsection under the Fedosov connection. Then Q BV (cid:16) [ O f ] ∞ · e ˜ R ∇ /2¯ h · e γ ∞ /¯ h (cid:17) = . APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 27
The trace.
For the above BV quantization, we can define the correlation function of a local quan-tum observable:
Definition 3.18.
The correlation function of a formal smooth function f ∈ C ∞ ( X )[[ ¯ h ]] isdefined by (cid:104) f (cid:105) : = (cid:90) X (cid:90) Ber [ O f ] ∞ · e ˜ R ∇ /2¯ h · e γ ∞ /¯ h .The proofs of the following propositions, which we omit, are the same as that in [15]: Proposition 3.19 (cf. Proposition 2.43 in [15]) . Let f , g ∈ C ∞ ( X )[[ ¯ h ]] be two smooth functionson X, and let h = [ f , g ] (cid:63) α denote the commutator of f , g under the Fedosov star product (cid:63) α . Then [ O h ] ∞ is exact under the differential (3.2) and hence (cid:104) f ∗ α g (cid:105) = (cid:104) g ∗ α f (cid:105) . Proposition 3.20 (cf. Proposition 2.44 in [15]) . The correlation function of f ∈ C ∞ ( X )[[ ¯ h ]] isof the form (cid:104) f (cid:105) = (cid:90) X f · ω n + O ( ¯ h ) . Corollary 3.21.
The association
Tr : f (cid:55)→ (cid:104) f (cid:105) is the trace of the Fedosov star product (cid:63) α . To summarize, the BV quantization method gives rise to a way to explicitly computean integral density of the trace of a Wick type star product (cid:63) α : for any smooth function f ∈ C ∞ ( X ) , we(1) compute O f using the iterative equation (2.14),(2) obtain γ ∞ as a Feynman graph expansion (see Lemma 3.22 below),(3) compute [ O f ] ∞ via Theorem 3.16, and(4) finally take the Berezin integral of [ O f ] ∞ · e ˜ R ∇ /2¯ h · e γ ∞ /¯ h to obtain the density forTr ( f ) .It follows easily that this density for Tr ( f ) satisfies a locality: at every point x ∈ X , it onlydepends on the Taylor expansion of f , the curvature of X and the formal cohomologyclass [ α ] at x .3.3. One-loop exactness and the algebraic index.
We present the explicit computation of Tr ( ) as an example, whose formula is alsoknown as the algebraic index theorem. First of all, a solution γ ∞ of the QME (3.1) isobtained by running the homotopy group flow. This can be expressed as a Feynmangraph expansion: For some basic facts on Feynman graph expansion and Feynman weights, see Appendix A. For moredetails and a proof of this lemma, see Costello’s book [7].
Lemma 3.22.
Let γ ∈ A • X ( W X , C ) and γ ∞ be defined as in Definition 3.11. Then γ ∞ can beexpressed as a sum of Feynman weights: γ ∞ = ∑ G : connected ¯ h g ( G ) | Aut ( G ) | W G ( P , d TX γ ) , where the sum is over all connected, stable graphs G , and g ( G ) denotes the genus of G . It turns out that, in the K¨ahler case, if γ is the solution of the Fedosov equation (2.8)obtained by quantizing Kapranov’s L ∞ -algebra structure, then the Feynman graph ex-pansion of the QME solution γ ∞ involves only trees and one-loop graphs ; in other words, γ ∞ gives a one-loop exact BV quantization of the K ¨ahler manifold X . This is in sharp con-trast with the general symplectic case [15], in which the BV quantization involves all-loopquantum corrections. The same kind of one-loop exactness was observed for the holo-morphic Chern-Simons theory by Costello [8] and for a sigma model from S to the target T ∗ Y (cotangent bundle of a smooth manifold Y ) by Gwilliam-Grady [14].As a corollary, we obtain a succinct explicit expression of the algebraic index Tr ( ) = (cid:82) X (cid:82) Ber e ˜ R ∇ /2¯ h e γ ∞ /¯ h of the star product (cid:63) α . The latter is a cochain level enhancement of theresult in [15]: there the technique of equivariant localization was applied to show that allgraphs of higher genera ( ≥
2) give rise to exact differential forms after the Berezin inte-gration and thus do not contribute after integration over X , while the Feynman weightsassociated to these graphs in our QME solution γ ∞ vanish already on the cochain level.3.3.1. A weight on the BV bundle and one-loop exactness.
The key to one-loop exactness lies in the existence of a suitable weight on the BV bun-dle:
Definition 3.23.
We define a weight on the BV bundle A • X ( ˆ Ω −• TX )[[ ¯ h ]] by setting • | ¯ y i | = | d ¯ y i | =
1, and • | y i | = | dy i | = | dz i | = | d ¯ z i | = | ¯ h | = Lemma 3.24.
For every closed (
1, 1 ) -form α , let D F , α = ∇ − δ + h [ γ , − ] (cid:63) be the associatedFedosov connection obtained in Theorem 2.17. Then in the ¯ h power expansion of γ : γ = ∑ i ≥ ¯ h i · γ i , the weight of each term in γ is either or .Proof. The only term in γ of weight 0 is ω i ¯ j d ¯ z j ⊗ ¯ y i . All the other monomials in γ haveexactly one ¯ y i ’s, and are thus of weight 1. (cid:3) APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 29
Lemma 3.25.
The following operators preserve the weight: ¯ h ∆ , ¯ h ∂ P , ¯ h {− , −} ∆ . In particular, the homotopy group flow operator also preserves the weight.
Here is one of the main discoveries in this paper:
Theorem 3.26 (One-loop exactness) . Let γ be a solution of the Fedosov equation (2.8) . ThenW G ( P , d TX γ ) = whenever b ( G ) ≥ As a result, the graph expansion of the QME solution γ ∞ , as defined in Definition 3.11, involvesonly trees and one-loop graphs: γ ∞ = ∑ G : connected , b ( G )= ¯ h g ( G ) | Aut ( G ) | W G ( P , d TX γ ) . Proof.
Let G be a connected graph with first Betti number b ( G ) ≥
2. Since every term in γ has at most weight 1, the total weights of those decorated vertices are bounded aboveby | V ( G ) | . On the other hand, each propagator P is of weight −
1. Thus the internal edgesof G decorated by P has total weight − ( | V ( G | + b ( G ) − ) . Hence we must have | V ( G ) | − ( | V ( G | + b ( G ) − ) ≥ b ( G ) ≤ (cid:3) Remark . The argument here is similar to the proof of one-loop exactness of the holo-morphic Chern-Simons theory by Costello [8].3.3.2.
A cochain level formula for the trace.
To derive a formula for the algebraic index Tr ( ) = (cid:82) X (cid:82) Ber e ˜ R ∇ /2¯ h e γ ∞ /¯ h of the star prod-uct (cid:63) α , we first extend the symbol map to the BV bundle σ : A • X ( ˆ Ω −• TX ) → A • X y i , ¯ y j , dy i , d ¯ y j (cid:55)→ Lemma 3.27.
The Berezin integral (cid:82)
Ber e ˜ R ∇ /2¯ h e γ ∞ /¯ h can be expressed as follows: (cid:90) X (cid:90) Ber e ˜ R ∇ /2¯ h e γ ∞ /¯ h = (cid:90) X σ (cid:16) e ¯ h ι Π ( e ˜ R ∇ /2¯ h e γ ∞ /¯ h ) (cid:17) Proof.
A simple observation is that every term in ˜ R ∇ and γ ∞ has the same degree in dz i , d ¯ z j ’s and dy i , d ¯ y j ’s. Since the integration (cid:82) X only takes the top degree term in dz i , d ¯ z j ’s,the only term that matters in e ¯ h ι Π is n ! ( ¯ h ι Π ) n . (cid:3) Then we compute: e ¯ h ι Π ( e ˜ R ∇ /2¯ h e γ ∞ /¯ h ) = e ¯ h ι Π (cid:18) e ˜ R ∇ /2¯ h · Mult (cid:90) S [ ∗ ] e ¯ h ∂ P + D e ⊗ γ /¯ h (cid:19) ( ∗ ) = e ¯ h ι Π (cid:18) Mult (cid:90) S [ ∗ ] e ¯ h ∂ P + D ( e ˜ R ∇ /2¯ h d θ ) ⊗ e ⊗ γ /¯ h (cid:19) = Mult (cid:90) S [ ∗ ] e ¯ h ( ι Π + ∂ P ) ◦ e D (cid:16) ( e ˜ R ∇ /2¯ h d θ ) ⊗ e ⊗ γ /¯ h (cid:17) = exp (cid:32) ¯ h − · ∑ G connected ¯ h g ( G ) | Aut ( G ) | W ( P + ι Π , d θ ⊗ ( d TX γ +
12 ˜ R ∇ )) (cid:33) ;here the equality ( ∗ ) follows from the facts that ∂ P cannot be applied to ˜ R ∇ /2 by typereasons and that D ( ˜ R ∇ ) = ( ) can be ex-pressed as a sum of Feynman weights. But note that they are different from those for γ ∞ because the propagator is now ¯ h ( ∂ P + ι Π ) . Similar to Theorem 3.26, this graph sum alsoinvolves only trees and one-loop graphs: Proposition 3.28.
In the Feynman graph expansion ∑ G connected ¯ h g ( G ) | Aut ( G ) | W ( P + ι Π , d TX γ +
12 ˜ R ∇ ) , only terms which correspond to graphs with first Betti number and are non-vanishing. We now proceed to compute the Feynman weights associated to trees and one-loopgraphs. To clarify the computation, we first decompose the terms labeling the verticesand edges respectively. For the vertices, recall that the term γ in the Fedosov connectionhas the ¯ h -power expansion: γ = ∑ i ≥ ¯ h i γ i .For later computations, we give a detailed description of these γ i ’s: • For γ , we know that γ = ω i ¯ j ( dz i ⊗ ¯ y j − d ¯ z j ⊗ y i ) + γ (cid:48) ,where all terms of γ (cid:48) are at least cubic, with the leading term R i ¯ jk ¯ l d ¯ z j ⊗ y i y k ¯ y l . • For every i >
0, the leading term of γ i is given by ( δ ) − ( α i ) .For the propagators, notice that ∂ P and ι Π are respectively tensor products of forms on S [ ∗ ] and tensors on the BV bundle of X . They correspond to the analytic and combinatorial parts of the propagators respectively.We assign colors to both the vertices and edges, according to the previous decomposi-tion of the functionals and propagators. For edges, we have(1) a blue edge is labeled by ∂ P ;(2) a black edge is labeled by ∂ P ;(3) a red edge is labeled by the operator ι π ; APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 31 (4) a yellow vertex is labeled by ω i ¯ j ( dz i ⊗ d ¯ y j − d ¯ z j ⊗ dy i ) ;(5) a purple vertex is labeled by d TX γ (cid:48) ;(6) a green vertex is labeled by ∑ i > d TX γ i ;(7) a a blue vertex is labeled by ˜ R ∇ .In a graph, when we do not distinguish ∂ P or ∂ P , we assign black color to this edge.Moreover, since the analytic parts of ∂ P , ∂ P and ι Π are all given by contractions using theinverse of the K¨ahler form, we assign an orientation on each internal edge, going fromthe holomorphic derivatives to the anti-holomorphic derivatives.Some sample pictures are listed here: Lemma 3.29.
The vertices in our Feynman graphs have the following properties: (1)
A purple vertex is at least trivalent with exactly one incoming tail and at least two outgo-ing tails. (2)
The tails on a green vertex must be outgoing. (3)
A blue vertex has exactly one incoming tail and one outgoing tail. (4)
Every yellow, purple or green vertex can be connected to at most one red internal edge.Proof.
All the statements follow by considering the types of the sections of the BV bundlelabeling the corresponding colored vertices. For instance, ( ) follows from the fact thatthe Weyl bundle component of γ i ( i > ) lives in W X . (cid:3) In terms of Feynman weights, the procedure of taking the symbol map σ in Lemma3.27 corresponds to taking only those connected graphs with no tails (external edges) butonly internal edges. Let us first consider trees. Proposition 3.30.
Suppose G is a tree which contains at least one vertex labeled by γ (cid:48) (i.e., apurple vertex). Then this tree has at least one outgoing tail (external edge) which can be contractedby ∂ P or ∂ P (i.e., blue or black).Proof. We have seen that a purple vertex has to be at least trivalent. It is clear that everyunivalent vertex can only be connected by an internal edge labeled by ι Π . Let v i denote thenumber of vertices in G which are i -valent. Let G (cid:48) denote the graph obtained from G bydeleting those blue and yellow vertices, and also red edges (both internal and external). Inparticular, every internal and external edge must be either blue or black, and the numberof n -valent vertices in G (cid:48) is v n + , for n ≥ v =
0. Then a simple counting shows that there exists in G (cid:48) at least oneoutgoing external edge. If v >
0, we choose one of these univalent vertices in G (cid:48) , whose tail must be outgoing. We can draw a line starting from this vertex along the edges withdirections and go as far as possible. Now we need the following simple fact: Suppose avertex is at least bivalent in G , then either all tails are outgoing, or at least one outgoingand exactly one incoming. This says that the line that we are drawing has to stop at anoutgoing tail on a vertex which is at least bivalent. (cid:3) As a corollary, we have a full list of trees without external edges:
Corollary 3.31.
The only trees which survive under the symbol map are listed below, which ex-actly give rise to ω ¯ h . ω ¯ h = Proof.
According to Proposition 3.30, these trees cannot include a purple vertex. We firstshow that they cannot include blue vertices either: A blue vertex has exactly one incomingtail and one outgoing tail. Since a green vertex has only outgoing tails, the outgoing tailof a blue vertex has to be connected to a yellow vertex, as shown in the following picture: = √− R i ¯ jk ¯ l dz i ∧ d ¯ z j ∧ dz k ⊗ d ¯ y l .But then this has to vanish since R i ¯ jk ¯ l is symmetric in i and k . Therefore such a graph canonly contain yellow and green vertices. These green vertices must be univalent becausethere is no way to contract its blue or black tails. Hence the graphs satisfying the conditionin this corollary have to be as stated. (cid:3) Next we turn to one-loop graphs. Since every one-loop graph can be obtained by at-taching trees to a wheel (see Definition A.2), we first look at all the possible wheels (possi-bly with tails). It is clear that there are three types of wheels, according to the labeling ofthe internal edges of the wheel:(1) All the edges on the wheel are labeled by either P or P .(2) All the edges on the wheel are labeled by ι Π .(3) The labelings of edges on the wheel include both P (either P or P ) and ι Π . Proposition 3.32.
The symbol of a one-loop Feynman weight coming from a wheel of type ( ) vanishes.Proof. Let G denote such a one-loop graph. A simple observation is that every vertex onthe wheel of G is either labeled by γ (cid:48) or ˜ R ∇ . We now choose a vertex v such that the twoedges on the wheel adjacent to v are labeled by ι Π and P respectively, as shown in thefollowing picture: APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 33
In particular, v must be labeled by γ (cid:48) , and is at least trivalent. Thus v must have at leastone outgoing tail (the dotted line in the above picture) which is not on the wheel. In orderfor the symbol of this Feynman integral to be non-vanishing, we have to attach a treeto this outgoing tail. Such a tree must include at least one vertex labeled by γ (cid:48) , so thatthis vertex can be connected to the dotted line in the above picture. Thus by Proposition3.30, this tree must also contain an outgoing tail. But this implies that the symbol of theFeynman integral of G (cid:48) vanishes. (cid:3) Corollary 3.33.
A one-loop graph which survives under the symbol map (equivalently, withoutexternal edges) must be of the following two types:Proof.
Let G be such a one-loop graph. If G is of type ( ) , then every vertex on the wheelmust be labeled by ˜ R ∇ , as shown on the left. If G is of type ( ) , then every vertex on thewheel must be labeled by γ (cid:48) . A similar argument as in Proposition 3.32 shows that thesevertices must be exactly trivalent, as shown on the right. (cid:3) We can also see that if the edges of the wheel on such a one-loop graph are labeled by ∂ P , then they have to be of the same color: Proposition 3.34.
If a wheel contains edges labeled by both ∂ P and ∂ P , then the correspondingFeynman weights vanish. In other words, the following Feynman weights vanish:Proof. We focus on the vertex v in the above picture. Notice that both the red and greenedges incident to v only contribute constants to the analytic part of the propagator. Theanalytic part of the propagator P labeling the blue edge between v and v is u − as in (3.3). Thus the Feynman weight of the above graph is a multiple of the followingintegral (cid:90) u = ( u − ) du = (cid:3) To summarize, the only one-loop graphs which contribute non-trivially to the integrandwhose integral gives rise to Tr ( ) are of the following 4 types:(3.4)(3.5)In order to have a more precise computation of the Feynman weights, we first computethe Feynman weights of the following two types of line graphs with m ≥ m purple vertices in the left picture):(3.6)We label both the internal edges and vertices to avoid the issue of automorphisms ofgraphs and ordering of the contraction of propagators with vertices. Lemma 3.35.
The Feynman weight for the graph on the left of (3.6) is given by − m · ( ω k ¯ l ω k ¯ l · · · ω k m − ¯ l m )( R i ¯ j k ¯ l R i ¯ j k ¯ l · · · R i m ¯ j m k m ¯ l m )( dz i d ¯ z j · · · dz i m d ¯ z j m ) ⊗ ( d ¯ y l ∧ dy k m ) while that for the graph on the right of (3.6) is given by m − · ( ω k ¯ l ω k ¯ l · · · ω k m − ¯ l m )( R i ¯ j k ¯ l R i ¯ j k ¯ l · · · R i m ¯ j m k m ¯ l m )( dz i d ¯ z j · · · dz i m d ¯ z j m ) ⊗ ( ¯ y l y k m ) . APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 35
Proof.
When m =
2, the Feynman weight of the left picture can be explicitly computed asfollows: ω p ¯ q (cid:16) ι ∂ zp ( ˜ R ∇ /2 ) ⊗ ι ∂ ¯ zq ( ˜ R ∇ /2 ) (cid:17) = ω p ¯ q · Mult (cid:16) ι ∂ zp ( R i ¯ j k ¯ l dz i ∧ d ¯ z j ⊗ dy k ∧ d ¯ y l ) ⊗ ι ∂ ¯ zq ( R i ¯ j k ¯ l dz i ∧ d ¯ z j ⊗ dy k ∧ d ¯ y l ) (cid:17) = − ω k ¯ l · Mult (cid:16) ( R i ¯ j k ¯ l dz i ∧ d ¯ z j ⊗ d ¯ y l ) ⊗ ( R i ¯ j k ¯ l dz i ∧ d ¯ z j ⊗ dy k ) (cid:17) = − ω k ¯ l · ( R i ¯ j k ¯ l R i ¯ j k ¯ l dz i ∧ d ¯ z j ∧ dz i ∧ d ¯ z j ⊗ ( d ¯ y l ∧ dy k )) .The statement for general m > R i ¯ jk ¯ l dz i ∧ d ¯ z j ⊗ y k ¯ y l :And the Feynman weight of the right picture of (3.6) follows from a straightforward com-putation which we omit here.Thus we indeed get the cancellation. For wheels with more vertices, the cancellationcan be proved by an induction on the number of vertices on the wheels. (cid:3) A simple consequence of this computation is the following
Proposition 3.36.
The first type (i.e., the left picture in (3.4) ) and third type (i.e., the left picturein (3.5) ) one-loop Feynman weights cancel with each other.Proof.
The first type and third type one-loop graphs are exactly obtained by connectingthe starting and ending tails of the graphs shown in the picture (3.6). It follows from theprevious lemma that the Feynman weight for the first type is of the form − C · m · ( ω k ¯ l ω k ¯ l · · · ω k m − ¯ l m ω k m ¯ l )( R i ¯ j k ¯ l R i ¯ j k ¯ l · · · R i m ¯ j m k m ¯ l m )( dz i d ¯ z j · · · dz i m d ¯ z j m ) ,where the constant C arises from the combinatorics of graphs, while that for the third typeis of the form C · m · ( ω k ¯ l ω k ¯ l · · · ω k m − ¯ l m ω k m ¯ l )( R i ¯ j k ¯ l R i ¯ j k ¯ l · · · R i m ¯ j m k m ¯ l m )( dz i d ¯ z j · · · dz i m d ¯ z j m ) . (cid:3) The second type one-loop graphs, i.e., connected wheels with only blue edges on thewheel (the right picture in (3.4)), give rise precisely to the logarithm of ˆ A genus, as wasshown by Grady and Gwilliam [14]: Proposition 3.37 (Corollary 8.6 in [14]) . The Feynman weights corresponding to the followingone-loop graphs give rise to the logarithm of the ˆ A genus of X.
Finally, the contribution from the fourth type one-loop graphs, i.e., a tadpole graph (theright picture in (3.5)), is described by the following lemma:
Lemma 3.38.
The Feynman weight of the tadpole graph is given by Tr ( R + ) , i.e.,
12 Tr ( R + ) = , where R + is given in (2.1) , (2.2) .Proof. The Feynman weight of the tadpole graph is explicitly given by¯ h ι Π (cid:18) h ˜ R ∇ (cid:19) = ι Π (cid:32) √− R i ¯ jk ¯ l dz i ∧ d ¯ z j ⊗ dy k ∧ d ¯ y l (cid:33) = − √− ω k ¯ l R i ¯ jk ¯ l dz i ∧ d ¯ z j =
12 Tr ( R + ) . (cid:3) Combining Proposition 3.28, Corollary 3.31, Proposition 3.36, Proposition 3.37 and Lemma3.38, we arrive at our second main result, which is a cochain level formula for the alge-braic index:
Theorem 3.39.
Let γ be a solution of the Fedosov equation (2.8) and γ ∞ be the associated solutionof the QME as defined in Definition 3.11. Then we have σ (cid:16) e ¯ h ι Π ( e ˜ R ∇ /2¯ h e γ ∞ /¯ h ) (cid:17) = ˆ A ( X ) · e − ω ¯ h ¯ h + Tr ( R + ) = Td ( X ) · e − ω ¯ h ¯ h + Tr ( R + ) , where Td ( X ) is the Todd class of X.Proof. We only need to show the second equality, which follows from the formula Td ( X ) = ˆ A ( X ) · e − Tr ( R + ) . (cid:3) Applying Lemma 3.27 gives the algebraic index theorem : APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 37
Corollary 3.40.
The trace of the function is given by Tr ( ) = (cid:90) X ˆ A ( X ) · e − ω ¯ h ¯ h + Tr ( R + ) = (cid:90) X Td ( X ) · e − ω ¯ h ¯ h + Tr ( R + ) .As we mentioned in the introduction, when α = ¯ h · Tr ( R + ) or ω ¯ h = − ω + ¯ h · Tr ( R + ) ,we will prove in the forthcoming work [6] that the associated star product (cid:63) α is exactlyequal to the Berezin-Toeplitz star product studied in [3, 4, 20]. In this case, the algebraic indextheorem is formulated as: Tr ( ) = (cid:90) X Td ( X ) · e ω /¯ h .A PPENDIX
A. F
EYNMAN GRAPHS
In this section, we describe the basics of Feynman graphs. For more details, we referthe reader to [7].
Definition A.1. A graph G consists of the following data:(1) A finite set V ( G ) of vertices ,(2) A finite set H ( G ) of half edges ,(3) An involution σ : H ( G ) → H ( G ) . The set of fixed points of this map is called theset of tails of G , denoted by T ( G ) ; a tail is also called an external edge . The set oftwo-element orbits of this map is called the set of internal edges of G , denoted by E ( G ) ,(4) A map π : H ( G ) → V ( G ) sending a half-edge to the vertex to which it is attached,(5) A map g : V ( G ) → Z ≥ assigning a genus to each vertex. Remark
A.1 . In a picture of a graph, we will use solid lines and dotted lines to denoteinternal edges and tails respectively.It is obvious how to construct a topological space |G | associated to a graph G . A graph G is called connected if |G | is connected. The genus is a graph is defined by g ( G ) : = b ( G ) + ∑ v ∈ V ( G ) g ( v ) .Here b ( G ) denote the first Betti number of |G | . Remark
A.2 . We call a graph G a tree if b ( G ) =
0, or a one-loop graph if b ( G ) = Definition A.2.
A one-loop graph G is called a wheel if removing any of its internal edgeswill give rise to a tree, as in the following picture:Let E be a graded vector space over a base ring R . Let E ∗ denote its R -linear dual (orcontinuous dual when there is a topology on E ). Letˆ O ( E ) : = ∏ k ≥ Sym kR ( E ) ,denote the space of formal functions on E . We define a subspace O + ( E ) ⊂ ˆ O ( E )[[ ¯ h ]] consisting of those formal functions which are at least cubic modulo ¯ h and the nilpotentideal I in R . Let F ∈ O + ( E ) which we expand as F = ∑ g , k ≥ F ( k ) g ,where F ( k ) g : E ⊗ k → R is an S k -invariant (continuous) map.We will fix an element P ∈ Sym ( E ) which we call the propagator . With F and P , wewill describe for every connected stable graph G the Feynman weight : W G ( P , F ) ∈ O + ( E ) .Explicitly, we label each vertex v ∈ V ( G ) of genus g ( v ) and valency k by F ( k ) g ( v ) . Thisdefines an element: F v : E ⊗ H ( v ) → R ,where H ( v ) denotes the set of half-edges incident to v . We label each internal edge by thepropagator P e = P ∈ E H ( e ) ,where H ( e ) denotes the two half edges which together give rise to the internal edge e . Wecan then contract the tensor product of vectors in E (from E ( G ) ) with the tensor productof formal functions on E (from V ( G ) ) to yield an R -linear map: W G ( P , F ) : E T ( G ) → R . APRANOV, FEDOSOV AND 1-LOOP EXACT BV ON K ¨AHLER MANIFOLDS 39
Definition A.3.
We define the homotopic renormalization group flow operator (HRG) withrespect to the propagator
P W ( P , − ) : O + ( E ) → O + ( E ) by W ( P , F ) : = ∑ G ¯ h g ( G ) | Aut ( G ) | W G ( P , F ) ,where the sum is over all connected graphs, and Aut ( G ) denotes the automorphismgroup of G . We can equivalently describe the HRG operator formally by the simple for-mula: e W ( P , F ) /¯ h = e ¯ h ∂ P ( e F /¯ h ) ,where ∂ P denotes the second order differential operator on O ( E ) given by contractingwith P . R EFERENCES [1] S. Axelrod and I. M. Singer,
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Universality of Fedosov’s construction for star products of Wick type on pseudo-K¨ahler mani-folds , Rep. Math. Phys. (2003), no. 1, 43–80.[24] L. Rozansky and E. Witten, Hyper-K¨ahler geometry and invariants of three-manifolds , Selecta Math. (1997), no. 3, 401–458.D EPARTMENT OF M ATHEMATICS , T HE C HINESE U NIVERSITY OF H ONG K ONG , S
HATIN , H
ONG K ONG
E-mail address : [email protected] T HE I NSTITUTE OF M ATHEMATICAL S CIENCES AND D EPARTMENT OF M ATHEMATICS , T HE C HINESE U NIVERSITY OF H ONG K ONG , S
HATIN , H
ONG K ONG
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OUTHERN U NIVERSITY OF S CIENCE AND T ECHNOLOGY , S
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