moduli space of connections g - university of notre dame
TRANSCRIPT
Moduli space of flat connectionstin a principal bundle t.PL
G
a connection A or P Q G is a splittingnap for1on
theexact seq o v TP T't TM oe
Thorp cTpA
KerA
require UgEG pep Ap Tpp Vp
Ap vg ApgCvg G equivarace
Let g LieCGAdg Apa Apjug
pxg Vp z ftp.ocp expctzs
Taking HomaC P x G we see
go Home TM g Hong CTP g KengCg g
Aidg HongCT TM g
IT TM TMso Fiso Home HTM g 1 TN TN xnPE gT Moru.rs mCTM Pxgggg
I TM
uR car Adp Icm TCM PE 8r car g byabuse of notations
A connection da CM g n Msg2 1 3 da i A
wedge ofhornpart E ofA s z kepar2 8 z
TA D Sas Ez zda dat daCda Tana
d Agde And It A a 1A a
dana AAA an 7a didat'ztaxAT a 2
for A Fa dAt TAXA cu naturebyFrobenius thA is Qatif Fa o Thor CTP
is an integrabledistribute
def A map betweenprincipalbundles Pdpis a G equivariant bundlearap
a gaugetransform is a bundle isom P IPKp p g yep h 61ps h p.gl ph 1h g l
Erp GQ P P withactorbyo P g TP a TM e E p Esgoth
feet 14 140din Pgtpg loecpg
ygp.geTP TM o
Aif A B are a connectuson P AND if z E P sp
sit y A DNotes if Fa o then Fe o e A TP Pig
he f AEDG R ut M G p Mx G
only corrections n th atthetrue G bindle
ICM R R M Fa da A AT o
let M R AER m l dA o
An A daT
4 A 4 95 d9Holonomy n P with connection Arecall 8 o D M and P CPrcoPtr Poco Pro rcoA principal correction PtsCp p g
Pt Cph pt Cp ha p gh pLehigh
Help Air g HotCA r EGq p s P Holecp
ConstructionofillcinMCM G a CP A principal G budles over M
with a flat connector AT
gauge trashUI µ M G HomCri M actsbyyugatoP AT 1 tr Help Air
Chair this is abijectionneed to check i gaugeequivclasses
set1Tdk all M G principalbundles P ith feet A hygauge
equivalenceThe MCM G Is HomG M G IG
acts via conjugation
P AT a CERT Hot Cairgtoget a greyhem
peD.tlolpCA r onlydepends on RTbijective
E G abelianrun
the conj trivial UCM G HomCT M G C discrete PdHomCH G M
H M G NEM f eEmBETif G IR 112
HondtM GKgG Uci o
Marcum G tflet bundles41 HYM Va torsionpetof myan I Een LIF n H C iRssDllbund6s3L
H2CayI71RtPMC1Rp3uc
pt s m oftsi Uh
Ga Suh M S Suh a
C Ucr M torus c
G iMcm Saas L a.be SUN lab ba a
4,1 i lC i i
M G fixedT a cell decenp UCT G LU a cells GI ruled
Yeezy
u p fFgiou3GI 1 threadedge e
ugue p U e gasU e guy
Symplecticstructure an Llc 2 G
Atiyah Bott 2 form M 2 compact oriented sa face withoutboundaryG Liegreup with bi invariantmetric
Fix a Iso class Pym D P A correctors d CA Adp
AE A Ta D d cm AdpAB 2 form crab cap Ld f p where a p ET A
GCP Aut PLie GCPD ICM Adp
Acm Adp ICM Adp RF h 1 7 CF h
Lie G P Erich AdpA Lie GCPDP A 1 3 Fa
claim it 5 GCP is Hamiltonianwith moment nap f
µ o flat connections in PSyrup reduction moduli space µ 2 G
IAD syrup structure
The for X compact Kahler C Lie group admitting bi 2 ua metricher ke CT ll GHq has a syny structure
Yael KarstenAndy preet of
syrupstr a
medal space