More on integrality

SupposeAgidg+SynCA)

and(Ford > 2In,FA

Lecture 12=Infact,

itis

finite.

0conearChau (a]shownbecanthate+gidg

11 - 1asaconnection

aprincipal Gbundle Pa,

aconnectionon

Wo and An
Do on

Pon

is

Wois

1

proved below

someKy.
st.

=.

means

[Wo Wol of So withandtheresugidg ).

thatthegauge potential defines

principal G-bundle

aon+

at

that

Spa Chau (a]
AFA-0

that

so,

implies Ashown

it was,Spalasters

asisgiven bymap

900

given byrelatedby the gauge transformation by:

onanS&Indeed,

open coverVon Uo + G,

gauge potentials
onGoothe,overlap Von No
Ao=A300s.

·In the presentcase

Vo =R!

Un=grodefines

a=a

Ao

=

&

=1SUS,&Aneighborhoodof 0An,=Og

G-bundle with

It also defines

Then,

99

ach(E)/joch(E)/0oa

connection

E

Vector bundle=Chr(A),=0=onSaPX,V,andSchE) Schu(A) ==d#)chaAwith FiberV.

At this pointuse

Atiyah-Singer formula for thetheDirac operator

index ofmethodwe,(which can beby Fujikawa's

derivedS&M path-integral) :orindex (D SY(E):+SYE))=Sch(E)(TS").

ga(TS %Claim
frof

If

tangent

bundleTI(**is=

areboth&+

trivial as TIR

On theOther hand,*onaA

So-can+thedecomposition

hasis,

Soand theexpressed( 4-1R*.
topologically trivial

trivialization ofbeinunit+4)

normalthe latter .

inas

1

IR*t ,

sphere inNN

field provides

Pontjagin classes

unit

normal bundle ofIga andThe former

theof IR* restricted

theisTIR

vectorS"as/ga = TS

where NNote :1realizewe*=+...

terms of(The total

Pontiagin classP(3 on)Applying

this3to=

TS9

**(a)

1 + P , + Put=p(5)p()=

TSPEN (ETIRofP.

M,Ige).

.

andN

using triviality
we,

find,

u
~

1p(TSY=)This

provesThus

theindex (Nand

1ie=p(TS) + (N)=Satisfies

...

that,

Pj(TSisACTS 1=

Fjz1.

0Il.

index formula says:

S4(E)Since the index+SY(E))=Sch(E).

ga

isanintegerSache) Sira#(,char(A]isaninteger.

Remark

In Lecture 12

it was,

SpaCha(a)This
isisfor theassumed

Wo & WowhichS

todivides

ch(E)

ga

configuration (#)

The same.

canbederived

So

wecan

general G-bundleonbeneighborhoods ofballs&" along=Si

d2oa(d-1)-sphere,

EDo

&-%

I

Don D:= ga.

-Il,

J Do Way,v(to) + We (AddNSanga8+"-&D8 -ga=(Wa v(Ao)-Wav(Amb),DodWa v(Ad),&takechar(Ao] +Cho v(An]dWr v (Ao]=which An = oinaSo=War(gg].

assumption :anTo constructlim)

Reoparticularinwithout such=shown-HererecallweCS form

Ao=

An00

andgauge transformations

underchange ofthe:

Wat v(AS]-Wa v(A],,Way v(5dg] + daz v(9 A).

=

Then

we

find thattheright hand side isv (900'd900)(WatSan+,& &-z /900 Ao),,

asThus,.

,& S4=p(O,Sch(E)sa=SWar vlgoodSor),.

say-(However,

the assumption (#) does

We can acheive

Woarethatvanishingly

notloseany generality:

by taking the limit where Do andsmallneighborhoodsofDeS ?

(Proof ofSuppose FAfor=For Face Ordraw

Ig·If Gise.

any path V,on

Op Su RRP/1kR]
from Vo =1=

matrixatoo

U=group like SUCM SO(),,,USy(n),the solution is-9t
where=(fat VI.A (v)Pexp
theSince ORESMxIR

issimply connectedFAEOpath Utmap.

onORAnd we,gecang : Om - G.
exponential

path-ordereddoes not.

if d > 2

dependdefine g(x)tI,

isaOp.

doe.

Pexp

as+

=

gi = A(t) with theinitial condition 90.

i.

,.
gidg if d > 2+

Pick a fix any point.

solve ODEeimplies A

conearo=near D0large R

someandFA:

gidg:==on92.

,the choiceThisAlonIofdefinesA