A representation formula for maps on supermanifolds

A representation formula for maps on supermanifolds

Frédéric Hélein^a兲

Institut de Mathématiques de Jussieu, UMR 7586, Université Denis Diderot-Paris 7, Case 7012, 2 place Jussieu, 75251 Paris Cedex 5, France

共Received 17 September 2007; accepted 11 January 2008;

published online 12 February 2008

兲

We analyze the notion of morphisms of rings of superfunctions which is the basic concept underlying the definition of supermanifolds as ringed spaces

共i.e., follow-

ing Berezin, Leites, Manin, etc.兲. We establish a representation formula for all

共pull-back兲

morphisms from the algebra of functions on an ordinary manifolds to the superalgebra of functions on an open subset of a superspace. We then derive two consequences of this result. The first one is that we can integrate the data associated with a morphism in order to get a

共

nonunique

兲

map defined on an ordinary space

共

and uniqueness can be achieved by restriction to a scheme

兲

. The second one is a simple and intuitive recipe to compute pull-back images of a function on a manifoldMby a map from a superspace toM. ©2008 American Institute of Physics.

关

DOI:10.1063/1.2840464

兴

I. INTRODUCTION

The theory of supermanifolds, first proposed by Salam and Strathdee¹⁶ as a geometrical framework for understanding the supersymmetry, is now well understood mathematically and can be formulated in roughly two different ways: either by defining a notion of superdifferential structure with “supernumbers” which generalizes the differential structure of R^p and by gluing together these local models to build a supermanifold. This is the approach proposed by Dewitt⁵ and Rogers.^14,10 Alternatively, one can define supermanifolds as ringed spaces, i.e., objects on which the algebra

共

or the sheaf

兲

of functions is actually a superalgebra

共

or a sheaf of superalge- bras

兲

. This point of view was adopted by Berezin,⁴Letes,¹¹and Manin¹²and was recently further developed¹³ by Deligne and Morgan,⁷Freed,⁹and Varadarajan.¹⁷The first approach is influenced by differential geometry, whereas the second one is inspired by algebraic geometry. Of course, all these points of view are strongly related, but they may lead to some subtle differences

共see

Batchelor,³ Bartocci et al.,² and Bahraini¹

兲. For a synthetic overview and a comparison of the

various existing theories, see Ref.15.

The starting point of this paper was to understand some implications of the theory of super- manifolds according to the second point of view,4,11,12,8,9,17

i.e., the one inspired by algebraic geometry. The basic question is to understand R^p兩q, the space with p ordinary

共bosonic兲

coordi- nates andqodd

共fermionic兲

coordinates. There is no direct definition nor picture of such a space beside the fact that the algebra of functions onR^p兩qshould be isomorphic to C^⬁

共R

^p

兲关

␩^{1, . . . ,}␩^q

兴,

i.e., the algebra overC^⬁

共R

^p

兲

spanned byqgenerators␩^{1, . . . ,}␩^qwhich satisfy the anticommutation relations␩ⁱ␩^j+␩^j␩ⁱ= 0. Hence,C^⬁

共R

^p

兲关

␩^{1, . . . ,}␩^q

兴

is isomorphic to the set of sections of the flat vector bundle overR^pwhose fiber is the exterior algebra

⌳

^*R^q. To experiment further R^p兩q, we define what should be maps from open subsets of R^p兩q to ordinary manifolds. We adopt the provisional definition of an open subset of

⍀

of R^p兩q to be a space on which the algebra of functions is isomorphic toC^⬁

共兩⍀兩兲关

␩^{1, . . . ,}␩^q

兴, where 兩⍀兩

is an open subset ofR^p. So let us choose such an open set

⍀

and a smooth ordinary manifoldNand analyze what should be maps␾^from

a兲Electronic mail: helein@math.jussieu.fr.

49, 023506-1

0022-2488/2008/49共2兲/023506/19/$23.00 © 2008 American Institute of Physics

⍀

toN. Again, there is no direct definition of such an object except that by the chain rule it should define a ring morphism␾^*from the ring C^⬁

共N兲

of smooth functions on N to the ring C^⬁

共兩⍀兩兲

⫻关

␩^{1, . . . ,}␩^q

兴. The morphism property means that

␾^*

共␭f

+␮g兲=

␭

␾^*f+␮␾^*g, ∀

␭,

␮

苸

R, ∀f,g

苸

C^⬁

共N兲 共1兲

and

␾^*

共

fg兲=

共

␾^*f兲共␾^*g兲, ∀f,g

苸

C^⬁

共N兲. 共2兲

We restrict ourself toevenmorphisms, which means here that we impose to␾^*fto be in the even partC^⬁

共兩⍀兩兲关

␩^{1, . . . ,}␩^q

兴

0ofC^⬁

共兩⍀兩兲关

␩^{1, . . . ,}␩^q

兴.

In the Sec. II, we prove our main result

共Theorem 1.1兲

which shows that, for any even morphism␾^*, there exists a smooth map␸^from

兩⍀兩

toNand a family of vector fields

共⌶

x

兲

x苸兩⍀兩

depending on x苸

兩⍀兩

and tangent to N and with coefficients in the commutative subalgebra R关␩^{1, . . . ,}␩^q

兴

0such that

␾^*f=

共1 ⫻

␸

兲

^*

共e

^⌶f兲, ∀f

苸

C^⬁

共N兲. 共3兲

One may interpret the term e^⌶ as an analog with odd variables of the standard Taylor series representation,

g共x兲=

兺

k=0

⬁ ⳵^kg

共

⳵^x

兲

^k

共x

0

兲 共x

−x₀

兲

^k

k! =

共e

^{兺i=1n 共xi−x0i兲共⳵/⳵xi兲}g兲共x0

兲

for a functiongwhich is analytic in a neighbourhood ofx₀. We also show that the vector field

⌶ 共which is not unique兲

can be build as a combination of commuting vector fieds. Then, the rest of this paper is devoted to the consequences of this result.

The second section explores in details the structure behind relation

共3兲. First, we exploit the

fact that one can assume that the vector fields which compose

⌶

commute, so that one can integrate them locally. This gives us an alternative description of morphisms. Eventually, this study leads us to a factorization result for all even morphisms as follows. First, let us denote by

⌳

₊^2*R^q the subspace of all even elements of the exterior algebra

⌳

^*R^q of positive degree

共

i.e.,

⌳

₊^2*R^q

⯝

R^2q−1−1

兲

. We construct an idealI^q

共兩⍀兩兲

of the algebraC^⬁

共兩⍀兩 ⫻⌳

₊^2*R^q

兲

in such a way that if we consider the quotient algebraA^q

共兩⍀兩兲

ªC^⬁

共兩⍀兩 ⫻⌳

₊^2*R^q

兲

/I^q

共兩⍀兩兲, then there exists a canoni-

cal isomorphism T_⍀^*:A^q

共兩⍀兩兲

→C^⬁

共⍀兲

. By following the theory of scheme of Grothendieck, we associate to A^q

共兩⍀兩兲

its spectrum SpecA^q

共兩⍀兩兲

, a kind of geometric object embedded in

兩⍀兩

⫻⌳

₊^2*R^q. Then, for any even morphism ␾^{* from} C^⬁

共N兲

to C^⬁

共兩⍀兩兲关

␩^{1, . . . ,}␩^q

兴, there exists a

smooth map

⌽

from

兩⍀兩 ⫻⌳

₊^2*R^q toN, such that

␾^*=T_⍀^*ⴰ

⌽

_兩쐓^*,

where ∀f

苸

C^⬁

共N兲, ⌽

_兩쐓^*f=fⴰ⌽modI^q

共兩⍀兩兲. So by dualizing we can think of the map

⌽

_兩쐓: SpecA^q

共兩⍀兩兲→N

as the restriction of

⌽

to SpecA^q

共兩⍀兩兲. Hence, we obtain an interpretation

of a map on a supermanifold as a function defined on an

共almost兲

ordinary space. This reminds somehow the theory developped by Vladimirov and Volovich¹⁸who represent a map on a super- space as a function depending on many auxiliary ordinary variables satisfying a system of so- called “Cauchy–Riemann-type equations.” However, their description in terms of ordinary func- tions satisfying first order equations differs from our point of view.

The last section is devoted to applications of our results for understanding the use of super- manifolds by physicists. First, we explain briefly how one can reduced the study of maps between two supermanifolds to the study of maps from a supermanifold to an ordinary one by using charts.

Second, we recall why it is necessary to incorporate the notion of the functor of point

共

as illus- trated in this framework in Refs.7,9, and19

兲

in the definition of a map␾between supermanifolds

in terms of ring morphisms. Then, we address the simple question of computing the pull-back image␾^{*f of a map f} on an ordinary manifoldNby a map␾from an open subset ofR^p兩qtoN. For instance, consider a superfield ␾⁼␸⁺␪¹␺1+␪²␺2+␪¹␪^{2F from} R^3兩2

关

with coordinates

共

x¹,x²,t,␪^1,␪²

兲兴

toRand look at the Berezin integral,

Iª

冕

_R^{3兩2d3xd2␪␾*f,}

where f:R→R is a smooth function. Such a quantity arises for instance in the action

兰

_R³兩2d³xd²␪

共

₄¹_⑀^abDa␾^Db␾⁺␾^*f

兲

and then fplays the role of a superpotential. Following Berezin’s rules, the integralIis equal to the integral overR³of the coefficient of␪¹␪²in the development of

␾^*f, which is actually

␾^*f=fⴰ␸⁺␪¹

共f

⬘^ⴰ␸

兲

␺1+␪²

共f

⬘^ⴰ␸

兲

␺2+␪¹␪²

关共

f⬘^ⴰ␸

兲F

−

共f

⬙^ⴰ␸

兲

␺1␺2

兴, 共4兲

so thatI=

兰

_R³d³x

关共

f⬘ⴰ␸

兲

F−

共

f⬙ⴰ␸

兲

␺1␺2

兴

. The development

共

4

兲

is well known and can be obtained by several approaches. For instance in Ref. 6 or 9, one computes the coefficient of ␪¹␪² in the development of␾^*f by the rule␫^*

共

−¹₂

共D

1D₂−D₂D₁

兲

␾^*f

兲

, whereD₁ andD₂ are derivatives with respect to␪^1and␪², respectively, and␫is the canonical embeddingR³R^3兩2. Here, we propose a recipe which, I find, is simple, intuitive, but mathematically safe for performing this computation

共this recipe is of course equivalent to the already existing rules!兲. It consists roughly in the

following: we reinterpret the relation␾⁼␸⁺␪¹␺1+␪²␺2+␪¹␪^{2F as}

␾^*=␸^{*e␪1␺1+␪2␺2+␪1␪2F=}␸^*

共1 +

␪¹␺1

兲共1 +

␪²␺2

兲共1 +

␪¹␪²F兲,

共5兲

where

• ␺1,␺2, and Fare first order differential operators which acts on the right, i.e., for instance,

∀f

苸

C^⬁

共R兲,

␺af=df共␺a

兲

=f⬘␺a, and so␸^*␺af=

共f

⬘^ⴰ␸

兲

␺a;

• ␺1,␺2, andF areZ2-graded in such a way that␾^*is even, i.e., since␪^{1 and}␪^{2are odd,} ␺1

and␺2are odd andF is even; and

• all the symbols␪^1,␪^2,␺1,␺2, andF supercommute.

Let us use the supercommutation rules to develop

共5兲, we obtain

␾^*f=␸^*f+␪¹␸^*␺1f+␪²␸^*␺2f+␪¹␪²␸^*Ff−␪¹␪²␸^*␺1␺2f, ∀f

苸

C^⬁

共R兲.

Then, we let the first order differential operators act and this gives us exactly

共4兲.

All these rules are expounded in details in the Sec. IV of this paper. Their justification is precisely based on the results of Sec. II.

II. EVEN MAPS FROMR^p円qTO A MANIFOLDN

Our first task will be to study even morphisms ␾^{* from} C^⬁

共N兲

to C^⬁

共兩⍀兩兲关

␩^{1, . . . ,}␩^q

兴, i.e.,

maps between these two superalgebras which satisfy

共1兲

and

共2兲. Let us first precise the sense of

even. If A=A₀丣A₁ and B=B₀丣B₁ are two Z2-graded rings with unity, a ring morphism ␾^:B

→Ais said to beevenif it respects the grading, i.e.,∀b

苸

B_␣,␾

共b兲 苸

A_␣ for␣= 0 , 1. In the case at hand,B=C^⬁

共N兲

is purely even, i.e.,B₁=

兵0其, and so

␾^*is even if and only if it mapsC^⬁

共N兲

to C^⬁

共兩⍀兩兲关

␩^{1, . . . ,}␩^q

^兴

0, the even part ofC^⬁

共兩⍀兩兲关

␩^{1, . . . ,}␩^q

兴. We then say that

␾^{is anevenmap from}

⍀

toN. In the following, we shall denote the rings C^⬁

共兩⍀兩兲关

␩^{1, . . . ,}␩^q

兴

andC^⬁

共兩⍀兩兲关

␩^{1, . . . ,}␩^q

兴

0

by, respectively,C^⬁

共⍀兲

andC^⬁

共⍀兲

0 and we shall denote by Mor

共C

^⬁

共N兲

,C^⬁

共⍀兲

0

兲

the set of even morphisms fromC^⬁

共N兲

toC^⬁

共⍀兲

.

We observe that because of hypothesis

共1兲, any such morphism is given by a finite family

共a

i₁. . .i_2k

兲

of linear functionals on C^⬁

共N兲

with values inC^⬁

共兩⍀兩兲, where 共i

1¯i_2k

兲苸 冀1 ,q冁

^2kand 0

艋

k

艋 关

q/2

兴 共关

q/2

兴

is the integer part ofq/2

兲

, by the relation

␾^*f=

兺

k=0 关q/2兴

1艋i₁⬍¯⬍

兺

i_k艋q

ai₁¯i_2k

共f兲

␩ⁱ¹¯␩^i2k=a_쏗

共f兲

+

兺

1艋i₁⬍i₂艋q

ai₁i₂

共f兲

␩ⁱ¹␩ⁱ²⁺ ¯.

Here, we will assume that theai₁¯i_2k’s are skew symmetric in

共i

1¯i_2k

兲. At this point, it is useful

to introduce the following notations: For any positive integer k, we let I^q

共

k

兲ª 兵共

i₁¯i_k

兲 苸 冀1 ,

q冁^k

兩

i₁

⬍

¯

⬍

ik

其, we denote by

I=

共i

1¯ik

兲

an element of I^q

共k兲

and we then write ␩^I ª␩^{i1. . .}␩^ik. It will be also useful to use the convention I^q

共0兲

=

兵쏗其. We let

I^qª

艛

_k=0^q I^q

共k兲,

I₀^q ª

艛

_k=0^关q/2兴I^q

共

2k

兲

,I₁^qª

艛

_k=0^{关共q−1兲/2兴}I^q

共

2k+ 1

兲

, andI₂^qª

艛

_k=1^关q/2兴I^q

共

2k

兲

. Hence, the preceding relation can be written as

␾^*f=

兺

k=0 关q/2兴

I苸I

兺

^q共2k兲

aI

共f 兲

␩^I=

兺

I苸I0 q

aI

共f兲

␩^I

共6兲

or

共

␾^*f

兲共x兲

=

兺

I苸I0 q

aI

共f 兲共x兲

␩^I, ∀x

苸 兩⍀兩.

A. Construction of morphisms

We start by providing a construction of morphisms satisfying

共

1

兲

and

共

2

兲

. We note ␲^:

兩⍀兩

⫻

N→Nthe canonical projection map and consider the vector bundle␲^*TN: the fiber over each point

共x,

q兲

苸 兩⍀兩 ⫻

Nis the tangent spaceTqN. For anyI苸I₂^q, we choose a smooth section ␰Iof

␲^*TNover

兩⍀兩 ⫻

N and we consider theR关␩^{1, . . . ,}␩^q

兴

0-valued vector field,

⌶

ª

兺

I苸I2q

␰I␩^I.

Alternatively,

⌶

can be seen as a smooth family

共⌶

x

兲

x苸兩⍀兩 of smooth tangent vector fields onN with coefficients inR关␩^{1, . . . ,}␩^q

兴

0. So each

⌶

xdefines a first order differential operator which acts on the algebra C^⬁

共N兲

丢R关␩^{1, . . . ,}␩^q

兴

0, i.e., the set of smooth functions on N with values in R关␩^{1, . . . ,}␩^q

兴

0, by the relation

⌶

xf=

兺

I苸I₂^q

共共

␰I

兲

x·f

兲

␩^I, ∀f

苸

C^⬁

共N兲

丢R关␩^{1, . . . ,}␩^q

兴

0.

Here, we do not need to worry about the position of ␩^I since it is an even monomial. We now define

共letting ⌶

⁰= 1兲

e^⌶ª

兺

n=0

⬁

⌶

ⁿ n! =

兺

n=0 关q/2兴

⌶

ⁿ n!,

which can be considered again as a smooth family parametrized byx

苸 兩⍀兩

of differential operators of order at most

关q

/2兴acting on C^⬁

共N兲

丢R关␩^{1, . . . ,}␩^q

兴

0. Now, we choose a smooth map ␸^:

兩⍀兩

→Nand we consider the map

1

⫻

␸:兩⍀兩→

兩⍀兩 ⫻

N x

哫 共x,

␸

共x兲兲,

which parametrizes the graph of ␸. Lastly, we construct the following linear operator on C^⬁

共N兲 傺

C^⬁

共N兲

丢R关␩^{1, . . . ,}␩^q

兴

0:

C^⬁

共N兲 苹

f

哫 共1 ⫻

␸

兲

^*

共e

^⌶f兲

苸

C^⬁

共⍀兲,

where

共1 ⫻

␸

兲

^*

共e

^⌶f兲共x兲ª

共e

^⌶xf

兲共

␸

共x兲兲

=

兺

n=0 关q/2兴

冉 ^共⌶兲

^n!xnf

冊 ^共

^␸

^共x兲兲,

^∀x

^{苸 兩⍀兩.}

We observe that actually, for anyx苸

兩⍀兩, we only need to define ⌶

xon a neighborhood of␸

共x兲

in N, i.e., it suffices to define the section

⌶

on a neighborhood of the graph of␸ⁱⁿ

兩⍀兩 ⫻

N

关

or even on their Taylor expansion inq at order

关

q/2

兴

around␸

共

x

兲兴

.

Lemma 1.1: The map f

哫 共

1

⫻

␸

兲

^*

共

e^⌶f

兲

is a morphism fromC^⬁

共N兲

toC^⬁

共⍀兲

0, i.e., satisfies assumptions(1)and (2).

Proof:Property

共

1

兲

is obvious, so we just need to prove

共

2

兲

. We first remark that for anyx

苸 兩⍀兩

,

⌶

xsatisfies the Leibniz rule,

⌶

x

共fg兲

=

共⌶

xf兲g+f共⌶xg兲, ∀f,g

苸

C^⬁

共N兲

丢R关␩^{1, . . . ,}␩^q

兴

0, which immediately implies by recursion that

⌶

x

n

共fg兲

=

兺

j=1

n n!

共

n−j

兲

!j!

共⌶

x n−jf

兲共⌶

x

jg兲, ∀f,g

苸

C^⬁

共N兲

丢R关␩^{1, . . . ,}␩^q

兴

0, ∀n

苸

N.

共7兲

We deduce easily that

e^⌶x

共fg兲

=

共e

^⌶xf兲共e^⌶xg兲, ∀x

苸 兩⍀兩,

∀f,g

苸

C^⬁

共N兲

丢R关␩^{1, . . . ,}␩^q

兴

0,

共8兲

by developping both sides and using

共7兲. Now, relation 共8兲

is true, in particular, for functions f,g

苸

C^⬁

共N兲

and if we evaluate this identity at the point␸

共x兲 苸

Nwe immediately conclude that

f

哫共1 ⫻

␸

兲

^*

共e

^⌶f兲satisfies

共2兲.

䊏

The following result says that actually all morphisms are of the previous type.

Theorem 1.1: Let ␾^*:C^⬁

共N兲

→C^⬁

共⍀兲

0 be a morphism. Then there exists a smooth map

␸:兩⍀兩→N and a smooth family

共

␰I

兲

I苸I₂^q of sections of␲^*TN defined on a neighborhood of the graph of␸ ⁱⁿ

兩⍀兩 ⫻

N,such that if

⌶ª 兺

I苸I₂^q␰I␩^I,then

␾^*f=

共1 ⫻

␸

兲

^*

共e

^⌶f兲, ∀f

苸

C^⬁

共N兲. 共9兲

Proof:Let ␾^*:C^⬁

共N兲

→C^⬁

共⍀兲

0 which satisfies

共1兲

and

共2兲. We denote by

aI the functionals involved in identity

共6兲. We also introduce the following notation: for any

N苸N, O共␩^共N兲

兲

will represent a quantity of the form

O共␩^共N兲

兲

=

兺

n=N

⬁

兺

I苸I^q共n兲

cI␩^I,

where the coefficientsc_I’s may be real constants or functions. The result will follow by proving by recursion onn

苸

N^*the following property:

共

P_n

兲

: There exists a smooth map ␸^:

兩⍀兩

→N and there exists a family of vector fields

共

␰I

兲

I, where I

苸

I^q

共

2k

兲

and1

艋

k

艋

n, defined on a neighborhood of the graph of␸ⁱⁿ

兩⍀兩 ⫻

N, such that if

⌶

nª

兺

k=1

n

兺

I苸I^q共2k兲

␰I␩^I, then

␾^*f=

共

1

⫻

␸

兲

^*

共

e^⌶nf

兲

+O共␩^共2n+1兲

兲

, ∀f

苸

C^⬁

共N兲

.

Proof of

共P

1

兲:

We start from relation

共2兲

and we expand both sides by using

共6兲: we first obtain

by identifying the terms of degree 0 in the␩^i’s,

a_쏗

共

fg

兲共

x

兲

=

共

a_쏗

共

f

兲共

x

兲兲共

a_쏗

共

g

兲共

x

兲兲

, ∀x

苸 兩⍀兩

, ∀f,g

苸

C^⬁

共N兲

, which implies that for anyx

苸 兩⍀兩

, there exists some value␸

共

x

兲 苸

Nsuch that

a_쏗

共

f兲共x兲=f

共

␸

共x兲兲,

∀x

苸 兩⍀兩,

∀f

苸

C^⬁

共N兲.

In other words, there exists a function␸:兩⍀兩→Nsuch thata_쏗

共f兲

=fⴰ␸^{. Sincea}쏗

共f 兲

must beC^⬁ for any smoothf, this implies that␸

苸

C^⬁

共兩⍀兩

,N兲. The relations between the terms of degree 2 in

共2兲

are

aI

共fg兲共x兲

=

共a

I

共f兲共x兲兲共a

_쏗

共g兲共x兲兲

+

共a

_쏗

共f 兲共x兲兲共a

I

共g兲共x兲兲

=

共a

I

共f兲共x兲兲g共

␸

共x兲兲

+f

共

␸

共x兲兲共a

I

共g兲共x兲兲,

∀x

苸 兩⍀兩,

∀f,g

苸

C^⬁

共N兲,

∀I

苸

I^q

共2兲,

which implies that for any x苸

兩⍀兩, each

aI

共·兲共x兲

is a derivation acting on C^⬁

共N兲, with support 兵

␸

共x兲其, i.e.,

∀I

苸

I^q

共2兲

there exist tangent vectors

共

␰I

兲

x

苸

T_␸共x兲N such that

aI

共

f兲共x兲=

共共

␰I

兲

x·f兲共␸

共x兲兲,

∀f

苸

C^⬁

共N兲

.

Sincea_I

共

f

兲

must be smooth for anyf

苸

C^⬁

共N兲

, the vectors

共

␰I

兲

xshould depend smoothly onx, i.e., x哫

共

␰I

兲

xis a smooth section of␸^*TN. It is then possible

共see the Proposition 1.1 below兲

to extend it to a smooth section of ␲^*TN on a neighborhood of the graph of ␸. If we now set

共⌶

1

兲

x

ª

兺

I苸I^q共2兲

共

␰I

兲

x␩^I, we have on the one hand, e^共⌶1兲xf=f+

兺

I苸I^q共2兲

共共

␰I

兲

x·f兲␩^I+O共␩^共3兲

兲,

∀x

苸 兩⍀兩,

∀f

苸

N,

and on the other hand,

共

␾^*f

兲共x兲

=f

共

␸

共x兲兲

+

兺

I苸I^q共2兲

共共

␰I

兲

x·f

兲共

␸

共x兲兲

␩^I+O共␩^共3兲

兲,

∀x

苸 兩⍀兩,

from which

共

P₁

兲

follows.

Proof of

共P

n

兲

⇒

共P

n+1

兲: We assume 共P

n

兲

so that a map␸

苸

C^⬁

共兩⍀兩

,N兲and a vector field

⌶

n

have been constructed. Let us denote bybI the linear forms onC^⬁

共N兲

such that

共1 ⫻

␸

兲

^*

共e

^⌶nf兲=

兺

k=0 关q/2兴

I苸I

兺

^q共2k兲

bI

共

f兲␩^I.

共10兲

Then, property

共P

n

兲

is equivalent to

a_I=b_I, ∀k

苸 冀

0,n

冁

, ∀I

苸

I^q

共

2k

兲

.

共

11

兲

We use Lemma 1.1: it says us that f

哫共1 ⫻

␸

兲

^*

共e

^⌶nf

兲

is a morphism; hence,

共1 ⫻

␸

兲

^*

共e

^⌶n

共fg兲兲

=

关共1 ⫻

␸

兲

^*

共e

^⌶nf

兲兴关共1 ⫻

␸

兲

^*

共e

^⌶ng兲兴, so by using

共10兲,

兺

k=0

n+1

兺

I苸I^q共2k兲

bI

共fg兲

␩^I=

兺

k=0

n+1

兺

j=0

k

兺

J苸I^q共2k−2j兲,K苸I^q共2j兲

bJ

共f 兲b

K

共g兲

␩^J␩^K+O共␩^共2n+3兲

兲. 共12兲

However, the morphism property

共

2

兲

for␾^*implies also

兺

k=0

n+1

兺

I苸I^q共2k兲

aI

共fg兲

␩^I=

兺

k=0

n+1

兺

j=0

k

兺

J苸I^q共2k−2j兲,K苸I^q共2j兲

aJ

共f 兲a

K

共g兲

␩^J␩^K+O共␩^共2n+3兲

兲. 共13兲

We now substract

共12兲

to

共13兲

and use

共11兲: it gives us

兺

I苸I^q共2n+2兲

共a

I

共fg兲

−bI

共fg兲兲

␩^I=

兺

I苸I^q共2n+2兲

关共a

I

共f兲

−bI

共f兲兲a

_쏗

共g兲

+a_쏗

共f兲共a

I

共g兲

−bI

共g兲兲兴

␩^I. Hence, if we denote␦^aIªaI−bI, we obtain that

␦^aI

共fg兲

=␦^aI

共f兲共g

ⴰ␸

兲

+

共f

ⴰ␸

兲

␦^aI

共g兲,

∀I

苸

I^q

共2n

+ 2兲.

By the same reasoning as in the proof of

共P

1

兲, we conclude that

∀I

苸

I^q

共2n

+ 2兲, there exist smooth sections␰I of␲^*TNdefined on a neighborhood of the graph of␸, such that

␦^aI

共

f兲共x兲=

共共

␰I

兲

x·f兲共␸

共x兲兲,

∀x

苸 兩⍀兩,

∀I

苸

I^q

共2n

+ 2兲.

Now let us define

⌶

n+1ª

⌶

n+

兺

I苸I^q共2n+2兲

␰I␩^I. Then, it turns out that

e^⌶n+1f=

兺

k=0

n+1

冉 ^⌶

^n+I苸I^q

^兺

共2k+2兲␰I␩^I

冊

^k

k! f+O共␩^共2n+3兲

兲

=

兺

k=0 n+1

⌶

n

k

k!f+

兺

I苸I^q共2n+2兲

␰I·f␩^I+O共␩^共2n+3兲

兲

=e^⌶nf+

兺

I苸I^M共2n+2兲

␰I·f␩^I+O共␩^共2n+3兲

^兲,

so that

共1 ⫻

␸

兲

^*

共e

^⌶n+1f兲=␾^*f+O共␩^共2n+3兲

兲.

Hence, we deduce

共

P_n+1

兲

. 䊏

Proposition 1.1: In the preceding result, it is possible to construct smoothly the vector fields

␰I’s in such a way that

关共

␰I

兲

x,共␰J

兲

x

兴

= 0, ∀x

苸 兩⍀兩,

∀I,J

苸

I₂^q.

Proof:Recall that in the previous proof, in order to build

⌶

n+1 out of

⌶

n, we introduced, for each I

苸

I^q

共

2n+ 2

兲

, an unique smooth section x

哫共

␰I

兲

x of ␸^*TN. We will explain here how to extend each such vector fields defined along the graph of␸to a neighborhood of the graph of␸ in

兩⍀兩 ⫻

N in order to achieve the claim in the proposition. For that purpose, we prove that for some set

Vª

兵共

x,␰^,q

兲 苸

␸^*TN

⫻

N兩x

苸 兩⍀兩

,␰

苸

T_␸共x兲N,q

苸

V_␸共x兲

其

, where eachV_␸共x兲 is a neighborhood of␸

共x兲

inN, there exists a smooth map

V →TN,

共x,

␰,q兲

哫 共q,V共x,

␰,q兲兲,

such that ∀共x,␰

兲 苸

␸^*TN, V共x,␰^,␸

共x兲兲

=␰ ^and ∀x苸

兩⍀兩

fixed, ∀␰^,␨

苸

T_␸共x兲N,

关V共

x,␰^{, ·}

兲

,V共x,␨^{, ·}

兲兴

= 0, i.e., the vector fieldsq

哫V共

x,␰^,q

兲

andq

哫V共

x,␨^,q

兲

commute onV_␸共x兲.

Then, the proposition will follows by extending each vector

共

␰I

兲

x

苸

T_␸共x兲N on V_␸共x兲 by q

哫

V共x,

共

␰I

兲

x,q

兲

.

The construction is the following. Let

共

U_a

兲

a苸Abe a covering ofNby open subsets, let

共

␹a

兲

a苸A

be a partition of unity, and let

共y

a

兲

a苸Abe a family of charts associated with this covering. For any x苸

兩⍀兩, let

Axª

兵a 苸

A

兩

␸

共x兲苸

Ua

其. For any

a

苸

Ax and for any linear isomorphism

ᐉ

:T_␸共x兲N

→Rⁿ, where n= dimN, letR_x,ᐉ,a be the unique linear automorphism ofRⁿ such that R_x,ᐉ,aⴰdy_{a兩␸共x兲}=

ᐉ

.

We then set

y_x,ᐉ

共

q

兲

ª_a

兺

_苸A

x

␹a

共

q

兲

R_x,ᐉ,aⴰy_a

共

q

兲

, ∀q

苸

N.

We observe that dy_{x,ᐉ兩␸共}x兲=

ᐉ

and hence, by the inverse mapping theorem, there exists an open neighborhoodV_␸共x兲 of␸

共x兲

inNsuch that the restriction ofy_x,ᐉtoV_␸共x兲 is a diffeomorphism. We then define

V共x,␰,q兲ª

共dy

_x,ᐉ兩q

兲

⁻¹

共ᐉ共

␰

兲兲,

∀q

苸

V_␸共x兲.

Because of the obvious relationy_x,uⴰᐉ=uⴰy_x,ᐉ for all linear automorphismu ofRⁿ, it is clear that the definition ofV共x,␰^,q兲does not depend on

ᐉ 共for the same reason

V_␸共x兲is also independant of

ᐉ兲. Moreover,

q

哫

V共x,␰^,q兲 is simply a vector field which is a linear combination with constant coefficients of the vector fields

共

⳵/⳵^yx,iᐉ

兲

i=1,. . .,n so that the property

关V共x,

␰^{, ·}

兲

,V共x,␨^{, ·}

兲兴

= 0 fol- lows. Note also that these vector fields are of course not canonical since they obviously depend on

the charts. 䊏

Remark 1.1: If we assume furthermore that the image of␸is contained in an open subset U ofNsuch that there exists a local chart y=

共y

¹, . . . ,yⁿ

兲:

U→Rⁿ, then it is possible to choose all the vector fields␰I such that

共

␰I

兲

x·

共

␰J

兲

x·y= 0, ∀x

苸 兩⍀兩,

∀I,J

苸

I₂^q.

共14兲

Indeed in this case the proof of Proposition 1.1 is much simpler since we do not need to use a partition of unity in order to build V. We just set Vª

兵共x,

␰,q兲

苸

␸^*TN

⫻

N

兩

x苸

兩⍀兩

,␰

苸

T_␸共x兲N,q

苸

U其 and define V by V共x,␰,q兲ª

共dy

_兩q兩

兲

⁻¹ⴰdy_兩␸共x兲兩

共

␰

兲. Then, for each 共x,

␰

兲苸

␸^*TN fixed, the vector field q

哫V共x,

␰,q兲 has constant coordinates in the variables y^␣. Hence, (14) follows.

Remark 1.2: We can write an alternative formula for e^⌶ by developping this exponential: in each term of the form

共兺

I␰I␩^I

兲

ⁿ, we can see that each monomial which appears contains at most one time any operator␰I, so we obtain

e^⌶=

兺

I苸I₀^q

␩^I

冉

ⁿ

^兺

^艌0n!1I1,. . .,I

^兺

_n苸I₀^q

⑀I

I₁¯I_n␰I₁¯␰I_n

冊

^,

^共

¹⁵

^兲

with the convention that the I₀^q

共0兲= 쏗

contribution is the identity. Here, we have introduced the notation ⑀I

I₁. . .I_n

: first all the ⑀_쏗^{I1. . .In}’s vanish except for ⑀_쏗^쏗= 1, so that e^⌶= 1 mod关␩^{1, . . . ,}␩^q

兴.

Second, for k

艌

1, if I₁=

共i

1,1, . . . ,i_1,2k

1

兲, . . .,

In=

共i

n,1, . . . ,i_n,2k

n

兲,

and I=

共i

1, . . . ,i_2k

兲

we write that I₁¯In=I if and only if k₁+¯+kn=k,

兵i

1,1, . . . ,i_1,2k

1, . . . ,i_n,1, . . . ,i_n,2k

n

其= 兵i

1, . . . ,i_2k

其

and∀j, I_j⫽

쏗

(i.e., k_j

⬎

0). Then,

• if I₁¯In⫽I,⑀I I₁¯I_n

= 0; and

• if I₁¯In=I, ⑀I I₁¯I_n

is the signature of the permutation

共

i_1,1, . . . ,i_1,2k

1, . . . ,i_n,1, . . . ,i_n,k

2n

兲哫 共

i₁, . . . ,i_2k

兲

.

The preceding expression of e^⌶can be recovered by another way: since all the operators␩^I␰I

commute, we have¹

e^⌶=e^{兺I苸I2q␩I␰I}=

兿

I苸I2 q

e^␩I␰I, which gives also the same result by a straightforward development.

III. A FACTORIZATION OF THE MORPHISM

␾

^* A. Integrating the vector fields

␰

I’s

In the same spirit as a tangent vector at a pointq to a manifoldN can be seen as the time derivative of a smooth curve which reachesq, we can describe the ␩^I-components of the mor- phism␾^*as higher order approximations of a smooth map from some vector space with values in N. Indeed, let ␾^*

苸

Mor共C^⬁

共N兲

,C^⬁

共⍀兲

0

兲: then by the preceding result

␾^* is characterized by a map␸

苸

C^⬁

共兩⍀兩

,N兲and 2^q−1− 1 vector fields¹␰I tangent toN defined on a neighborhood of the graph of␸ⁱⁿ

兩⍀兩 ⫻

N. By Proposition 1.1, these vector fields can moreover be chosen so that they pairwise commute whenx

苸 兩⍀兩

is fixed. So, for anyx

苸 兩⍀兩

, we can integrate simultaneously all vector fields

共

␰I

兲

xin order to construct a map,

⌽共x, · 兲:U

x

共⌳

₊^2*R^q

兲

→ N,

where

⌳

₊^2*R^q

⯝

R^2q−1−1is the subspace of even elements of positive degree of the exterior algebra

⌳

^*R^q andUx

共⌳

₊^2*R^q

兲

is a neighborhood of 0 in

⌳

₊^2*R^q, such that

⌽共

x,0

兲

=␸

共

x

兲 共

16

兲

and denoting by

共s

^I

兲

I苸I₂^q the linear coordinates on

⌳

₊^2*R^q,

⳵

⌽

⳵^sI

^共x,s兲

⁼␰I

共⌽共x,s兲兲,

∀s

苸

Ux

共⌳

₊^2*R^q

兲,

∀I

苸

I₂^q.

共17兲

We hence obtain a map

⌽

from a neighborhood of

兩⍀兩 ⫻ 兵0其

in

兩⍀兩 ⫻⌳

₊^2*R^q to N. By using a cutoff function argument, we can extend this map to an application

⌽: 兩⍀兩 ⫻⌳

₊^2*R^q

哫

N. Lastly, we introduce theR关␩^{1, . . . ,}␩^q

兴-valued vector field on 兩⍀兩 ⫻⌳

₊^2*R^q,

␽ª

兺

I苸I2 q

␩^I ⳵

⳵^si, so that by

共

17

兲 ⌽

_*␽⁼

⌶

=

兺

I苸I₂^q␩^I␰I. Then, relation

共

9

兲

implies

␾^*f共x兲=

共e

^␽

共f

ⴰ

⌽兲兲共x,0兲,

∀f

苸

C^⬁

共N兲,

∀x

苸 兩⍀兩,

or by letting␫^:

兩⍀兩→ 兩⍀兩 ⫻⌳

₊^2*R^q,x哫

共x, 0兲

to be the canonical injection,

␾^*f=␫^*

共e

^␽

共f

ⴰ

⌽兲兲. 共18兲

Alternatively by using

共

15

兲

, we have

␾^*f共x兲=

兺

I苸I0 q

␩^I

冉

^k

^兺

^{艌0k!1I1,. . .,I}

^兺

^k苸I0 q

⑀I

I₁¯I_k ⳵^k

共f

ⴰ

⌽兲

⳵^sI1¯⳵^sIk

^共x,0兲 冊

^{, ∀x}

^{苸 兩⍀兩}

^.

^共19兲

It is useful to introduce the differential operatorsD_쏗ª1 and

1Note that cardI^q共2k兲=q!/共q− 2k兲!共2k兲! and兺k=0

关q/2兴q!/共q− 2k兲!共2k兲! = 2^q−1.