Basic properties of Fedosov supermartifolds Текст научной статьи по специальности «Физика»

Geyer B.,1 e:) Lavrov PM.2 a)'b BASIC PROPERTIES OF FEDOSOV SUPERMANIFOLDS3

a) Center of Theoretical Studies, Leipzig University, Augustusplatz 10/11, D-04109 Leipzig, Germany b) Tomsk Stale Pedagogical University, 834041 Tomsk, Russia

1 Introduction

The formulation of fundamental physical theories, classical as well as quantum ones, by differential geometric methods nowadays is well established and has a great conceptual virtue. Probably, the most prominent example is the formulation of general relativity on Riemannian manifolds, i.e., the geometrization of gravitational force; no less important is the geometric formulation of gauge field theories of primary interactions on fiber bundles. Another essential route has been opened by the formulation of classical mechanics — and also classical field theories - nn symplectic manifolds and their connection with quantization. The properties of such kind of ire widely studied.

investigated in detail (see, e.g., F Especially, let

us mention that for any Fedosov manifold the scalar curvature K is trivial, K =0, and that the specific relation mtjM = 1/3RkU), in terms of normal coordinates, holds between the symplectic structure cotf. and the curvature tensor RkUj.

The discovery of supersymmetry [3] enriched modern quantum field theory with the notion of supermanifoids being studied extensively by Berezin [4]. Systematic considerations of supermanifoids and Riemannian supermanifoids were performed by De Witt [5 ]. At present, symplectic supermanifoids and the corresponding differential geometry are widely involved and studied in consideration of some

problems of modern theoretical and mathematical physics [6,7].

However, the situation concerning Fedosov

supermanifoids is quite different. Only flat even Fedosov supermanifoids have been used in the study of a coordinate-free scheme of deformation quantization [8], for an explicit realization of the extended antibrackets [9] and for the formulation of the modified triplectic quantization in general coordinates, see, [10] and references cited therein. Here, on the basis previous results [11,12], we give an overview on the present status concerning the structure of arbitrary Fedosov supermanifoids, especially the properties of their curvature tensor and scalar curvature as well as the relations between the supersymplectic structure, the connection and the curvature in normal and general local coordinates.

The paper is organised as follows. In Sect. 2, we give a brief review of the definition of ter

i1, mifolds. In Sect. 3, we ■: >

■ dons on a supermanifold ano • - .

In Sect. 4, we present the not;

. - supermanifoids and of even (ode curvature tensors. In Sect. 5, we study the of the Ricci tensor and the property the scalar curvature which is iion-trivial for odd Fedosov supermanifoids. In Sect, 6, we introduce normal coordinates on supermanifoids and affine extensions of the Christoffe! symbols as well as tensor fields. In Sect. 7, we derive the relation existing between the first order affine extension of the Christoffe! symbols and the curvature tensor for any Fedosov supermanifold both in normal coordinates and arbitrary local coordinates. In Sect. 8, we present a relation between the second order affine extension of the symplectic structure and the curvature tensor. In Sect. 9 we give a few concluding remarks.

We use the condensed notation suggested by De Witt [5]. Derivatives with respect to the coordinates x‘ are understood as acting from the left and for them the

1 E-mail: QeverQitD.uni-leiDzia.de

2 E-mail: lavrovitSDU.eciu.ru: lavrov@itp.uni-ieipzig.de

3 Contribution to Special Issue of Vestnik of Tomsk State Pedagogical University devoted to 70th Anniversary of Physical and Mathematical Department

notation 3fi4 = 3A/9x‘ is used. Right derivatives with respect to x' are labelled by the subscript “r” or the notation Au = drA/dx' is used. The Grassmann parity of any quantity /1 is denoted by e(A).

2 Tensor fields on supermanifolds

To start with, we review explicitly some of the basic definitions and simple relations of tensor analysis on supermanifolds which are useful in order to avoid elementary pitfalls in the course of the computations. Thereby, we adopt the conventions of DeWitt [5].

Let the variables x',e(x') = ei be local coordinates of a supermanifold M, dimM = N in the vicinity of a

point P , Let the sets

:= —- > and f e' := dx'} be dx' J

coordinate bases in the tangent space TpM and the cotangent space TpM , respectively. If one goes over to another set x‘ =x‘(x) of local coordinates the basis

vectors in TPM and TpM transform as follows:

d,.xJ

(1)

' J dx‘ dxJ

For the transformation matrices the following relations hold:

dxk <ff'

• = §2.

dxJ dxk

d,.x' rs,.xk

==82,

dx* dx' M'a?

(2)

: 6' ,..

ar a.r'

Introduce the Cartesian product space II"

Щ =TpX---x TpX Tp x • • • x Tp

Let T be a mapping T : П"

(3)

■ A that sends every

(4)

(5)

....'Vy........<■,.).

T„. 2""4-w*,..........«,,•«'......

Then a tensor field of type (n,m) with rank n + m is defined as a geometric object which, in each local coordinate system (x) = (x‘,...,xw), is given by a set

of functions with n upper and m lower indices obeying definite transformation rules. Here we omit the transformation rules for the components of any tensor under a change of coordinates, (x) —> (x), referring to

[11], and restrict ourselves to the case of the second order tensor only. From (1), (4) and (5) it follows

¿Fij „ rpmn ^

3x”aX;

f ,= j 3rx' drxm ,6,(6,+6„)

dxJ dx‘ ' J

j" dxJ dxm

rn j fr n 3x SrX +et>! )

' “ " ax"■‘'a?

Note that the unit matrix §’. is connected with unit tensor fields 5'; and 8,' as follows

(6)

(7)

S' = S' = (-1)6' 5.. ‘ = (-- l)tj S2

S)

From a tensor field of type (n,m) with rank n * m , where n 5* 0, m # 0, one can construct a tensor field of type (n-l,m-l) with rank «-+-»? -2 by the contraction of an upper and a lower index by the rules Th-Ui~x-i, : (9)

4-і)

..+€. + Є„ +...4Є, , )

h¡ h Sq- S

(10)

element (co‘‘,..., d", X J(,..., X . ) 4 II“m into super-number T((fl'l,...,(o’',,XJ|,...,Ii)eA where A is the Grassmann algebra. This mapping is said to be a tensor of rank (n,m) at P if for all co,oe TpM, all X,Ye TPM and all oce A it satisfies the multilinear laws

T(...co+c...) = T(...a>...) + T(...o...),

T(... X + F...) = T(... X...) + T(... F...),

T(. ..coa,a...) = T(. ..(o,ao...),

T(...caa,X...) = T(...03,aX...),

T(...Xa,F...) = T(...X,aF...),

T(...Xa) = T(...X)a.

It is useful to work with components of T relative to the bases {e'\ and led

in particular, for the tensor fields of type (1,1) the contraction leads to the supertraces,

F ,.(-l)e; and 7’1 (11)

From two tensor fields T1'"'1’ and of types

(«,()) and (0,m) one can construct new tensor fields of type using the multiplication procedure

in the following way:

НУ

ЄІPXe,, +...+sk4 +Ej К J.¡,,,

Vi-О»,-I

(12)

(13)

In particular, for the second rank tensor fields T‘J and PtJ

(-;|)£(p)(€i+e‘H€*T‘kPy or PikTtj, (14)

Furthermore, taking into account (14), the unique inverse of a (non-degenerate) second rank tensor field of type (2,0) will be defined as follows:

(-1)'

(€;+et)6(T)+et

TIK (T

%=8‘

-%T*= ÔJ

(15)

t'j =t:

dxH dx'

€i'-e.+€«) _

±t:

3br ax"

: ±(-l)£'£' fjl

Thus, the notion of generalized (anti)symmetry of a tensor field of type (2,0) is invariantly defined in any coordinate system.

Now, suppose that T,!i is n< allowing for the introduction of inverse tensor fields of type (0,2)

From tin; one gets ee.i-

3, thus

lOIlding

.0 (15)'

t(--J) • '{'I',')

tl should metry is inva>i.»i<

! ; /) general* zed

rv. =T\- +r*r(i(.(-l)6‘<e'+1),

(19)

e(i;:r1) = e(rï) = e(r)+€I+eJ.,

and correspondingly for tensor fields of type (0,2).

Let us emphasize that, the inclusion of the correct sign factors into the definitions of contractions, (9) and (10), and of the inverse tensors, (15), is essential. Namely, let us consider a second rank tensor field of type (2,0) obeying the property of generalized (anti)symmetry,

r/=±(-l)e‘e'2T/. (16)

Obviously, that property is in agreement with the transformation law (6), dxJ dx‘

,, ,• (20)

and on second-rank tensor fields of type (2,0), (0,2) and (1,1) by the rules

(21)

rv( = T\k +r/r/tt(“l)e (—-])e‘€j+e^6i +€^+1'>

^Vt=^~7;rit-7’r',*(-l)^+6'e% (22)

T'/P* = T'M +r'/fs(-1)e'6j+£,<6’+e'+n. (23)

Similarly, the action of the covariant derivative on a tensor field of any rank and type is given in terms of their tensor components, their ordinary derivatives and the connection components.

As usual, the affine connection components do not transform as mixed tensor fields, instead they obtain an additional inhomogeneous term:

r,=(-iy

6„(6,„+€j) 9,*' p; d,.xm 3.x'1 . d,.x‘ d;.v"'

3 Affine connection on stipermanifoids and curvature

In analogy to the ease of tensor analysis on manifolds, on a supermanifold M one introduces the covariant derivation (or affine connection) as a mapping V (with components V|5 e(V,.) = e,. from the set of tensor fields on M to itself by the requirement that it should be a tensor operation acting from the right and adding one more lower index and, when it is possible locally to introduce Cartesian coordinates on M , that it should reduce to the usual (right-) differentiation. For arbitrary supermanifolds the covariant derivative V (or connection F) is defined through the (right-) differentiation and the separate contraction of upper and lower indices with the connection components accompanig definite numerical factors which depend on the Grassmann parities of local coordinates. More explicitly, they are given as local operations acting on scalar, vector and co-vector fields by the rules

rvf=rjt (18)

dx1 m" dx'1 dx* dxm dxJdxk

(24)

In general, the connection components F' jk do not

have the property of (generalized) symmetry w.r.t. the

lower indices. The deviation from this symmetry is the torsion,

r;t :=r jk -(-\f’eiV kJ, (25)

which transforms as a tensor field. If the superrnaniioM M is toisionless, i.e., T2 =0, then one says that a

symmetric connection is defined on M . Here, with the ait dosov superman;folds, we consider

on nections.

i lit- iviviiiciituia n tensor field R: , according to

Ref. [5], is defined in a coordinate basis by the action of the commutator of covariant derivatives, [V,,V,] = V.V( - (-if'?,?., on a vector field T‘ as

L ! j J 1 J v J 1

follows:

nv, Vt ] = ~K~l)£"(e'+l)rn,/?i„#. (26)

A straightforward calculation yields

Rimjk =~rLj.t +r;,«,j(-i)€,e* + +r;ri(-i)^^ri,trii(-i)£,i£”+v.

(27)

pi rnk V S kn mj'

The Riemannian tensor field possesses the following generalized antisymmetry property,

«U=-H)6/t*W (28)

furthermore, it obeys the (super) Jacobi identity, i-De*c* R‘nik + H)v* R'Pn + (-if'J = 0 (29)

and the (super) Bianchi identity,

H)*e' R'\njk, + (-D** * V + (-D£t6i R^js o> (30)

with the notation := RnmJlV).

4 Fedosov supermanifolds

Suppose now we are gi ven an even (odd) symplectic supermanifold, (M ,co) with an even (odd)

symplectic structure o>, e(to) = 0 (or 1). Let V (or F) be a covariant derivative (connection) on M which preserves the 2-form to, coV = 0. In a coordinate basis this requirement reads

H)€‘e' =0. (31)

If, in addition, F is symmetric then we have an even (odd) symplectic connection (or symplectic covariant derivative) on M. Now, a Fedosov supermanifold (M, a>,T) is defined as a symplectic supermanifold with a given symplectic connection.

Let us introduce the curvature tensor of a

symplectic connection with all indices lowered,

Rijki = ®Jn^'jki> e(/iya ) = 6(G)) + €,. + 6 j +ek+e,, (32)

where RnjU is given by (27). This leads to the

following representation,

Rmjk =-~il\Xnmi.k +

mj,k in mk,j ^

(33)

+rimr",Bi(-i)E/e“-rfcrv^)€i(e”+£/>>

where we used the notation

! ¡¡k = ®„,I" lt i €(f ¡.k) -- 6(d)) + €; + + €j, (34)

Using this, the relation (31) reads

(Ci.; j =r;, -r^., (.l)e,t' . Furthermore, from Eq. (33) it

is obvious that

**,=-(-1)^*,,*. (35)

and, using (32) and (29), one deduces the (super) Jacobi identity for RiJU,

Ryu + H)e,6i Rm + Rm = 0. (36)

In addition, the curvature tensor R.

■¡¡и

is

(generalized) symmetric w.r.t, the first two indices (see [11]),

RljU=(-if^Rjm. (37)

For any even (odd) symplectic connection there holds the identity

e,€*+e,€, p

' KIiJk +

«ш+Н)^^=о.

(38)

"H"1) ' "uy

This is proved by using the Jacobi identity (36) together with a cyclic change of the indices. In the identity (38) the components of the symplectic curvature tensor occur with cyclic permutations of all the indices (on R). However, the pre-factors

depending on the Grassmann parities of indices are not obtained by cyclic permutation.

5 Ricci and scalar curvature tensors

Having the curvature tensor, Rijld, and the tensor field of, which is inverse to co,7,

w»m^(_l)€I+€№X€,«t) =gif

Н)6'+е(м,™ш^=8/,

af ='_(_i

(39)

(40)

G(eo'/)=e(Cû) + 6i + €j.

one can define the following three different tensor fields of type (0,2),

R, = atnRMj(-l)ime"e') = R%(~ ir, (41)

K, = = R%(-l)e‘«+t), (42)

6,- +€;)i.6i -r€i( )+(e(0>)+l)(S4, +e;l )

Qu — ioRjjrik (-1)

(43)

€(Rf) = e(Kÿ) = e(QIj) = €,+€,.

From the definitions (41), (43) and the symmetry properties of Rljkl, it follows immediately that for any

symplectic connection one has Ry = -(-l)t,fcj RJt and Qij ~ -(-I)6'6' Qfi. Moreover we obtain the relations [1 + (-1)E№]^ = 0, [1 - (~lfw }QiJ = 0. (44)

From (38) and (4l)-(43) it follows the relations Rv + Qij + H)€;€j Kÿ + (-l)«“' Ky = 0, (45)

[l + (-l)e(“,](^-(-l)e,ejA:jï) = 0. (46)

Therefore for any even symplectic connection we obtain

Kh ==(-])*'*' KJit Ry=0, (47)

while for any odd symplectic connection we have a(=0, ^=^-(-1)^^. (48)

The tensor field K!: should be considered as the only independent second-rank tensor which can be constructed from the symplectic curvature. We refer to Kfj as the Ricci tensor of an even (odd) Fedosov

supermanifold. Notice that in the odd case Kif has no

special symmetry property.

Let us define the scalar curvature K of a Fedosov supermanifold by the formula

K = (üilKÿ(-lf*ei

= OyV'!J?

(49)

From the symmetry properties of Ri]U and of, it follows that on any Fedosov supermanifold one has [l + (-l)e(<!>)]£' = 0. (50)

Therefore, as is the case for ordinary Fedosov manifolds [2], for any even symplectic connection the scalar curvature necessarily vanishes. But the situation becomes different for odd Fedosov supermanifold where no restriction on the scalar curvature occurs. Therefore, in contrast to both the usual Fedosov manifolds and the even Fedosov supermanifolds, any odd Fedosov supermanifolds can be characterized by the scalar curvature as an additional geometrical structure [11]. This basic property of the scalar

curvature can be used to formulate the following statement [13]: In both, the even and odd cases there exists the relation K2 = 0, and therefore any regular function of the scalar curvature on any Fedosov supermanifolds belongs to class of linear functions <t>M = f[K{x)] = a + |3A%x).

«• Affine extensions of Christoffel symbols and tensors on symplectic supermanifolds

In Ref. [2] the virtues of using normal coordinates for studying the properties of Fedosov manifolds were demonstrated. Here, following Ref. [12], we are going to extend this method on Fedosov supermanifolds (M,m,F) [12]., Normal coordinates j/} within a point pe M can be introduced by using the geodesic equations as those local coordinates which satisfy the relations ( p corresponds to y ~ 0)

I ‘jk (>')>’* yJ = 0, 6(1 iJk) = €((*)) + 6,- + 6;. + . (51)

It follows from (51) and the symmetry properties of Tijk w.r.t. (j.k) that

rtlk( 0) = 0. {52)

In normal coordinates there exist additional relations at p coniaining the partial derivative,; of

F,,t. Namely, consider the Taylor expansion of Fia( y)

at v — 0 ,

vhere

>M-j,

= Am...jSP') =

a:r¥S

àyJ'... dyJ

Jh-k =rpi,"-i*

il"-/,, I»,..J»,,.

dT^J‘ (0) = —

dy}'... dyJ"

(55)

where are normal coordinates associated

with (x\...,x2*) at p. The first extension of any tensor coincides with its covariant derivative because i"'jk (0) = 0 in normal coordinates.

In the following, any relation containing affine extensions are to be understood as holding in a neighborhood II of an arbitrary point pe M . Let us also observe the convention that, if a relation holds for arbitrary local coordinates, the arguments of the related quantities will be suppressed.

7 First order affine extension of Christoffel symbols and curvature tensor of Fedosov supermanifolds

For a given Fedosov supermanifold (M,co,F), the symmetric connection F respects the symplectic structure co [10]:

=ra- -F,»(-!)**'. (56)

Therefore, among the affine extensions of co5 and TiJk there must exist some relations. Introducing the affine extensions of ft);; in the normal coordinates

(y\...,y2'') at pe M according to,

(0).

Using the symmetry properties of îo,.-(i f (0) one easily obtains the Taylor expansion for tty , :

oy;,(}-)

(58)

is called an affine extension of F^ of order n = 1,2.... The symmetry properties of A»,.../, are evident from their definition (53), namely, they are (generalized) symmetric w.r.t. (j,k) as well as O'; . The set of

all affine, extensions of Fijk uniquely defines a

symmetric connection according to (53) and satisfy an infinite sequence of identities [12]. In the lowest nontrivial order they have the form

4;« + Ajik H)** + 4ffW)e,(6'+6‘> = 0- (54)

Analogously, the affine extensions of an arbitrary tensor T = (T'‘"Jtmi_mi) on M are defined as tensors on M whose components at pe M in the local coordinates (x1,..., x2h) are given by the formula

Taking into account (56) and comparing (53) and (58) we obtain

£1

(-D

6j€; .

(59)

ly-0

in particular,

= (60)

Now, consider the curvature tensor RijM in the

normal coordinates at pe M. Then, due to

rijk(p)~0, we obtain the following representation of

the curvature tensor in terms of the affine extensions of the symplectic connection

Rsum=-Ava+Aim(-ir\ (6i)

Taking into account (54) and (61) a relation containing the curvature tensor and the first affine extension of F can be derived. Indeed, the desired relation obtains as follows

4m s r#r (0) = (0) + Rw 1, (62)

where, the antisymmetry (35) of the curvature tensor were used.

Notice, that relation (62) was derived in normal coordinates. It seems to be of general interest to find its analog relation in terms arbitrary local coordinates (x) because the Christoffel symbols are not tensors while the r.fa.s. of (62) is a tensor. Under that change of coordinates (x) -4 (y) in some vicinity U of p the

Christoffel symbols transform according to the rule

r,*(30 =

/ ^ d2rXP

+C0«wW

r fr'dr*'d,** r ,,6,(€,4-6,)

dy* 3/

(63)

cly'

In its turn the matrix of second derivatives can be expressed in the form

. = ^.r,jt(y)_nfciW^^(_i)^(^«-)

dyW dvl

dyk dy1

(64)

In particular at pe M (y ~ 0) we have the relation

dy %*

= -F«

dyk

/o \

dy

X I

(65)

Differentiating (63) with respect to v we find ^ , T, , a xs drxr drxq d,.xp ,

r„.(v) = I

x(_r

(€; ■?-£{. •? £; )<,£,. -t€;i rH€j. )(€, +6^. )+€; (€-. ■*-£,.)

. . drxr 'd*xv d xp

+iOwAA.

x(—iy€* J’Cj J< €' *€p i+e' ('e> *€/+e> +£,J +€f'

32 q :)2rP

n dy>dyk dy'dy1 +£0 ____d,XP ( n(e^€t+e,)(er)-ef)

dytyBy1 3/ ( 1}

where the covariant derivative (for arbitrary local coordinates) is defined by _

x pqn rs pur qsv •'

r =r

pgr; .v pqr.

-F F" (-])'

(€,+€, ){6,+e,,)

(66)

Restricting to the point pe M we get r ,kJ m - (rst;/ (X0) -Fib (x0)rjk (/)(-lf) +

«*,<*>)

3 y

ydy^dy^dy' /0

(67)

1 f.

3)%*3/

¿[(r^-r^x-i)8'8' (68)

■Kri(,,t-ri*1rJt)(-i)e'e*

+^>«0 ~ rjfc,r"|j.)(-l)€'€j ](x0)(-l)Ej,S!.

With the help of (68) we get the following

transformation law for F coordinates at the point p

r*,(0) =

with the abbreviation

7 =r,w-fr,;i(-l)^+ritf;i(-lf‘+€'J6

¡jk.j under change of

(69)

+2rilnr'/t(~l

' iß',k x {€*+€;)€/

__r p -F F'

1 ¡to1 j/V ^ 1 im1 id *

(70)

In straightforward manner one can check that the relations (67) reproduce the correct transformation law for the curvature tensor RijU. Therefore, relation (62) is to be generalized as

1 „ 1

¡jkj

(71)

The last equation gets an identity" when using the definition (70) of Zw(x) and relation for R^ix) on the r.h.s.

8 Second and third order affine extension of symplectic structureand curvature tensor on Fedosov supermanifolds

Now, let us consider the relation between the second order affine extension of symplectic structure and the symplectic curvature tensor. It is easily found by taking into account (60) and (62). Indeed, using the Jacobi identity (36), we obtain

®^(0)=A,« - A„«(-1)^ =r««„(0)(-if'+6'X6,+e'),

\lkl ‘ lfiki

Again, since pe M is arbitrary, we finally obtain its generalization for any local coordinates x:

Due to (67) and the identity (54), the matrix of third derivatives at p obeys the following relation,

. (72)

Furthermore, using the second Bianchi identity [11] one gets a relation between the first derivative of the curvature tensor and the affine connections,

V« = -V + 'W(-1>**- (73)

as well as the third affine extension of the symplectic structure 1 ,

co„

+ iW/ H)

€,6,+€*(€,+€„,) p / jx«, (€*+€/)

In local coordinates (jc) the following identity can be

proven:

Km (" 1 )€j <6‘+£*1 - Rmijhk H)’

+^/;i(-i)€‘(6i+ei+e,)-W-1)“'“"'"*' =o.

e* (€,+£,)

<e.+£*> ..

^Sjie,4-ej;-r€t5

For the derivation of these relations, see. Ref. [12].

9 Summary

We have considered some properties of tensor fields defined on supermanifolds M . It was shown that only the generalized (anti)symmetry of tensor fields has an invariant meaning, and that differential geometry on supermanifokls should be constructed in terms of such tensor fields.

Any supermanifold M can be equipped with a symmetric connection F (covariant derivative V). The Riemannian tensor R',u corresponding to this

symmetric connection F satisfies !

Jacobi . ’ 1 '”•! : . 1 I

'super)

■'sctic

ci ■ ■ i is cat. ■ i. The

trij..-_ ... .*,1 • - jdosov

supermanifoki. The curvature tensor R:M of a

symplectic connection obeys the property of

generalized symmetry with respect to the first two indices, and the property of generalized antisymmetry with respect to the last two indices. The tensor Rijkl

satisfies the Jacobi identity and the specific (for the symplectic geometry) identity (see (38)) containing the sum of components of this tensor with a cyclic permutation of ail the indices, which, however, does not (!) contain cyclic permuted factors depending on the Grassmann parities of the indices.

On any even (odd) Fedosov manifold, the Ricci tensor KtJ can be defined. In the even case, the Ricci

tensor obeys the property of generalized symmetry and gives a trivial result for the scalar curvature. On the contrary, in the odd case the scalar curvature, in general, is nontrivial

Using normal coordinates on a supermanifold equipped with a symmetric connection we have found relations among the first order affine extensions of the Christoffel symbols and the curvature tensor, the second order affine extension of symplectic structure and the curvature tensor. In similar way it is possible to find relations containing higher order affine extensions of sypmlectic structure, the Christoffel symbols and the curvature tensor. We have established the form of the obtained relations in any local coordinates (see (71). (72)). It was shown that F(>i;;U) ~T/3Z((W(jt) is a tensor field in terms of which the s obtained for

general loea! coordinates can be f i, ci. Eq. (69).

Acknowledgements:

PI..,, thanks Leipzig University, Graduate College Quantum Field Theory, for kind hospitality. The work was supported under grant DPG 436 RUS 17/15/04.

References

t. Fedosov B.V. // J. Ditf. Geom. 1994. V. 40. P. 213; Deformation quantization and index theory. Akademie Verlag, Berlin, 1996.

2. Gelfand I., Retakh V., Shubin M, II Advan. Math. 1998. V. 136. P. 104, dg-ga/9707024.

3. Golfand Yu.A., Likhtman E.P. IIJETP Lett. 1971 .V. 13. P. 323; Wess J„ Zumino B. II Phys. Lett. 1974. V. B49. P. 52.

4. Berezin F.A. II Yad. Fiz. 1979. V, 29. P. 1670; ibid. 1979. V. 30 P. 1168; Introduction to superanalysis. Reidel, Dordrecht, 1987.

5. DeWitt B. Supermanifolds. Cambridge University Press, Cambridge, 1984.

6. Batalin I.A., Vilkovisky G.A. II Phys, Lett. 1981. V. B102. P. 27; Phys. Rev. 1983. V. D28. P, 2567.

7. Witten E. II Mod. Phys. Lett. 1990. V. A5. P. 487; Khudaverdlan O.M. II J. Math. Phys. 1991. V. 32. P. 1934; Khudaverdian O.M., Nersessian

A.P. II Mod. Phys. Lett. 1993. V. Ä8. P. 2377; J. Math. Phys. 1996. V. 37. P. 3713; Schwarz A. II Commun. Math. Phys. 1993. V. 155. P. 249;

ibid. 1993. V. 158, P. 373; Batalin I,A., Tyutin I.V. II Int. J. Mod. Phys, 1993. V. A8. P. 2333; Mod. Phys, Lett. 1993. V. A8. P. 3673; ibid, 1994.

V. A9. P, 1707; Hata H., Zwiebach B. II Ann. Phys. (N.Y.). 1994. V. 322. P. 131; Alfaro J., Damgaard P.H. II Phys, Lett. 1994. V. B334. P. 369;

Khudaverdian O.M. II Commun. Math. Phys, 1998. V, 198, P, 591; Laplacians in Odd Symplectic Geometry. math.DG/0212354.

8. Batalin I.A., Tyutin I.V. II Nucl, Phys. 1990.1990. V, B345. P. 645.

9. Grigoriev M,A„ Semikhatov A.M. II Theor. Math. Phys, 2000. V. 124. P. 1157,

to, Geyer B„ Lavrov P.M. II Ini. J. Mod. Phys. 2004. V. AW. P. 1S39.

11. Geyer B„ Lavrov P.M. Fedosov supermanilolds: Basic properties and the difference in even and odd cases II hep-th/0306218; to appear: Int. J. Mod. Phys. 2004, V. A19. P. 3195.

12. Geyer B., Lavrov P.M. Fedosov supermanifolds: II. Normal coordinates II hep-th/0406206.

13. Radchenko O.V. II Russ. Phys. J. (in press).