Asymptotics in dynamics, geometry and PDEs; Generalized Borel Summation. / vol. II

289 Pages·2011·2.009 MB·English

by  Costin O., Fauvet F., Menous F., Sauzin D. (eds.)

12**
CRM
SERIES
OvidiuCostin
MathematicsDepartment
TheOhioStateUniversity
231W.18thAvenue
Columbus,Ohio43210,USA
[email protected]
Fre´de´ricFauvet
De´partementdemathe´matiques-IRMA
Universite´deStrasbourg
7,rueDescartes
67084StrasbourgCEDEX,France
[email protected]
Fre´de´ricMenous
De´partementdemathe´matiques,Baˆt.425
Universite´Paris-Sud
91405OrsayCEDEX,France
[email protected]
DavidSauzin
CNRSParis
and
ScuolaNormaleSuperiore
PiazzadeiCavalieri7
56126Pisa,Italia
[email protected]
[email protected]
Asymptotics in Dynamics,
Geometry and PDEs;
Generalized Borel
Summation
vol. II
edited by
O. Costin, F. Fauvet,
F. Menous, D. Sauzin
(cid:2)c 2011ScuolaNormaleSuperiorePisa
ISBN: 978-88-7642-376-5
e-ISBN:978-88-7642-377-2
Contents
Introduction xi
Authors’affiliations xv
ChristianBognerandStefanWeinzierl
Feynmangraphsinperturbativequantumfieldtheory 1
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Perturbationtheory . . . . . . . . . . . . . . . . . . . . 2
3 Multi-loopintegrals . . . . . . . . . . . . . . . . . . . . 6
4 Periods . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5 AtheoremonFeynmanintegrals . . . . . . . . . . . . . 8
6 Sectordecomposition . . . . . . . . . . . . . . . . . . . 10
7 Hironaka’spolyhedragame . . . . . . . . . . . . . . . . 12
8 Shufflealgebras . . . . . . . . . . . . . . . . . . . . . . 13
9 Multiplepolylogarithms . . . . . . . . . . . . . . . . . 17
10 FromFeynmanintegralstomultiplepolylogarithms . . . 19
11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 22
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
JeanEcalle
withcomputationalassistancefromS.Carr
Theflexionstructureanddimorphy:
flexionunits,singulators,generators,andtheenumeration
ofmultizetairreducibles 27
1 Introductionandreminders . . . . . . . . . . . . . . . . 30
1.1 Multizetasanddimorphy . . . . . . . . . . . . . 30
1.2 Fromscalarstogeneratingseries . . . . . . . . . 32
1.3 ARI//GARIanditsdimorphicsubstructures . . . 35
1.4 Flexionunits,singulators,doublesymmetries . . 36
vi
1.5 Enumerationofmultizetairreducibles . . . . . . 37
1.6 Canonicalirreduciblesandperinomalalgebra . . 38
1.7 Purposeofthepresentsurvey . . . . . . . . . . 38
2 Basicdimorphicalgebras . . . . . . . . . . . . . . . . . 40
2.1 Basicoperations . . . . . . . . . . . . . . . . . 40
2.2 Thealgebra ARI anditsgroupGARI . . . . . 45
2.3 Actionofthebasicinvolutionswap . . . . . . . 48
2.4 Straightsymmetriesandsubsymmetries . . . . . 49
2.5 Mainsubalgebras . . . . . . . . . . . . . . . . . 52
2.6 Mainsubgroups . . . . . . . . . . . . . . . . . . 53
2.7 Thedimorphicalgebra ARIal/al ⊂ ARIal/al. . . 53
2.8 ThedimorphicgroupGARIas/as ⊂ GARIas/as . 54
3 Flexionunitsandtwistedsymmetries. . . . . . . . . . . 54
3.1 Thefreemonogenousflexionalgebra Flex(E) . 54
3.2 Flexionunits . . . . . . . . . . . . . . . . . . . 57
3.3 Unit-generatedalgebras Flex(E) . . . . . . . . 61
3.4 Twistedsymmetriesandsubsymmetries
inuniversalmode . . . . . . . . . . . . . . . . . 63
3.5 Twisted symmetries and subsymmetries in polar
mode . . . . . . . . . . . . . . . . . . . . . . . 67
4 Flexionunitsanddimorphicbimoulds . . . . . . . . . . 70
4.1 Remarkablesubstructuresof Flex(E) . . . . . . 70
• •
4.2 Thesecondarybimouldsess andesz . . . . . 79
• •
4.3 Therelatedprimarybimouldses andez . . . . 87
4.4 Somebasicbimouldidentities . . . . . . . . . . 88
4.5 Trigonometricandbitrigonometricbimoulds . . 89
4.6 Dimorphicisomorphismsinuniversalmode . . . 94
4.7 Dimorphicisomorphismsinpolarmode . . . . . 95
5 Singulators,singulands,singulates . . . . . . . . . . . . 99
5.1 Someheuristics. Doublesymmetriesandimparity 99
5.2 Universalsingulatorssenk(ess•)andseng(es•) . 101
5.3 Propertiesoftheuniversalsingulators . . . . . . 102
5.4 Polarsingulators: descriptionandproperties. . . 104
5.5 Simplepolarsingulators . . . . . . . . . . . . . 105
5.6 Compositepolarsingulators . . . . . . . . . . . 105
5.7 Fromal/al toal/il. Natureofthesingularities . 106
6 Anaturalbasisfor ALIL ⊂ ARIal/il . . . . . . . . . . 107
6.1 Singulation-desingulation: thegeneralscheme . 107
6.2 Singulation-desingulationuptolength2 . . . . . 111
6.3 Singulation-desingulationuptolength4 . . . . . 112
6.4 Singulation-desingulationuptolength6 . . . . . 112
6.5 Thebasislama•/lami•. . . . . . . . . . . . . . 116
vii
6.6 Thebasisloma•/lomi• . . . . . . . . . . . . . . 116
6.7 Thebasisluma•/lumi• . . . . . . . . . . . . . 117
6.8 Arithmeticalvsanalyticsmoothness . . . . . . . 118
6.9 Singulatorkernelsand“wandering”bialternals . 119
7 Aconjecturalbasisfor ALAL ⊂ ARIal/al.
Thethreeseriesofbialternals . . . . . . . . . . . . . . . 120
7.1 Basicbialternals: theenumerationproblem . . . 120
7.2 Theregularbialternals: ekma,doma . . . . . . . 120
7.3 Theirregularbialternals: carma . . . . . . . . . 121
7.4 Main differences between regular and irregular
bialternals . . . . . . . . . . . . . . . . . . . . . 121
7.5 Thepre-domapotentials . . . . . . . . . . . . . 123
7.6 Thepre-carmapotentials . . . . . . . . . . . . . 124
7.7 Constructionofthecarmabialternals . . . . . . 126
7.8 Alternative approach . . . . . . . . . . . . . . . 127
7.9 Theglobalbialternalidealandtheuniversal
‘restoration’mechanism . . . . . . . . . . . . . 129
8 Theenumerationofbialternals.
Conjecturesandcomputationalevidence . . . . . . . . . 130
8.1 Primary,sesquary,secondaryalgebras . . . . . . 130
8.2 The ‘factor’ algebra EKMA and its subalgebra
DOMA . . . . . . . . . . . . . . . . . . . . . . 132
8.3 The‘factor’algebraCARMA . . . . . . . . . . . 133
8.4 ThetotalalgebraofbialternalsALAL
andtheoriginalBK-conjecture . . . . . . . . . . 133
8.5 Thefactoralgebrasandoursharperconjectures . 133
8.6 CelldimensionsforALAL . . . . . . . . . . . . 135
8.7 CelldimensionsforEKMA . . . . . . . . . . . . 135
8.8 CelldimensionsforDOMA. . . . . . . . . . . . 136
8.9 CelldimensionsforCARMA . . . . . . . . . . . 136
8.10 Computationalchecks(SarahCarr) . . . . . . . 137
9 Canonicalirreduciblesandperinomalalgebra . . . . . . 139
9.1 Thegeneralscheme . . . . . . . . . . . . . . . 139
9.2 Arithmeticalcriteria . . . . . . . . . . . . . . . 144
9.3 Functionalcriteria . . . . . . . . . . . . . . . . 144
9.4 Notionsofperinomalalgebra. . . . . . . . . . . 147
•
9.5 Theall-encodingperinomalmould peri . . . . 149
9.6 Aglimpseofperinomalsplendour . . . . . . . . 150
10 Provisionalconclusion . . . . . . . . . . . . . . . . . . 152
10.1 Arithmeticalandfunctionaldimorphy . . . . . . 152
10.2 Mouldsandbimoulds. Theflexionstructure . . . 154
viii
10.3 ARI/GARI andthehandlingofdoublesymme-
tries . . . . . . . . . . . . . . . . . . . . . . . . 158
10.4 Whathasalreadybeenachieved . . . . . . . . . 160
10.5 Looking ahead: what is within reach and what
beckonsfromafar . . . . . . . . . . . . . . . . 164
11 Complements . . . . . . . . . . . . . . . . . . . . . . . 165
11.1 Originoftheflexionstructure . . . . . . . . . . 165
11.2 From simple to double symmetries. The scram-
bletransform . . . . . . . . . . . . . . . . . . . 167
11.3 Thebialternaltesselationbimould . . . . . . . . 168
11.4 Polar,trigonometric,bitrigonometricsymmetries 171
11.5 TheseparativealgebrasInter(Qi )andExter(Qi )175
c c
11.6 Multizetacleansing: eliminationofunitweights 180
11.7 Multizetacleansing: eliminationofodddegrees . 189
11.8 GARI andthetwoseparationlemmas . . . . . 192
se
•
11.9 Bisymmetralityofess : conceptualproof . . . . 193
•
11.10 Bisymmetralityofess : combinatorialproof . . 195
12 Tables,index,references . . . . . . . . . . . . . . . . . 198
12.1 Table1: basisfor Flex(E) . . . . . . . . . . . . 198
12.2 Table2: basisfor Flexin(E) . . . . . . . . . . 202
12.3 Table3: basisfor Flexinn(E) . . . . . . . . . . 202
•
12.4 Table4: theuniversalbimouldess . . . . . . . 205
•
12.5 Table5: theuniversalbimouldesz . . . . . . . 206
σ
12.6 Table6: thebitrigonometricbimouldtaal•/tiil• 207
12.7 Indexoftermsandnotations . . . . . . . . . . . 207
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
AugustinFruchardandReinhardSchäfke
Ontheparametricresurgenceforacertainsingularly
perturbedlineardifferentialequationofsecondorder 213
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 213
2 Thedistinguishedsolutions . . . . . . . . . . . . . . . . 216
3 Stokesrelationsforthefunctions . . . . . . . . . . . . . 221
4 Behaviorneartheturningpoints . . . . . . . . . . . . . 223
5 Theresidue . . . . . . . . . . . . . . . . . . . . . . . . 226
6 Stokesrelationsforthewronskians . . . . . . . . . . . . 228
7 Stokesrelationsforthefactorsofthewronskians . . . . 229
8 Resurgenceoftheseries(cid:2)r(ε)and(cid:2)γ± . . . . . . . . . . . 232
9 ResurgenceoftheWKBsolution . . . . . . . . . . . . . 238
10 RemarksandPerspectives . . . . . . . . . . . . . . . . 240
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
ix
ShingoKamimoto,TakahiroKawai,TatsuyaKoike
andYoshitsuguTakei
OnaSchrödingerequationwithamergingpair
ofasimplepoleandasimpleturningpoint
—AliencalculusofWKBsolutionsthroughmicrolocal
analysis 245
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
YoshitsuguTakei
Ontheturningpointproblemforinstanton-typesolutions
ofPainlevéequations 255
1 Backgroundandmainresults . . . . . . . . . . . . . . . 255
2 Transformationnearadoubleturningpoint . . . . . . . 261
2.1 ExactWKBtheoreticstructureof(PII,deg) . . . . 261
2.2 Transformation theory to (PII,deg) near a double
turningpoint . . . . . . . . . . . . . . . . . . . 265
3 Transformationnearasimplepole . . . . . . . . . . . . 269
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273