• Aucun résultat trouvé

An introduction to Frobenius manifolds, moduli spaces of stable maps and quantum cohomology

N/A
N/A
Protected

Academic year: 2021

Partager "An introduction to Frobenius manifolds, moduli spaces of stable maps and quantum cohomology"

Copied!
66
0
0

Texte intégral

(1)

HAL Id: hal-00129568

https://hal.archives-ouvertes.fr/hal-00129568

Preprint submitted on 8 Feb 2007

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

An introduction to Frobenius manifolds, moduli spaces of stable maps and quantum cohomology

Michèle Audin

To cite this version:

Michèle Audin. An introduction to Frobenius manifolds, moduli spaces of stable maps and quantum cohomology. 1997. �hal-00129568�

(2)

"&('%) *+,-". ,'/

012 % 31 4 4" 56

78:9<;>=

?.@A?%B%CED 8AF

GHFEIKJL8AJ

C J DM?ONP?

9<;

?.Q 9R;

?

7SJT;VU

?.W S#JT8YX

CZ?[B]\

SFE9^U

?.?

_%F`8

\!?.Q IK8aJ`U

?cbEdeC

8YI]fgShIKJ

?.C`Q%?

J[i2j

N[k

l

QTC`?ON?

FU

?nmpo]?

IL9.S

Q J ? I

q>rtsulwvexy

k J Q SeI{z

dhC`QT|

i

?DM?.}

W S C~D

8AF^

W

SJT;>€

CMm IKJ

Q

SeI{z

|

€‚

Q

W

S.ƒ…„.I{J„†e†!l

(3)

!"$#%"&#('*),+-',./012'$.3 "&54678. 9

„ k

J<S#z

@A?c|e?.QLW

I d  b S

|eQ S#F

|h?3:`QTd;T?

9nJT8

d

FEI=<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</< y

„h€A„

b S

|eQ S#F

|

8:SF-I

C z W SF`8

dh@:D

I8AF S9

d

J<SF

|e?

FMJz

C F DM@A?

</<@</<@<><?<@<><?<><@</<@</<@</< y

„h€BA

k

8AF

|hC`@

S Q 8aJT8

?

IS#F

DC FM

dh@:D 8aF

| I </<@<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</< C

„h€BDFE

} S WG:`@A?

I d  b S

|eQ S#F

|h?3:`QTd;T?

9nJT8

d

FEIH<?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</< l

„h€y k J

QLC 9nJ

CZQT?

I d

FJI>KL<@<@</<@</<@</<@<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</< x

A*7

CZ@

JT8

:`@

8Y9.S#JL8

d F d 

\!?

9nJ

deQ M~?.@:D ISF

D I :E?

9nJ

Q S @ 9 d\!?.Q

IN</<@</<@</<@<><?<@<><?<><@</<@</<@</< „.v

At€A„

kO:E?

9nJ

Q S @ 9 d\!?.Q

I </<@<@</<@</<@</<@<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</< „.v

At€BAFPSe9RQJ

dC Ft

de@:D 8AF

| I <@</<@</<@</<@<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</< „ny

At€BDTS]JL;

?.Q?.}

S

WG:`@A?

I d

2I

:E?

9nJ

Q S @ 9 d\!?.Q

IU<?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</< „.s

D*q

QLd z ?

F`8

C I W S#FZ8

de@YD IWV

?.}

S

WX:`@a?

IY<@</<@<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</< „.†

Dt€A„

o]?WM F`8aJT8

d F d

2S,q

QLd z ?

F`8

C I W S#F`8

de@:D

<?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</< „.†

Dt€BA q

QLd z ?

F`8

C I W S#F`8

de@YD

I d  D 8

W?

FEI{8

d

FEIAS#F

D

DZ<@</<@</<@</<@<><?<@<><?<><@</<@</<@</<[AD

Dt€BDT\

m

S#F

D*]PmWdtDM?.@

I^<@<@</<@</<@</<@<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</<[AC

_("`178'a "&).3"&cb"& Od 78#e f

y 7

dtDtC`@

8+I

:

Se9

? I d

2I{J<S#z

@A?cW S :

IWV$g8aJT;

?.}

S

WX:Z@A?

IY<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</<[Aex

yZ€A„

kO:

Se9

? I d

I{J<Sz

@A?cW S :

I</<@</<@</<@<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</<[Aex

yZ€BA

NP?.@

S#JL8

d

Fhg8aJT;…JT;

? I : Se9

?cd

2IKJ S#z

@a? W S Q Q

?wD

9

C`QL\!?

Ii</<@</<@<><?<@<><?<><@</<@</<@</<[Dev

yZ€BDi

d F

\!?.}W SF`8

dh@:D I2S#F

D I Wd!d

JT;`F

?

ITI

:`QLd:E?.Q JL8

?

I></<@</<@</<@<><?<@<><?<><@</<@</<@</<[DD

Ckj

QLdeWd\!mml 8aJTJ

? F :Ed

J ? FMJL8:S

@

SF

D X C S#FtJ

C`W 9 d ;

dhWde@ade|

ƒ </<@</<@<><?<@<><?<><@</<@</<@</<[Des

Ct€A„nj

QLdeWd\!mml 8aJTJ

?

F8AF

\ S Q

8:S#FtJ<Io<@</<@<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</<[Des

Ct€BATp

C

S#FtJ

C`W 9 d ;

dhWde@ade|

ƒ <@</<@</<@<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</< y!v

Ct€BDi

deWX:Ed I{8AJL8

d F QLC`@A?

S#F

D

SeILI

d

9n8YS#JL8

\

8AJ{ƒ <><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</< y`„

Ct€y[q;

? q QLd

z ? F`8

C

I%IKJ

QLC 9nJ

C`QT?

</<@</<@<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</< yey

Ct€BCTj

? F

?.Q S @

8YITS#JL8

d

FEIT</<@<@</<@</<@</<@<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</< yMl

s

o%?.|eQL?.?nm A,9

@

ShITI

?

IS#F

D

JT;

? I W S

@A@

X C S#FtJ

C`WZ:ZQTdtDMC

9 J </<@</<@</<@<><?<@<><?<><@</<@</<@</<[Cev

st€A„np

C

S#FtJ

C`W WC`@

JT8

:Z@

8:9S#JT8

d

FSJ[S

| 8

\!?

F

:Ed

8aFMJ>rts2

deQTW S @ I

?.Q 8?

In<?<@<><?<><@</<@</<@</<[Cev

st€BAFE

} S WG:`@A?

I d

2X

C

SFMJ

C`W 9 d ;

dhWde@ade|

ƒ Q 8AF

| I <><@</<@</<@</<@<><?<@<><?<><@</<@</<@</<[CA

uX))2'$wv xx

B E

@a@

8: JT8:9%9

C`QL\!?

ISF

Dy:Z@

S#F

? 9 C

z`8:9I?<@</<@<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</<[CC

P

B%@A|e?

z Q S8:9

@ad!d:

I : Se9

?

IS#F

D 9 d F ME|eC`Q

S#JL8

d

FI

:

Se9

?

IL<@</<@</<@</<@<><?<@<><?<><@</<@</<@</<[Cul

i f d

8:ILI

d

F9

dhWWC JT8F

|cQT?.@

S#JL8

d

F~I </<@</<@<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</<[Ce†

o

q;

? I

?.W 8m

I{8

WX:`@

8Y9n8JKƒ SeILI

C`WX:

JT8

d

F0<@</<@<><?<><@<?<><@</<@</<@</<@<><?<@<><?<><@</<@</<@</< sev

z381788#%"&)de {(

(4)

q;

?

S8

W*d

>JL;

? I ? I C`QT\!?

ƒ F d J ?

I8YI J d J Q

ƒJ

d[Wd JT8

\

SJ

?

SF

D J d 8AFtJ

QLdtDMC

9 ? J g d z ? S C JL8

C`@`|h?.deW ? J Q 8:9

9 d FE9

?-:

J<IJT;ESJ[;ES

\!?

S

@AQL?

S D ƒ

:`QLd#\!?wD

V

DM?

I : 8AJ

?

JT;

? 8 Q%QT?.@

S#JL8

\!?

ƒ

dhC JT;VEJ

d z

?,Q 8:9<; S#F

DC I ? 

C`@

V FES

W?.@

ƒ

o]C

z

QLd#\

8AFI2q

QLd z ?

F`8

C I W S#FZ8

de@YD ISF

D

d

FtJ<I

?\

8Y9R;I

WdtDMC`@

8>I

:

Se9

? I d

2IKJ S#z

@a?cW S : I €

B

9.9

deQ<D 8AF

| J d S :

;`8

@ad

I

d:

;tƒ g;Z8:9<;…G ;ES

\!?[@a?

S Q F

?D 8aFNj]8

\!?

FtJ<S

@ I : S : ?Q

I At„OV&A DmVMJL;

?.QL?

S

QL?

J g d

z`8

|  S W 8@8? I d

q

QLd z ?

F`8

C I W S#F`8

de@YD

I

JL;

d I

?cDM?WM

F

?D ztƒ

C

FM

dh@:D 8aF

| I b S#F

D S CMm

j]8aF.z

CZQT|

V

deQ

\

mHW dtDM?.@

IVtSe9.9

dhQ<D 8aF

| J d JL;

? :

;MƒZIK8Y9.S

@

J

?.QTW 8AF

dh@Ade|

ƒ V

JL;

d I

?ODM?WM

F

?wD ztƒ…X

C

S#FtJ

CZW 9 d ;

deWdh@Ade|

ƒ

M~?.@:D JT;

?.deQ

ƒ

deQ ]Pm WdtDM?@

I

SF

D

JL;

?  S WdhC

I W 8

QTQLdeQ 9 d F

;T?

9 J

C`QL?2:Ed IKJ

C`@

SJ

?

IJT;

? ?}

8YIKJ

?

FE9

?[d

S9

deQLQT?

I

:Ed

F

DM?

FE9

? z ? J g

?.?

F JT;

? I ?

J g d  S W 8@8? I €

q;

? J g dX:

S Q J I d

2JL;

? I ? F d J ?

Ic9.SFz

? 9 d FEIK8

DM?.QT?wD ShI S

DM?

IT9

Q 8: JL8

d F d

JL;

? I ? J g d  S W 8@8? I QT?nm

I : ? 9nJT8

\!?.@

ƒRV!JT;

QTdeCZ|

;9

d

FE9

QL?

J ? ?.}

S

WX:`@a?

In€

B]@

JT;

dhC`|

;JT;

?Dt?WM F`8aJT8

d F d

%S…q

QTd z ?

F`8

C I W SF`8

dh@:D

I

deW ? ; d g

@Ad!d QZI

@8Q ? I

deW ? q

QT?

F~9R;9

d!d QM8aF

|

QL?

9n8

:E?

I W

SFMƒO8AF

|hQT?D 8?

FtJ<IgSF

D S Q

SJT;

?.Q 8aFMJ

Q

8:9S#J

?W 8} J

C`QL?d

 JT;

?W V!I

??

€ „#l VE„-A VeJT;

? 9 d FE9

?-:

J

8aJ<I

?@

(8:I%X

C

8aJ

?

FESJ

C`Q S @ V S#F

D

V^8AF1I

dhW?

I ? FEI

? 9 @ SeILIK8:9S

@ € k 8AF

|hC`@

S Q 8YIKJ<IV~

dhQ

8aFEIKJ S#FE9

?

V^S

QT? DM?

S @ 8AF

|

g8AJL;

q

QTd z ?

F`8

C I W SF`8

dh@:D I 

deQ

S

|e?

I

deQ V`S#J

@a?

SeI{JI{8AFE9

? € k

S8AJ

d I : S :E?.Q

yZ„ <€ q;

? q QLd

z ? F`8

C

II{J

QTC 9nJ

C`QL?

8YI g;ESJƒ

deC

I

?.?

g;

?

F…ƒ

dhC@Ad!d

Q…S#JJL;

?=:`QLdtDMC 9nJ

d F JL;

?!

Sh9

d

z`8YS#F S

@a|e?

z Q S,S#F

D

SJJT;

? QT?

I{8

DMC`?



deQTW

ShI z ? 8aF

| S Q

8AF

|

IKJ

QTC 9nJ

C`QT?,d FJL;

?

J<S#F

|e?

FtJcIK;

?

SwVJ

d

JT;

? I : Se9

?,d

 : S Q S W ? J

?.Q I d

2JL;

?Dt?



deQLW S#JL8

d F

SF

D

m deQLW d

FJL;`8:IgI

:

Se9

?

V!JT;

?%W S#8aF

:Ed 8AFtJz

?

8AF

|,k S8AJ

d IgJL;

?.deQL?.W

VJT;ESJ8YIWV!JT;ESJJT;

?%QT?

I{8

DMCZ?



deQTW

SF

D

JL;

?@:`QTdtDMC 9nJ

Dt?WM F ?

S#"%$'&

W?

J Q 8Y9e€

B q

QLd z ?

F`8

C

II{J

QLC 9nJ

CZQT?d

F S W

SF`8

dh@:D)(

8YIWV+8AF

DM??D V+S

Q

8AF

|

IKJ

QLC 9nJ

C`QT?d F-JL;

? J S#F

|h?

FMJOI{;

?

Sw

J

de|e?

JT;

?.Q

g8AJT; S+* SJ

W?

J Q 8:9 I C z

;T?

9 JJ

d

S-9

dhC&:`@A? d  9 dhWX:

S#JL8Az`8

@

8aJKƒ9

d F D 8aJT8

d

FEI € GHF

\!?.Q

ƒ

DMd gF

m

J

d#m ? S Q JT; J

?.QLW IWV>JL;

?b S#F

D S CMm

j]8aF.z

CZQT|WdtDM?.@

I S

QT?

JT;

d I ?

8AFg;`8:9<;JT;

?WC`@

JT8

:Z@

8:9S#JT8

d F 8:I

Q

SJT;

?Q

I{8

WX:`@a?

V>z

C

J,

deQ

g;`8Y9R; 8aJ8:I;~S

Q<D

J

daM

F D * SJ9

d!deQ D 8AFESJ

?

IVg;`8

@A?

8aF X C SFMJ

C`W 9 d ;

deWdh@Adh|

ƒOVgJL;

?

* SJO9

d!deQ D 8AFESJ

?

IcS

QT?

JT;

?

FESJ

C`Q S @ 9

d!deQ D 8AFESJ

?

IVg;`8

@a?

JT;

?3:`QLdtDMC 9nJO8:I

\!?Q

ƒ…;ES

Q D J d DM?M F ?

€,GHFJL;

?

@

S#F

|hC

S

|h?

g;`8Y9R;8YI

C I

?wD 8aF…JT;

? q QTd

z ? F`8

C

Ig

deQL@:D V`8aF

b

S#F

D S CMm

j%8AF.z

C`QT|W dtDM?.@

IVMƒ

dhC

S

QT?c|

8

\!?

FJL;

?

9.S#F

d

F`8:9S

@ 9 d!dhQ<D

8aFES#J

? I g;Z8

@A?

8AF X C S#FtJ

C`W 9 d ;

dhWde@ade|

ƒ VMƒ

deC

S

QT?|

8

\!?

F JT;

?,

* S#J 9

d!deQ D 8AFESJ

? I €

_]FM

dh@:D 8aF

|

I+z

?@Ad F | J d

JL;

? g dhQT@YDd

 b S

|hQ S#F

|

8:SF…IK8aF

|eCZ@

S Q 8AJL8

? I 9.S

C

IKJL8:9.I-^z

C

JgJT;Z8:I 8YIgS

@ I d JL;

?

9.SeI

?

d

+S

@a@

q

QLd z ?

F`8

C I W S#FZ8

de@YD I.+S

:`QTdtDMC 9nJ

d F

\!?

9 J

deQ ME?@:D

I

DM?WM F ?

IS/103254& 6$7348:92-6

d

+JT;

?]W SF`8

dh@:D VhS

I C z \ S Q 8? JKƒ

d

3;=<

(

VOg;`8Y9R;,8YI

\!?.Q

ƒ d J ? F b S

|hQ S#F

|

8:SFVES#F

D QLdeC`|

; @

ƒI

: ? S Qt8AF

|ODM?

IL9

Q

8az

? I JT;

?

gS.ƒJL;

?

Q

8AF

|

I>;?

( I

:`@

8AJcShI I

C`W I d

VIK8

WX:`@a?

I

C`WW SF

D

I%Se9.9

deQ<D 8AF

| J d JT;

?,\

S

@aC`?

I d

A@u€ q;

C

IV

b S

|eQ S#F

|

8:SF

I C z \ S Q 8? JT8

? I DM?WM

F`8AJ

?.@

ƒz

?.@Ad F | J d

JT;

? q QTd

z ? F`8

C I @ S#F

D

IL9.S

:E?

€

q;

? M~Q IKJ%9<;ES

: J

?.Q2d

(JT;

? :`QT?

I ? FtJI

CZQT\!?

ƒ8:I

DM?.\!d

J

?D J d

JT;

?Dt?

IT9

Q 8: JL8

d F d

gJL;

?b

S

|hQ SF

|

8YS#FI

C z m

\ S Q 8? JL8

? I DM?WM

F

?wD ztƒ

C

FM

dh@:D 8aF

|

IV

d  I :E?

9nJ

Q S @ 9 d\!?.Q

IgS#F

D J d JL;

? DM?WM F`8aJT8

d F d

+SOq

QTd z ?

FZ8

C I W S#F`8

de@YD

€

G2I

:E?

F

DW SFMƒ

: S

|e?

I d F

?.}

S

WG:`@A?

IV

€ d FNA

m{D

8

W?

FEI{8

d

FES

@>?.}

S

WG:`@A?

I2S#F

D…d FJL;

?c\!?.Q ILS

@+C FM

de@YD 8AF

|

d

CBDFE

€

q;

? I ? 9 d F D 9<;ES

: J

?.Q 8:IS

DM?

IT9

Q 8: JT8

d F d

JL;

? q QTd

z ? F`8

C

I]I{J

QTC 9nJ

C`QL?d FJT;

? 9 d ;

deWdh@Adh|

ƒ d

2I

deW?

:`QLd;T?

9nJL8

\!?[\

S Q 8? JT8

? I | 8

\!?

F ztƒJL;

? X C S#FtJ

CZWZ:`QTdtDtC

9nJ.€%GHJ%8:Ig

?@A@m QtF

d

gF JL;ES#J JT;

? 8 Dt?

S d

2X

C

S#FtJ

C`W

9 d ;

deW de@Adh|

ƒ9

dhW?

I 

QTdhW JL;

? g deQ

Q d  :

;tƒ`I{8:9n8YIKJ<IgJ

Q

ƒt8AF

| J d,Dt?



deQLW JL;

? 9

C&:Mm :`QTdtDMC 9nJ

C

IK8aF

|Q S#JL8

d

F~S

@

GIH5JKL=MNJOKP3QIRSL-TT UVQWLRXYJOKM[Z5RSL-\ UVJ]_^`T PIP>aWbdcOe%fgFh5^5ikj e

Références

Documents relatifs

To test whether the vesicular pool of Atat1 promotes the acetyl- ation of -tubulin in MTs, we isolated subcellular fractions from newborn mouse cortices and then assessed

Néanmoins, la dualité des acides (Lewis et Bronsted) est un système dispendieux, dont le recyclage est une opération complexe et par conséquent difficilement applicable à

Cette mutation familiale du gène MME est une substitution d’une base guanine par une base adenine sur le chromosome 3q25.2, ce qui induit un remplacement d’un acide aminé cystéine

En ouvrant cette page avec Netscape composer, vous verrez que le cadre prévu pour accueillir le panoramique a une taille déterminée, choisie par les concepteurs des hyperpaysages

Chaque séance durera deux heures, mais dans la seconde, seule la première heure sera consacrée à l'expérimentation décrite ici ; durant la seconde, les élèves travailleront sur

A time-varying respiratory elastance model is developed with a negative elastic component (E demand ), to describe the driving pressure generated during a patient initiated

The aim of this study was to assess, in three experimental fields representative of the various topoclimatological zones of Luxembourg, the impact of timing of fungicide

Attention to a relation ontology [...] refocuses security discourses to better reflect and appreciate three forms of interconnection that are not sufficiently attended to