j. differential geometry
50 (1998) 387-413
FEDOSOV CONNECTIONS ON KAHLER
SYMMETRIC MANIFOLDS AND TRACE DENSITY
COMPUTATION
DMITRY E. TAMARKIN
Abstract
A Fedosov connection on a Kahler locally symmetric manifold in a compact
explicit form was found. The trace density for the corresponding quantization was directly computed.
0. Introduction
In this paper we introduce a Fedosov connection on Kahler locally
symmetric manifolds.
Contrary to the case of an arbitrary manifold, this connection appears to be written in a very compact form. A similar construction
for an arbitrary Kahler manifold was independently elaborated in 3].
We also compute the trace density for this connection. (See Theorem
2.6). As a result of our computation, the trace density is expressed
via geometrical data of the manifold. The nal expression resembles a
stationary phase approximation for some Feynman path integral. Easy
calculation (see section 2) shows that this formula gives the same value
as the index theorem for deformations 2], 4]. According to the result
of the author 5] which says that any topological invariant on a symplectic manifold expressed in terms of curvature and symplectic form is
a polynomial in characteristic classes and symplectic form, to prove the
index theorem for deformations it suces to prove it for Kahler symmetric manifolds. Thus, this yields a new proof of the index theorem for
deformations that is parallel to Atiyah-Bott-Patody heat kernel proof.
Received February 12, 1998.
387
388
dmitry e. tamarkin
Also, it would be interesting to generalize the formula for trace density
obtained in this paper to arbitrary manifold and to reveal its physical
sense.
Acknowledgments. The author would like to thank B.V. Fedosov,
B.L. Tsygan, and R. Nest for the concern to this paper and for useful
discussions. The author also greatly appreciates the work of referee
whose very useful remarks helped a lot.
0.1. Brief content of the paper
In sections 1.1-1.3 we briey review Fedosov quantization. Section 1.4
contains a collection of identities for the curvature tensor on Kahler
locally symmetric manifolds that we need in the sequel. Section 1.5
gives a construction of a Fedosov connection on a locally symmetric
Kahler manifold. In section 1.6 we quantize the Hamilton functions
of symmetry transforms. In section 1.7 on our manifold, we inroduce
Darboux coordinates, which are expressed in terms of the Hamilton
functions of symmetry transforms.
In sections 2.1-2.2 we develop some technique for trace density computation, which is applied to the case of a locally symmetric Kahler
manifold in section 2.3. In section 2.4 we apply the obtained formula
for the trace density to C P n , and we get the same number as in the
index theorem for deformations 2],4].
1. Deformation quantization on Kahler symmetric manifolds
We will briey describe B.V. Fedosov construction of deformation
quantization 1] and construct a Fedosov connection on a symmetric
Kahler manifold.
1.1. Deformation quantization
Let M be a symplectic manifold. A deformation quantization on M
is an associative algebra denoted by A h (M ) whose elements are formal
series from C 1 (M )h]]. The additive operation is dened simply as
the sum of functions, and the multiplication is supposed to satisfy the
Correspondence principle
f g = fg ; 21 ihff gg + o(h)
(1.1.1)
fedosov connections on kahler symmetric manifolds
and the Locality property
f g=
X
389
hiDi (f g)
where Di are bidierential operators of nite order. The last condition
means that the multiplication is well dened for C 1 -jets at any point
of M .
1.2. The Weyl Algebra
Let V ! be a symplectic vector space. Dene the Weyl algebra W (V )
whose elements are formal series
X
ay
where is a multiindex, a 2 C h]], and y1 y2 : : : y2n form a basis in
V . The product is given by the formula
kj @ @ !
a b(x) = exp ;ih 2 @y @z a(y)b(z)jy=z=x
k j
where !ij =< yi yj >, and < > is the symplectic scalar product on
V . The Weyl algebra provides a deformation quantization for jets of
functions on V .
A Symbol of an element s(y) 2 W (V ) is by denition its value at
zero (s) = s(0).
The Weyl algebra is naturally graded in the following way. Set
deg h = 2, deg y = 1 for y 2 V . The symplectic group Sp(V ) acts on
W (V ) naturally preserving the grading and the symbol. This means
that we can change the ber in the symplectic bundle T M from R2n
to W (R 2n ). Thus the new bundle of algebras is called the bundle of
Weyl algebras W (M ). Sections of this bundle form an algebra with
the berwise addition and multiplication. The grading and the symbol
of section of W (M ) are also well dened berwise. Any element f 2
C 1(M )h]] denes a section of W (M ), and these sections form the
center of W (M ).
1.3. Flat connections
A connection in W (M ) is a linear map
D : ;(W (M )) ! ;(W (M ) T M )
390
dmitry e. tamarkin
such that
D (a b) = D a b + a D b
Df
= df f 2 C 1(M )h]] W (M ):
Any such a connection can be extended to the forms W (M ) T M .
Let r be a symplectic connection on T M (i.e., r! = 0 and r is torsion
free). Obviously, r can be extended to be a connection on W (M ). Let
;0 2 W (M ) T M . Then any connection in W (M ) is of the form
D s = rs +
i ;0 s]
h
(the condition in (1.1.1) implies that any commutator is a multiple of
h).
Consider the symplectic form ! 2 T M T M . The rst tensor
factor of ! can be regarded as an element of W (M ), and we get the
form 0 2 W (M ) T M . Put = ;0 .
The main point in the Fedosov construction is a at connection
r + ;0 in W (M ) such that
;0 = + ;
where ; = o(2), i.e., all summands in ; have their grading not less
than 3. The atness means that the curvature form = D2 is a purely
scalar form. This form is called the Weyl curvature. Such at connections are called Fedosov connections. They do always exist and can
be constructed by means of a certain iteration process 1]. Covariant
constant sections of W (M ) form a subalgebra W (M )D W (M ), and
the symbol map : W (M )D ! C 1(M )h]], s 7! (s) is invertible.
Furthermore, the map denes a one to one correspondence between
the space of C 1 -jets at a point x and the ber of W (M ) at the same
point (jet of a function f goes into the section ;1 f restricted to x).
We have obtained a one to one map
(1.3.1)
quant : W (T Mx ) ! Jx1 (M )h]]:
Now we are in a position to construct a deformation quantization. Put
f g = (;1 f ;1g):
Note that the map quant denes an isomorphism between the Weyl
Algebra and the jet algebra Jx1 (A h ) corresponding to our quantization.
fedosov connections on kahler symmetric manifolds
391
1.4. Some formulas for symmetric Kahler manifolds
In this subsection we shall establish some sign conventions and collect
the identities for the curvature tensor that we will use in the sequel. All
these identities are well known and some of them can be found in 6].
Let M be a Kahler locally symmetric manifold with metrics g and
symplectic form !. Let x0 be a point in M and e1 e2 : : : en be a unitary
basis in T@ Mx0 . Then we have bases e1 e2 : : : en e1 e2 : : : en , and
e1 e2 : : : en in T@Mx0 , T@Mx0 , and T@Mx0 respectively. We have
X
X
g = ek ek ! = 1 ek ^ ek :
i
The symplectic product < > on T M is given by
< ek ej >= ;ikj < ek ej >=< ek ej >= 0:
Respectively, on TM we have
< ek ej >= ;ikj < ek ej >=< ek ej >= 0:
The hermitian connection r corresponding to g is torsion free there-
fore it is a symplectic connection and coincides with the Levi{Civita
connection. Let R(ei ej )ek = Rklij el . Since R(X Y ) is antihermitian
for real X and Y , we have
Rklpj = Rlkjp:
(1.4.1)
Using this, we nd
R(ep ej )ek = ;Rklpj el :
The Riemann identity R(X Y )Z + R(Y Z )X + R(Z X )Y = 0 yields
(1.4.2)
Rklpj = Rplkj = Rkjpl:
The identity r2 R = 0 reads
X
(1.4.3)
Rakpj Rcdkb ; Rkbpj Rcdak + Rckpj Rkdab ; Rkdpj Rckab = 0:
k
P
Let Rij = k Rkkij be the Ricci tensor. Let us prove that
X
X
X
X
(1.4.4)
Rak Rcdkb = RkbRcdak = Rck Rkdab = RkdRckab:
k
k
k
k
dmitry e. tamarkin
392
Contracting (1.4.3) with ij and using (1.4.2), we get
(1.4.5)
X
k
!
Rak Rcdkb ; RkbRcdak + Rck Rkdab ; RkdRckab = 0:
Contracting (1.4.3) with id and alternating j and b give
X
k
or
X
(1.4.6)
k
!
Rjk Rckab ; Rbk Rckaj = 0
Rjk Rckab =
X
k
Rbk Rcjak :
Taking the complex conjugate and using (1.4.1) lead to
X
(1.4.7)
k
Rkj Rkcba =
X
k
RkbRjcka:
Thus (1.4.4) follows from (1.4.5), (1.4.6), and (1.4.7).
1.5. Fedosov connection for symmetric Kahler manifolds
We will apply the Fedosov construction to a locally symmetric Kahler
manyfold M with the corresponding Kahler metrics g and the symplectic
form !. Our main goal is to construct a Fedosov connection on M . A
direct application of the algorithm from 1] causes certain diculties
and, probably, can not be completed in a compact form. Fortunately,
in our case, there exists another way to construct a Fedosov connection.
Consider the curvature tensor R as an element of T@ M T@ M T@ M T@M . We may consider the 2-nd, the 3-rd, and the 4-th tensor
multiples as elements of W (M ), and we get a form ; 2 W (M ) T M .
In a unitary basis e1 e2 en we have
X
; = R ej ek el ep = 1 ep ep ; ep ep
klpj
i
Set Ds = rs + hi ; 21i ; s].
Theorem 1.1. The connection
of this connection is equal to !.
D
is at, and the Weyl curvature
fedosov connections on kahler symmetric manifolds
393
Proof. For the sake of convenience set
!
X @s p @S p
i
s = h s] = ;
@ep e + @ep e
and = 21i ;. The curvature equals = ! + r ; + hi ], where
r = (1=i)Rklpj (ek el ep ^ ej ) 2 W (M ) 2 T M . Let us prove that
is a scalar and equals !. Note that ; r = 0. Let us show that
] = 0. This reads as
Rkbij ej ek eb Rcdal el ec ed ]
(1.5.1)
= 4hec ej eb ed (Rkbij Rcdak ; Rckij Rkdab ):
Now, take (1.4.3) with i and a alternated and contract it with
4hec ej eb ed . We will get (1.5.1), which proves the theorem. q.e.d.
Set
p = ep + 21 Rklpj ej ek el :
Then
1 Xep s] ep ; p s] ep :
(1.5.2)
D s = rs +
h
Let us construct explicitly the quantization map (1.3.1). Introduce
in a neighbourhood of a point x 2 M the geodesic coordinates
a1 a1
(1.5.3)
an an
so that dap jx = ep .
Theorem 1.2. We have
1 P
quant (s)(a) = e h ad ap p ;ap ep s
where ad x = x ], and the exponent is treated as a formal series.
Proof. Consider a geodesic line p = p t. Let us transport the basis
1
e : : : en along it, and write down the condition for a section s~(t) of
W (M ) to be a covariant constant along the geodesic line,
ds~ = 1 Xp p ; pep s~]:
(1.5.4)
dt
h
P
We have up to o(t1 ) that s~ = e h1 ad p p ;p ep s~(0). Now it suces to
set ap = tp and to calculate the symbol. q.e.d.
dmitry e. tamarkin
394
1.6. Quantization of Hamilton functions for innitesimal symmetry
transforms
Consider a simply connected open subset U in M . Innitesimal symmetry transforms are Hamiltonian vector elds on U , which correspond
to certain Hamilton functions. We will quantize these functions.
At each point x0 , let us dene in the ber W (U )x0 a subspace gx0 ,
generated over C by the elements ei i tij = ij + Rklij ek el . Each of
these subspaces forms a Lie algebra with respect to commutator i=h ].
Indeed,
(i=h)ep ej ] = (i=h)p j ] = 0
(i=h)ep j ] = ; itjp
(1.6.1)
(i=h)ep tkj ] = ; iRlpkj el
(i=h)p tkj ] = iRplkj l :
These identities follow from (1.4.3), (1.4.2). The commutator tij tkl ]
can be obtained from the Jacoby identity. All these algebras are isomorphic to the compexied Lie algebra g of symmetry transforms of M
and form a bundle P which is preserved by the at connection D and,
therefore, is trivial over U (because U is simply connected). The space
F of covariant constant sections of P has dimension dim g . Denote
= (F ).
Proposition 1.3. The space is closed with respect to the complex
conjugation.
Proof. In each ber gx0 of P dene a C -antilinear map J as follows:
J (ep ) = p J (p) = ep J (tpj ) = tjp:
One checks that J is an R-linear isomorphism of gx0 as a Lie algebra
over R. Using (1.5.2), we see that J is a covariant constant. Note that
(Js) = (s) for s 2 F , whence the proposition follows. q.e.d.
(1.6.2)
Theorem 1.4.
1. Let s t 2 F , and f g be the Poisson bracket on M . Then
( hi s t]) = f(s) (t)g:
2. Hamiltonian vector elds Xf , f 2 form the Lie algebra of innitesimal local symmetries of M .
fedosov connections on kahler symmetric manifolds
Proof. 1. Let
(1.6.3)
s = crs (a a)ers + drs (a a)r + prjs (a a)trj :
Eq.(1.5.4) implies that
@s = 1 k s]
@ak a=0 h
@s = ; 1 ek s]:
@ak a=0
h
(1.6.4)
Hence,
d(s)ja=0 = h1 (k s])ek ; h1 (ek s])ek
(1.6.5)
= cks ek + dks ek ja=0 :
Since the origin a = 0 can be chosen arbitrarily, this formula holds for
every point.P
Let t = cr (a a)er + dr (a a)r + prj (a a)t . Then
t
f (s)
t
t
t
rj
(t)gja=0 = ;cks dkt + ckt dkt = hi (s t])ja=0 :
2. Due to Proposition 1.3, it is enough to consider s 2 F such that
(s) is real. This means that Js = s and that
dk = ck prj = pjr :
Let X(s) be the Hamiltonian vector eld corresponding to (s). It
suces to prove that the map rX(s) : Y 7! rY X(s) maps ;(T@ M )
to ;(T@ M ) and that it is an antihermitian map. Indeed, in this case
LX (s) Y = rX (s) Y ; rY X(s) belongs to ;(T@ M ) if Y 2 ;(T@ M ) and
LX (s) (Y Z ) ; (LX (s) Y Z ) ; (Y LX (s) Z )
= (rY X(s) Z ) + (Y rZ X(s) ) = 0:
Eq. (1.6.5) implies that
X(s) = ;i(ck ek ; dk ek ):
Hence,
395
dmitry e. tamarkin
396
k
k
k ; @d ek )ja=0
= ;i( @c
e
t
@a
@at
rs
k
= iRtkrs p e :
We have used (1.6.5) and (1.6.1). We see that rX(s) maps holomorphic elds to holomorphic elds. Since prs = psr , the matrix kiRikrs prskik
is antihermitian. q.e.d.
Now we just need to prove that all symmetry transforms are Hamiltonian elds corresponding to functions from . This immediately follows from the coincidence of dimensions dim g = dim .
ret X(s) a=0
1.7 Darboux coordinates
We will construct Darboux coordinates in a neighbourhood of a point
x0 2 M that are expressed in terms of functions from .
Denote
(1.7.1)
p = quant ep
(1.7.2)
p = quant p
(1.7.3)
pj = quant tpj :
The system of variables (1.7.1)-(1.7.2) (which are functions from )
is non-degenerate in a neighbourhood V1 of x0 . The correspondence
(1.7.1)-(1.7.3) between generating functions and sections from P gives
us a hint how to construct Darboux coordinates. Note that i = ei =2 +
(1=2)tij ej . Let us specify new variables i by means of the system of
equations
(1.7.4)
p = p=2 + (1=2)pj j :
This system is non-degenerate in a neighbourhood V2 of x0 . Let us
choose a neighbourhood V of x0 so that V be dieomorphic to the ball
and V V1 \ V2 .
Theorem 1.5. We have in V
(1.7.5)
pj = pj + Rklpj k l X
! = 1i dp ^ dp:
(1.7.6)
Thus, p p are Darboux coordinates in V .
fedosov connections on kahler symmetric manifolds
Proof. Let us prove (1.7.5). Consider the ideal U
ated by the functions
2
397
C 1(V ) gener-
'pj = pj ; pj ; Rklpj k l :
Suppose we have proven that all the functions
f p
'jk g fp 'jk g
are in U . Then we have for 'pj a system of linear partial dierential
(1.7.7)
equations
f p
'jk g = apjklm'lm This system is of the form
f p
'jk g = bpjklm'lm :
@Fp = c (x)F :
@xj pjk k
One easily checks that 'jk (x0 ) = 0. The uniqueness theorem for systems
of the form (1.7.8) yields that 'jk = 0 which proves (1.7.5).
Let us prove that the functions in (1.7.7) are in U . The congruence
modulo U will be denoted as usual by . Theorem 1.4 yields f i j g =
(i=h)quant ep j ] = ;ijp. Compute the Poisson bracket with i and
(1.7.8)
both sides of the Equation in (1.7.4):
;ijp
hence
= 21 f p j g + 12 f p jk gk + 12 jk f p k g
= 21 (jk + jk )f p k g ; 2i Rlpjk l k
1 ( + )f p k g =
2 jk jk
=
1
l k
2 Rlpjk )
1
;i(jp ; Rlkjp l k )
2
1
; i(jp + jp ):
2
;i(jp ;
The matrix 12 i(jp + jp) is invertible therefore
(1.7.9)
f p
j g ;ipj :
dmitry e. tamarkin
398
Similarly, let us compute the Poisson bracket of j and both sides of
(1.7.4). We have
i
f j p g + pk f j k g + Rjlpk l k = 0:
(1.7.10)
2
Let us prove that
i
(1.7.11)
f p j g ; Rjlpk k l :
2
If we formally substitute (1.7.11) into (1.7.10), we will have on the lefthand side a function from U (to check it, it suces to substitute for ij
the congruent functions from the right-hand side of (1.7.5) and to use
(1.4.3),(1.4.2)). The matrix ij + ij is invertible, hence the congruences
(1.7.11) do hold. Using (1.7.9) and (1.7.11), one can prove by direct
computation that functions in (1.7.7) are in U . Thus, we have shown
that U = 0 and that all congruences modulo U are identities. Now
(1.7.6) easily follows from (1.7.9). q.e.d.
The coordinates p have another nice property.
Theorem 1.6. The functions p are antiholomorphic.
Proof. Symmetry transforms are known to preserve the complex
structure. On the other hand, the identities in (1.7.9), (1.7.11) imply
that holomorphic functions of 1 : : : n go to holomorpic functions of
1 : : : n under symmetry transforms. Furthermore, the action of the
symmetry group is transitive on M . Now the theorem follows from the
fact that the dierentials d1 : : : dn are antiholomorphic at x0 .
q.e.d.
2. Computation of trace density for symmetric Kahler
manifolds
2.1. The trace density
In this section we will recall the denition of the trace density and give
some formulas for its computation.
Let A h (M ) be a deformation of C 1 (M ) . For any local chart U
on M with Darboux coordinates X 1 X 2 : : : X 2n the algebra A h (U ) is
isomorphic to the standart Weyl deformation W (U ) with the multiplication given by
(f g)(X ) = e;ih=2~!pj @=@up @=@zj f (u)g(z ) ju=z=X
fedosov connections on kahler symmetric manifolds
399
where as usual
!~ i i+n = ;!~ i+n i = 1 otherwise !~ ij = 0
so that ! = !~ ij dxi ^ dxj . Moreover, the isomorphysm
: W (U ) ! A h (U )
can be chosen so that
(f ) = f +
X
hk Uk f
where Uk are dierential operators of nite order 4].
For elements of the Weyl deformation with compact support
f 2 C01(U ) the trace is well dened by the formula
Z
1
Trf = (2h)n f!n =n!:
U
This trace is transferred to A h0 (U ) by means of an arbitrary (innitely
dierentiable and continuous in h-adic topology) isomorphism W (U ) !
h
Ah
0 (U ). Then the trace is extended to A 0 (M ) by means of a partition
of unity, and thus the obtained trace does not depend on a partition
neither on a choice of the isomorphisms.
There exists a unique function R on M , which is the Laurent series
with respect to h, such that Trf = f!n=n!. The function is called
M
the trace density. One can easily show that = 1=(2h)n + o(1=hn ).
We need to modify this denition so as to express the trace density
in terms of jets. According to the locality property of deformations the
algebra A h (M ) can be restricted onto jets at a point x0 . The corresponding jet algebra Jx10 h]] is isomorphic to the Weyl algebra W on
R2n < y 1 : : : y 2n > with symplectic scalar product < y i y j >= !
~ ij
(see subsection 1.2). Our purpose is to express the trace density in
terms of such an isomorphism. To obtain a reasonable formula we need
to put some restrictions on these isomorphisms.
For further purposes we have to consider a wider class of coordinate
systems. Let x1 : : : x2n be "almost Darboux coordinates" in a neighbourhood of x0 , i.e., fxp xj g = !~ pj + o(1) (e.g. the normal geodesic
coordinates on a symmetric Kahler manifold). We can identify Jx10 h]]
and W by means of the map
: f (x h) 7! f (y h):
400
dmitry e. tamarkin
After that, given an isomorphism ' : W ! Jx10 h]] we can construct a
map = ' : Jx10 h]] ! Jx10 h]]. We want that were a dierential
operator. Let us specify what kind of operator d should be. Consider
the space DP of dierential operators that are polynomials in x @=@x
with the natural grading jxj = 1 j@=@xj = ;1. Consider the completion
D 0 of DP with respect to this grading. The action of operators from D 0
on Jx10 is well dened as well as the action of operators from D = D0 h]]
on Jx10 h]]. We require that 2 D and = 1 + O(1) + O(h), where
O(1) means sum of operators from D with their grading not less than
1. The isomorphisms ' such that corresponding operators satisfy
these two conditions are called suitable. Clearly, in this case, ;1 2
D . Note that the restriction to Jx10 h]] of the operator satises this
requirement. More precisely, let X = X (x) be transition functions from
the almost Darboux coordinates x to Darboux coordinates X such that
X = x + o(x). Then the formal series in X h can be viewed as elements
of W (if one substitutes y for X ), and the corresponding isomorphism
' : W ! Jx10 h]] can be written as follows
' : f (X ) 7! g(X ) = (f )(X ) 7! ' f (x) = g(X (x))
= ' : f (x) 7! g(X (x)):
Here, all functions are considered as functions of 2n variables rather
than functions of point in U . Of course, there corresponds a function
on U to each of these functions but F (x(P )) 6= F (X (P )), where P 2 U .
Note, that the maps f (x) 7! (f )(x) and g(x) 7! g(X (x)) are operators
from D. Therefore, 2 D. The condition = 1 + O(1) + O(h) can
also be checked easily. The operator ; 1 looks as follows. Let
(2.1.1)
h(X ) = ;1 f (x(X )):
Then ; 1 f = h(x).
For an operator D 2 D we can dene the conjugate by the same formula as in the case of conventional dierential operators: (x @=@x ) =
(;1)j j @x@ x for multiindices and . It can be easily checked that
the conjugation is well dened on D but it depends on a choice of coordinates.
Now we are in a position to nd the trace density.
Proposition 2.1. Let ' be a suitable isomorphism. Then
1 (;1 ) 1:
(x0) = (2h
)n
fedosov connections on kahler symmetric manifolds
401
Proof. First, let us prove the proposition for ' = ' . Let V and W
be neighbourhoods of x0 such that 8P 2 V 9!Q 2 W : xi (P ) = X i (Q).
Note that (2.1.1) species a map ~; 1 : C 1(W ) ! C 1(V ), its restiction
onto jets at x0 is ; 1 . Let f be a function with suppf 2 W . By the
denition of the trace,
Z
1
Trf = (2h)n (;1 f )(x)dx1 dx2 : : : dx2n
W
Z
1
= (2h)n (;1 f )(x)dx1 dx2 : : : dx2n
VZ
1
= (2h)n h(x)dx1 dx2 : : : dx2n
VZ
1
= (2h)n ~; 1 fdx1 : : : dx2n
VZ
1
= (2h)n f (~; 1 1)dx1 : : : dx2n
V
where h(x) is in (2.1.1). Therefore, (x0 ) = (1=hn ); 1 1 because is
a restriction onto jets of ~ . For arbitrary the map = ;1 ;1
is an automorphism of W . Any such an authomorphism is of the
form ei=had H for some H 2 W (see 4]). One can easily check that
(;1 ) ;1 = = e;i=had H . Note that 1 = ei=had H 1 = 1.
Therefore, (;1 ) 1 = 1 and ;1 1 = ; 1 (;1 ) 1 = (; 1 ) 1.
q.e.d.
Similarly one can prove the following modication of this proposition. Let J be a volume density, i.e., !n =n! = Jdx1 ^ : : : ^ x2n .
Proposition 2.2.
1 :
(J ) = (2h
)n
2.2. Exponents
We need to operate with expressions of the form e;k(x)=h where k(x)
is a positively denite quadratic form. In this section we legalize these
expressions and show how to apply them to the trace computation.
The corresponding formula also expresses the trace density in terms of
dmitry e. tamarkin
402
the isomorphisms of algebras of jets but it is not valid for all suitable
isomorphisms. Thus, we have to dene a narrower class of isomorphisms.
Consider the space Jx10 h;1 h]] which is a completion of the space
of polynomials in xp h h;1 with respect to the grading jxp j = 1 jhj =
2 jh;1 j = ;2. Dene the space U as a completion of the space of dierential operators which are polynomials in x @=@x h h;1 with the following grading on it: jxj = 1 j@=@xj = ;1 jhj = 2 jh;1 j = ;2. Then
operators from U act on Jx10 h;1 h]] and for U 2 U , f 2 Jx10 h;1 h]],
we have jUf j = jU j + jf j for the evaluations dened by the gradings.
For an X 2 U one can correctly dene an operator ek(x)=h Xe;k(x)=h .
To do it one suces to notice that this conjugation is obviously dened for X being a polynomial in x @=@x h h;1 and that it transforms
polynomial operators into polynomial operators preserving the grading.
Similarly, for f 2 Jx10 h;1 h]] one can dene the integral
Z
e;k(x)=hfdx1 : : : dx2n 2 C h;1 h]]:
For f being a polynomial on x h h;1 we put
Z
e;k(x)=hfdx1 dx2n =
Z
e;k(x)=h fdx1 dx2n
R2n Z
p
:= h1n e;k(y) f0 ( hy)dy1 dy2n
2n
(2.2.1)
R
where f0(y) = (f (y) + f (;y))=2 is the even part of f . This operation
preserves the gradings and is therefore well dened on U . We will apply
it to the trace density computation.
Proposition 2.3. Let ' be an isomorphism W ! Jx10 h;1 h]] such
that the corresponding dierential operator = ' is suitable and
belongs to U . Then
Z
e;k(x)=h (x)(ek(x)=h e;k(x)=h )1Jdx1 : : : dx2n
Z
1
= (2h)n e;k(x)=h dx1 : : : dx2n :
fedosov connections on kahler symmetric manifolds
403
Proof.
Lemma 2.4 For any D 2 U and 2 Jx10 h;1 h]] we have
Z
e;k(x)=h (ek(x)=h De;k(x)=h)1dx1 : : : dx2n
=
Z
e;k(x)=h Ddx1 : : : dx2n:
Proof of Lemma. One suces to check it for and D being polynomials. In this case the integrals can be treated as conventional integrals
over R2n , whence the statement of Lemma.
To prove the proposition, let us substitute instead of D. Thus we
have:
Z
e;k(x)=h (x)(ek(x)=h e;k(x)=h )1Jdx1 dx2n
Z
= e;k(x)=h (J )dx1 dx2n
Z
1
= (2h)n e;k(x)=h dx1 dx2n
by Propositon 2.2. q.e.d.
We will investigate the case of symmetric manifolds so that appeared to be a constant function on M . We can rewrite the above
proposition for this case.
Corollary 2.5. If is a constant function, we have
R ;k(x)=h k(x)=h ;k(x)=h 1
(e Z e
)Jdx : : : dx2n :
;
1
= e
1
;k(x)=hdx1 : : : dx2n
(2h)n e
2.3. Computation for symmetric manifolds
We will apply the tools developed in the previous section to compute
the trace density for the quantization which was constructed above for
locally symmetric Kahler manifolds. Let us outline the ultimate result.
Let # be an integral purely antiholomorphic form dened in a neighbourhood of a point x0 such that
(2.3.1)
d# = ! ; 2hi R
dmitry e. tamarkin
404
where R = Rkkij ei ^ ej is the Ricci curvature or, which is the same, the
curvature on T@ M . To dene uniquely, let us require that
(2.3.2)
#=
X
zp#p
for some holomorphic coordinates z , z jx0 = 0 and some 1-forms #i . Let
x0 !x be a geodesic line from xo to x in a small neighbourhood U of
xo. Denote by Rxx0 the reection of x0 with respect to x. Consider the
'action'
Z
S (x) =
#:
x0 !Rx x0
It can be easily seen that in geodesic coordinates (a a) in a neigbourhood of x0 , we have
S = 2i jaj2 + s
where s = o(jaj2 ) as a formal series from Jx10 h]]. This means that
e; hi s(x) 2 Jx10 h;1 h]]. Hence, we may dene the integral
Z
Z
i S (x) n def
;
e h ! = e;2jaj2 =he; hi s(a) !n
as in (2.2.1).
Theorem 2.6
R ; hi S(x) n
Z i
e
!
=n
!
n
1=(x0 ) = 1 R ;2jej2 =h n 1 1
e; h S(x)!n=n!:
n den = 2
e
i
de
de
:
:
:
de
n
(2
h)
This section is completely devoted to a proof of this theorem. We
will use Corollary 2.5. First of all we need an isomorphism ' such
that the corresponding operator belongs to U . We claim that the
isomorphism quant in Theorem 1.2 matches this requirement.
p
p
Proposition 2.7 Let xp + ixn+p= 2 = ap, xp ; ixn+p= 2 = ap be
geodesic normal coordinates in a neigbourhood of a point x0 2 M , and
' = quant. Then the corresponding morphism = ' is suitable and
can be expressed as a dierential operator from U .
Proof. Let C = C a a e e h]] with the grading jaj = jaj = jej =
jej = 1, jhj = 2. We can decompose quant = where : W !
;1
p p p p
C , (f ) = eh ad (a e ;a ) f , and : C ! Jx10 h;1 h]] is the simbol
fedosov connections on kahler symmetric manifolds
405
map. Note that jj = 0. Decompose h;1 (ap ep ; ap p) = A + B , where
A = h;1 (apep ; apep) and B = h;1 apRklpj ej el ek =2. Then jad Aj = 0,
jad B j = 2. Therefore, we can use the perturbation theory formula
ead A+ad B f =
where
0 = 1 n =
1
X
n=0
n ead Af =
Z
0t1 tn 1
X
nf (a + e a + e)
ad L(t1 ) : : : ad L(tn )dt1 : : : dtn
and L(t) = etad A B . Note that (ad A)4 B = 0. Therefore, L(t) is a
polynomial in t, and ad L(t) for any t is a polynomial dierential operator W ! C with grading 2. Hence, all n are polynomial dierential
operators W ! C with grading 2n. It easy follows that
%n : f (a) 7! (n f (a + e a + e))
are polynomial dierential operators
Jx10 h;1 h]] ! Jx10 h;1 h]]
with grading 2n: Finally,
(2.3.3)
= quant =
X
%k :
Thus, = quant 2 U because j%k j ! 1. To show that quant
is suitable one suces to check that h only appears in the dierential
operator in positive degrees and that = 1+ O(1)+ O(h). This easily
follows from Theorem 1.2. q.e.d.
Put k(e 2e) = 2jej22 . To calculate the trace density we need to calculate (e2jaj =h e;2ja j=h )1. To prove Theorem 2.6 it suces to show
that
(2.3.4)
e2jaj2 =he;2ja2 j=h1 = e2jaj2 =he; hi S(a) :
As usual, let us write down an intuitively obvious expression for this
quantity, explain how to understand this expression in strict terms of formal 2 series, 2 and prove that this expression does equal to
(e2jaj =h e;2ja j=h )1.
dmitry e. tamarkin
406
If e;2jaj2 =h were a usual function we would be able to write
1 P
(e2jaj2 =h e;2ja2 j=h )1 = e2jaj2 =h e h ad p (ap p ;ap ep ) e;2jej2 =h :
The last expression can be represented as a formal series in a so that
(e2jaj2 =h e;2ja2 jh)1 =
X
a (D e;2jej2=h)
where D are dierential operators that are polynomials in e e @=@e @=@e.
Hence, D e;2jej2 =h = P (e e h h;1 )e;2jej2 =h for some polynomials P .
By denition, set
(2.3.5)
D e;2jej2=h = P (0 0 h h;1 ) 2 C h;1 h]:
Thus, we have arrived at the following statement.
Proposition 2.8
X
(e2jaj2 =h e;2ja2 j=h)1 = a P (0)
(2.3.6)
:=e2jaj2 =h e h ad
1
P (ap p ;ap ep ) ;2jej2 =h p
e
:
Hence, the right-hand side converges in Jx10 h;1 h]].
Proof. Any element of Jx10 h;1 h]] is a well dened formal series in
a with coecients in C h;1 h]] ( but degrees of h may be no bounded
uniformly for all coecients of the series). Obviously, this formal series
converges in Jx10 h;1 h]]. Thus the expressions in the statement being
proved can be regarded as formal series in a, and it suces to prove
the coincidence of these series. In other words, if an element u from
Jx10 h;1 h]] is equal to a formal series v in a in the space of formal series,
then v converges in Jx10 h;1 h]] and u = v in Jx10 h;1 h]]. Obviously,
we only need to prove that the series coincide up to o(jajk ). The proof
of Propositon 2.7 implies that
(e2jaj2 =h e;2ja2 jh)1 = e2jaj2 =h
N
X
k=1
(k e;2ja+ej2=h) + o(N )
where the symbol is calculated in the same way as in (2.3.5) (Recall that
each k is a polynomial dierential operator). To obtain this relation,
one needs to take (2.3.3) into account and to check that any operator
fedosov connections on kahler symmetric manifolds
407
T that is a polynomial in a e a2 e @=@e @=@e
, denes
the operator2 T~ :
2
2
~ ;2jaj =h 1 = e2jaj =h (Te;2je+aj =h ).
f (a) 7! (Tf (a + e)) and e2jaj =hTe
Now it suces to notice that
X
P
k e;2ja+ej2=h = e h1 ad p(ap p;apep)e;2jej2=h
because of the denition of k . q.e.d.
Further, we will transform the expression on the right-hand side
of (2.3.6) so as to get (2.3.4). Note that the expression on the righthand side of (2.3.6) is a formal series in a. The proof of Propositon 2.8
yields that we only need to prove that the series (2.3.6) coincides with
the formal series on the right-hand side of (2.3.4). Thus, we need not
Jx10 h;1 h]] anymore. We will start with some standart facts about the
action of polynomial operators on the vacuum projector e;2jej2 =h in the
Weyl algebra.
First, recall the denition of the Weyl ordering. Suppose we have
a polynomial P (e e) in the Weyl algebra. Suppose to each e e we put
in correspondence some dierential operator e^ ^e respectively. Let P be
a monomial lk=1 tk , where all tk are equal to one of the elements from
fe eg. Dene
X
P (^e ^e) = l1! t^(k) :
2Sl
For an arbitrary polynomial, P (^e ^e) is dened by linearity.
Proposition 2.9 Let P and Q be polynomials.
1. e;2jej2 =h P (e e) = e;2jej2 =h fP (h@=@e e)1g (the Weyl ordering
is assumed)
2. (P (e e h)2 e;2jej2 =h Q(e e h))
= (e;2jej =h Q(e e h) P (;e ;e h))
3. (e;2jej2 =h P (e h)) = P (0 h).
Proof. Straightforward computation using the Moyal product formula (1).
Remark. Note that the correspondence e^ = h@=@e, ^e = e used
in Prop. 2.9,1 does not2 preserve the commutator. This is because the
vacuum projector e;2jej =h is multiplied from the right hand side, and if
we consider operators acting on the functions from the right hand side,
then h@=@e e] = ;h = e e], and the commutators are preserved. Since
dmitry e. tamarkin
408
we use the symmetric Weyl ordering, the operator P (h@=@e e) applied
from the left gives the same result as the one applied from the right.
Proposition 2.10
(e2jaj2 =h e;2ja2 j=h )1 = e2jaj2 =h e;2jej2 =h e;(2=h)(ap p ;ap ep ) :
Proof. We will use the identity ead A B = eA Be;A which allows us
to rewrite (2.3.6) as follows:
2jaj2 =h ;2ja2 j=h 2jaj2 =h n X (1=h(ap p ; apep))k e
e
1 =e
k!
k
X (1=h(ap p ; apep))l o
;
l!
l
p p
p p k
X
=e2jej2 =h fe;2jej2 =h (;2=h(a k!; a e )) g:
2
e;2jej =h The last identity follows from Proposition 2.9,2. q.e.d.
Using Proposition 2.9,1,3 to substitute h@=@e for e and to compute
the symbol, gives
(e2jaj2 =h e;2ja2 j=h)1 = e2jaj2 =h expf;2=h(ap ep ; h=2Rpj ap ej )
+2ap @=@ep + ap Rklpj ej el =2@=@ek )g1je=0
where Rij = Rklij kl is the Ricci curvature. Thus, we arrive at the exponent of a rst order dierential operator, which we may use characteristics to compute. Substituting for e in this operator the antiholomorphic
coordinates yields an operator in a neighbourhood of x0 . It can be
easily checked (using(1.7.9) and (1.7.11)) that this operator equals
(2=i)(X;ar r +ar r )antiholom ; p(x)
where Xf means the Hamiltonian vector eld corresponding to f and
p(x) = (2=h)ar (r + h=2Rij j ). Our vector eld is tangent to a geodesic
line (at) and has the constant length of 2jaj along this geodesic. This
means that we need to determine p(x) just on this geodesic and that
; R01 p(2at)dt
(e j j e; j j=h )1 = e
2 a 2 =h
Let us nd p(at).
2 a2
; 12 R02 p(at)dt
=e
:
fedosov connections on kahler symmetric manifolds
Proposition 2.11.
409
p(at) = hi rX #jat
where X = (2=i)X;ar r +ar r is the tangent eld to the geodesic line and
# is as in (2.3.1).
We need
Lemma 2.12.
(2.3.7)
# = 1i ( k dk ; h2 Rpj pdj ):
Proof. We have to show that the right-hand side # of (2.3.7) satises:
(2.3.8)
d# = ! + (h=2i)R
(2.3.9)
# satises (2.3.2):
Let us prove (2.3.8). We have
d# = (1=i)(dk ^ dk + 21 hRpj dp ^ dj )
= ! + 21i hRpj d p ^ dj :
The form 1i Rpj p ^ dj coincides with the Ricci form at the origin.
The Ricci form is invariant with respect to the symmetries. Therefore,
if 1i Rpj d p ^ dj is also invariant, then R = 1i Rpj d p ^ dj . Let us check
the invariance.
1 L R d p ^ dj = 1 R d p ^ df k j g = 0:
i
k pj
i
pj
1 L R d p ^ dj = 1 R d(fk p g) ^ dj
i k pj
i pj
+ 1i Rpj d p ^ d(fk j g)
= Rpj d(Rmnkp n m ) ^ dj
1
;Rpj d p ^ d( Rjmkl l m )
2
= Rpj Rmnkp m dn ^ dj
+Rpj Rmnkpn ^ d m ^ dj
;Rpj d p Rjmkl l ^ dm :
dmitry e. tamarkin
410
The last expression vanishes due to (1.4.4).
Let us prove (2.3.9). It suces to show that all k belong to the
ideal generated by 1 : : : n . Since k = k (see (1.6.2)),
k = k + (1=2)Rplkj j pl
and therefore k belongs to the ideal. q.e.d.
Proof of the Proposition. Since # is purely antiholomorphic,
(i=h)iX # = (i=h)iXantiholom #
= (i=h)(2ai @=@i
;ak Rklpj j l @=@p
(1=i)( p dp + (h=2)Rpi p dp ))
= (1=h)(2ap ; ak Rklpj j l )( p + (h=2)Rpi p ):
Consider the dierence
D = (i=h)iX # ; p = (i=h)iX # ; (2=has )(s + (h=2)Rsj j )
= (2=h)(as s ; (ak =2)Rklsj j l s ; as s + (h=2)Rps as p
+(h=4)Rklsj Rps j l p ak ; (h=2)Rsj as j )
= (2=h)(as s ; as s + (h=2)Rps as p
;(h=2)Rps ap (Rklsj j l k =2 + s )):
Here we have used (1.4.4). Since as s ; ass = 0 along the geodesic
line, we have
D = Rpsasp ; Rpsaps :
At the origin D = 0. Let us compute the derivative of D along the
geodesic line.
(i=2)LX D = fak k ; ak k Rps as p ; Rps aps g
= iak ap Rpssk ; iak asRps kp:
For the bracket f s j g, see (1.6.1). Eq.(1.4.4) implies that Rpskp =
Rkpps. Hence, LX D = 0 and D = 0. q.e.d.
Now, to prove (2.3.4) we only need to write
Z2t
0
Z2t 2i
p(at)dt = h iX #jat dt = 2hi
0
where x = a and, hence, Sx x0 = 2a.
Z
x0 !Sx x0
#
fedosov connections on kahler symmetric manifolds
411
2.4. Computation for the projective spaces.
In this subsection using our formula we compute the trace density for
the projective spaces.
Let (Z 0 : Z 1 : : : : : Z n) be homogeneous coordinates in C P n . We
choose the point (1 : 0 : : : : : 0) as an origin and set z s = Z s=Z 0 . We
will work in some small neighborhood of the otigin. Then the simplectic
form is
! = (1=i)@@ log(1 + jzj2 )
where jz j2 = z s z s . The Ricci form R equals ;i(n +1)!, and the integral
form (2.3.1) is # = (1=i)(1 ; h(n + 1)=2)@ log(1 + jz j2 ).We denote C =
(1=i)(1 ; h(n + 1)=2). Any geodesic line passing through the origin is
zi = ai t for some constant ai and the distance from the origin to a pont
z is arctan jzj. Compute the action. Let T be a positive number, such
that the distance from 0 to Tz is twice as much as the distance from 0
to z . Then
S (z) =
Z
0!Rz 0
ZT
= C
0
C@ log(1 + jzj2 )
d log(1 + jzj2 t2) = C log(1 + jzj2 T 2):
We have arctan T jz j = 2 arctan jz j and
jz j2
:
S (z) = C log 11 +
; jz j2
For the sake of convenience, let us compute
R ;iS=h n
n
R 1n
= 2 eR n !
! 0
! =n!
1
1
+
jz j2
@
R ;iC=h log 1 ; jzj2 A dzs ^ dz s
e
(1 + jz j2 )n+1 :
n
= 2
s
R dz ^ dz s
(1 + jz j2 )n+1
This integral should be understood in the formal sence, explained
above. But the integral makes sence if we take h to be a small positive number and integrate over a small enough ball jz j = . Thus the
dmitry e. tamarkin
412
obtained values of the integral coincide with the integral understood formally up to h1 for any small enough. Let N = iC=h = 1=h ; (n +1)=2.
Then we can assume that N is a large positive number. We have:
R 1 ; R2 N R2n;1
2
2 )n+1 dR
1
+
R
(1
+
R
1
0
n
R
R1 R2n;1
!n=n! = 2
2 n+1 dR
0 (1 + R )
R2 yields
where R = jz j. Substitution t = 11 ;
+ R2
R1 N
t (1 ; t)n;1dt
1
n
R
!n=n! = 2 R1
(1 ; t)n;1 dt
;1
N + 1);(n)
= 2n ;(;(
N + n + 1)2n =n
N + 1);(n + 1) :
= ;(;(
N + n + 1)
This expression can be written as an asymptotic series in h. Furthermore, for integer N we have:
Z
tr1 = !n =n! = CNn +n = jO(N )j
which implies that our trace density coincides with the one given by the
index theorem 4].
References
1] B. V. Fedosov, A simple geometric construction of deformation quantisation, J.
Dierential Geom. 40 (1994) 213-238.
2]
, Index Theorem for deformation quantization, Preprint.
3] A. V. Karabegov, On deformation quantization on a Kahler manifold, connected
with Berezin's quantization, to appear in Funct. Anal. Appl.
4] R. Nest & B. Tsygan, Algebraic index theorem for families, Adv. Math. 113 (1995)
151-205.
fedosov connections on kahler symmetric manifolds
413
5] D. E. Tamarkin, Topological invariants of connections over symplectic manifolds,
Funct. Anal. Appl. 29 (1995) 45-56.
6] N. Kobayashi & K. Nomizu, Foundations of dierential geometry, Vols. I, II, Interscience, New York, 1963-1969.
Pennsylvania State University