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