foresmall.Pmethat.int wehavelrlltimanypossibtities.BY

Geometryand Topdogīcd

^Fīeld

Thary

^F

^09221011

莊

秉 熏 ⼒

Week 1 Hmework

-

( Ìxti

^EP

)

Exerc.se/Z(x.E)i=fdxe foresmall.Pmethat.int

ZK.cc )

^⼆

ecomectedgraph.3-regular.de

g

^{= (}

-3! ⁱ Eǐ

.

1 V^{= #}

uevtiesoft

✗

E Aut(I) E ^#

edgesoftpmfUsmgTaylorexpanson.wegetZlx.si

_nw

Éfdxé ^-2

^(⼀n

^炒

! ^ˋ 3812

•

y

Nu that

fciiédx

⁼

^(f)

^。

⁽

^#

ofcontractnns.fr/Wewanttoamputethenumberof contractnns.fr

^{• , Notethat}

thecontractm.fr

^•

endsupnitha3-regulargraphwithrvertices.Theu.fr

^{each 3-} regulargraph

Eleītheramectdordnomected )

,

wewanttocomputethenumberofcontractimsof.to

T.ForeachedgeofI.weaddanewueruxonthnedge.Saythenewgra.ph ^É

^{. Gnsider the labded r •}

llahtheueruesandthreenearbyh.ge wehavelrlltimanypossibtities.BY

^{the" ) , BurnsidésTo}

^form

^{lemmaalabded,}

^É

^,

thenumberofcontractnnsofr.to#is(r!l(3!)rAut(I'-Hene,we

hae-txc.ie

)^{" ⼀}

☆

²

Zla E) ^⼆ 三

fdxe

ⁿ !

=

[

^""

fdxe ^必

^{)" ⼀}

neven na.eu

"!

(^⼀iE)^{" 列2} ₍n! )(³¹)^"

=

[ I il.nl

^.

(

_{1 Aut(I}^'₎ ^。

_李

nieven Ei

3-regularnithnverlosl-3.ie _ǐ

¹

⼆ 2丌

.

三

^.

X I^: 3^-

regular

^{XE Aut(I}

')

(

Nouthata3-regulargraphhasa.eu

^uertīas

autmatīcaly by ^theformula

3 V^{= 2}

E)

Fora3-regulargraphI.writeE-a.IT

taznt-taIK.whereEaretheumectedompmentsofI.Also.LI are3-regular.NU

^{that Aut (I)}

isgn.eu#lAut(Ii)l.(ai!l.Hene,

^T

_T

by

^…

吐 ^{= El}

Èn

!

⼀

El

(^-3!

ieǐ

¹

Therefre

^{, Zla E) ⼆ 2丌 .}

^三

^.

X I^:3^-

regular

^{XE Aut(I}

')

⼆

ǖe ^吾

^⽣ ,

Thisnotawn ^的 defmed

by Tykr

expansīm ^, #

Geometryand Topdogīcd

^Fīeld

Thary

^F

^0922011

莊

秉 熏 ⼒

Week 1 Hmework

Exerāse

^{3 Let}

Wlzi-Z~lbeaquasi-hmogeneaespdynomidwithwlizi.i.TN

_ZN ) ^{= 入} Wlz )

,

入比

, Also

, assume that ^Wlz…, ZN)⁼ 0

defmesanisdatedhypersurfacesigularity.lt

^commutatne

algebrafactstatesthatasequenea-EALEI.2-dmlAD.ua _{regular localrig}

ABregularifaudonlyifhghthl-dmA.Thus.TW

^}

formaregularsgeencemeh-ZNJifandonlyifAhiwis.FI

uw

dmensīond

^{, Hence}_,

weknowthatkwlformaregularsguenceofEG-ZNJ.Def.me ^Pi

^{= "不 不了}

gw_ jws.S.nu

^{孔了

sregular.wehaveshortexactugueuceio-spi.FR

^{" >}

_i

> 0

.

Bythe

additity ^of

^Pomcaré

^pdynmīal

^{and wt (我 W) = 1 - qi , wehae}

Plri

) ⁼ P (

我

^" ₎ ^- t

"

邸

( ^R"'

)

⁼ ( ^{1- t"}

到

P( R^") ,

Bymducton

^,

weget

^{PIR) = P}

(

^"<不_我W^到>

)

⁼

Ǚ

^{( 1- t"}

^到

^。

_PKGZND

=

fǐ l-tl-qyiill-ttij-Hene.lett-T.wegtdmR-l-1-G.FI ⁽

^G

Also, P (R) has

smeiymmetry

^{" , More}

precndy

^, theoeffīāentof

"

and that

of

^t"⼼ , where D ⁼

Ǐ

_1126i)

arethesame.EIIndeed.by _replac.mg tby

^{5' ,}

^weget

,

N N

NP.pl

^5'

⁾

⁼

^P

^.

Ell-s-a.IS?TSEi-s2Gi-lElSG-l

^{丌 "}

^ˇ

^{" _El}

⁵

^SEI比--11

11

P (^S ) 。

#

Geometryand Topdogīcd

^Fīeld

Thary

^F

^09221011

莊

秉 熏 ⼒

Week ² Hmework

ExerazeshowthatinphysicdsensiforT-sf.tt

^{p→ 0}

X(M) ⁼

trliē

^{" = 1}

(a)^"

21

PH-RIGauss-Bomet-chem.MX

pwi

^The

Lagrangīan

isgnenbyiliigiiifFDti-wiij-IRijkeiitkf.TO

getanEudideanactim.weapplyawickrotatonit-s-FT.de

m-fdt.tg.jii-s-fgjiirizgijliri-zit7-s-i.gg ^lūii

^-7

^球

⁴

^少

(

The

lasttemremamsunchangesmuitdoesnotmvdvei.JThen.tk

^{Eudidean actn beame}

lobtanedusigntegratnnbypat.JP SÉJ

^{Ldt ⼆}

% ^{igii +9g (II) tz Rjkeiiifdt}

^⼀

分

^只

Now, we can use

path mtegrd

^to compute SE

,

SEztS-SEN.tn#)tr(-ijFe-PH=fDXD4DFe.SineT=Sp,wehaveFourier expansims.fr

^{E 1,2- . - n}⼗^,

i

⁼

XǕ

^⼗ 三

xie-Tntlpn-oi-4.it _唟 4.ie?TT3.F=yi+ _唟 ^iétintlp

Smu the

Wittenmdexismdependntofp.wemayconsiderthehnitp-0.to

^{, wechoose}

Riemannnormdcoordnateatxoandwehauegj-S.jandI.io

^{at Xo , Theu, we have}

P

zifilm-ntlpwiimmzxix.ie

^{di = 2ˇ [}

^nixǔxi

SEF ! ⁱ

5 m.vn i P mo ^,

smceiarered , wehauexnixi

Eifiidt

P ^⼆

分 ^{( i}

⁺

^{產 iē}

^""

^{川 仁}

^mto

^⼆

^不

_P ^yiēinu

⼆

ZTTIl-mlFEy.im

^m

mto ^,

Then,

weobtanp.SE

^{= "2}

^zixǔxi

^+2T.fi

^{Itmlf 45}

⁺

z

Rijke

⁴

:

4

哈

⁺⁰⁽_p )

m m

P mto

mtoscahgti.uibypMThepathintgerdmeasure.is gnenby

^,

Dx ⁼ dxò^…dxd

丌 dxùidxi (a)^吃 mto (a)^"12

DY-dtid41Tdti-dti.DE ^{dtid T}

^T

^dtiidǜ

mto mto

Then ,

asp

^→

otwe _get ^trciē

^{" =}

1 ^DXDYDFEE

^""可

2丌

2

= mtoT

fdxi-dxmexp-p.mx

(

^vii)

^·

znnlz.IT/d4i-d4idti--diexpGrimi4i.mtojdxidx t.in/d4idEdU--dtexpfzRijke4iyi4Yf

^{。" B}

)

1

-

ⁿ

= T

mtofkiP-ET.im

^{1 .}

T

(²

Timifdtidtidhidūiǜtiti ⁾

mto.la ^岾 _fdi

^{。 …}

dididtapfi _Rjkeiiit

^。

^"

(^2丌)"12

ltkre.weapplychaugeofuariabksbyyi-fxi.7Efi.IE ^MT

and.tk

Bereziūan is

gnanby

^{1 .}

⁾

=

f

^{"12 .} ₇^-

I

^˙

czjnfpfl-R1.ph

^{1 ⼆}

mnnh

¹

^Pf

^{(- R 1}

^

x

M

x

.

Tkāfm₎ ⁼

é Üand ^Tnf

^-1圳

)

⁼ e

E

^F

#

m^--1 ml

'

T

^{T d}

miam

⼆⽉

²

_{by tahg}

_dss。 m

(f) ^5,5

^{(s) =}

^Ěl

^T

T

mn 2下m _

Geometryand Topdogīcd

^Fīeld

Thary

^F

^0922011

莊

秉 熏 ⼒

Week 3 Hmework

1

Rijkei ^Ìtkf

L-igiitjgfFD-ci-D.ci i)

^{- z}

① couaūantdanate

Dii

⁼ y ^+[

品 ^走

4

"

Super symmetry

^{: Si =}

EI

^-

āi

si-ai-iisi-Elni-Ij.it/sg.j=gij,kki-E4YsI=E(-Fi

i-Ij.it/sIii=Ii,kki-TYsi=E(rix-ip,kXkT-IiiP48-IpitP ^F)

si-E-fx-ip_xkft-Ii.it

^-

Ipi ^{扣 椡}

Show that

Sf

^{Ldt ⼆ 0,}

pnof

S ₂^'

gij.kki-TYiitgijeri.it

^'

^{) i}

n 12 11 13

szoiigij.de#iitEfIIiii-E4kii-E4Tiiii)

1 5 16

trigijd-riii-tjiii-riitiii-I.it

₁₃ 16

^{品 如}

⁴

^幾

^4"

+

T.elr-i-t.k-i-Ii.it iii)

11 ^{14 15}

+

Ǜmii

^-

āilxitǜmlei

^-

āuyi

16

⼗⼆点 ^她 Elrixm

^_

崐 ii)

¹⁴

"

1 1 =

igij.ee#ii+zg.jEii-gijiEi-g.jiIiiExm=i _gij

_,

KETiitzgijeiitgj.kiktiitgii.it integatnnbyparls

^,

-

ilgei.mtgim.e-gme.it ^Yieex

^{" = 0.}

-1

12 ⼆

zgij.it iitgāiǜmxěiigij

^,

^{kātkiit Ē}

lgie.mtgim.e-gme.iiii-o.BE

_0,14 ^⼆ 0 B dear,

15 ^⼆ 0

using 我 唬

⁼

^{Ì (}

^我

p.qtgiq.p-9pqil.16-fgij.koiii.fi gijāiiǘtitigijiǜmāii

-_-

Fgijiiiit

^Ē

lgjp.qtgjq.p-gpq.nl#48itE(9ie,mtgim,e-gme,i)Tif

^⼆0.

ˋ __ˋˋ ˋ _ __ .ˋ

________

的 ^{- _ -}

iskipqr

⁴

^Ttyr

⼆

T.Rijke.mff-etliitkftRijkeElfi-Ii.it/iiTtRijke ^iētrii

^-

_唬 ^Ǜtltkf

5 +

Rjkeii

Elfik-I.it#1TtRijkeitYE(-fxe-Iefi)PE7Notethat _Rijkem

^⼆

_Rijke

_,

m-tnRrjke-fnRirke-TnR.jre-l.in Rjke

^,

8 ^⼆

Rijke

^,

mftitliitkf

⼆

Rijke

^;

mktitniitt

⁺⁵

^品 Rrjdctitliitt

⁴

n-tiiitotjRirklti-tii.it ^⼗点 _Rjre

+5

品 R.jkrff.it/4ii4Tq'

7 10 =

( _Rijmeikt Rijkmie ^{) ( Eti 炒 4T 4kt}

⼆ E

Rjmeiktiiift

^E

Rjkmietiiif

Yby

Bmachiidentity

tā

Rjmeik

⁴

^吠 Fiftā _Rijkmie ^{4"4 Üǜt}

^{⼆ -10}

_mpt.es

10 ^⼆

0.to

_by Bmachiidentity

9 ^⼆ 0

smcewecanchangemdexmanditoget9-9.gl

issmilarto9Theremammgta.ms

^are

T

利

^,k

Ef

Iǎit^{" -}

āiiiiii

+

Tijtēiitiiiiipixktliiimii

^-

^ātii

-

迎 品 ^焱 崐 ^{武 州}

-

IRjkeEAiiff-RijkeiE.fi iykftR-jkeiiEAxkf-Rijkeiiii.it

Thete.ms

mvdmg

^{E are}

T

利

^,k

Ef

Iǎi

ⁱ⁾

tfgijliapixkittiii emiixi

^⼀

^{迎 品 焱} 崐 ^{V8 )}

'

( _Rjke Efiittt Rjkeii ^Efixk F)

-

z-FgisIjitgsjI.ie#FIiitm.

¹⁷

tfg.jlIEI.kxkTGIiim.a.ie ^{i) 我 迎 品 焱 崐 北 18}

-

gemlIii-I.tt Ijǐ iii)

^.EE

iitt

17 ^{⼆ 0}

smcewecaninterchangei-ktogetn-17-mii.it

- _ -

tiiiitt-igiji.it#f-EF9emik.jtEf9emIi!Ijix

Efgem

㗁

ii-gemliii-I.i.jtiiii-I-iil.EE

iitt-o.Sinilarly.thettermgwes-T9ij.ci _iiiiiiigjē

_唬

_呟 ^如

⁴

^幾

^{4" -}

^FIÜ

_em

,kēykxeym

tz

Rijkeiēfi

^T T

⁺

È

Rjkeiiiēfxe ^{= 0 .}

Hene _,

SSLdto.TogettheNethercharg.weassumeE-E.lt E-isafunctmoft.tn

^{, ) ,}

SGLdt-fgijei-eilitrgijiyni.it#)ttin(:t-et)i

^{- _ - pq}

bychangigl

^{的m ,}

tgijiii

^dt

______

neei-rerr-fgiii-g.jiiidt-friif.gg ii)

^-

^{Èfigiji :)}

Netherharge

^{: Q}

^E

^*

Geometryand Topdogīcd

^Fīeld

Thary

^F

^09221011

莊

秉 熏 ⼒

Week4 Hmework

Gnputatn ofthetwo-pmtcorrelatonfunctnonofvertexope.nu

^tor:

〈 eEKX (^七1,97 eikzxltz.sn _>

= <

ol Tlie

^{"9"⼤9) i: e}

ikzxltz.si ]

^|。>

Fist_, weassumeti-tzandth.us

thetmecrdemgpwductisieikxlti.SI/iieikzXltz.Sz):.3=ei(tj-gj

^ikzxltz.SI

ie

"""_""

iie ^: 1⁰⁾

g

⁼

eiltjtsj

⁾

expl-kiz-lx.nzi.int

E n_n✗ ^{e""" e"我 -}

^KÍ

¹

^xnzft.IE

^…

"

PEvnn.explilix.nzE.in

^{e"" e"⼀}

exp-kzztxnztizz-nksrim.in

"

10> sme 不 ,不 amilate ^⼼

andp.lk?=Klk2ilk

⁾。 ④ 10不 ^,

以1

Now_, wecomputetheope.no

-

KÍ

^{m 1 ⼼。些}

⁹

^{忶 ,}

印 _上

…

ixnzil-IT.expl-KE-lx.nzE.IE

_m.in

Themlynon-triidcmmutngrelatnnofh.tn

^{are [}

xn.x-nJ-n-kn.in?Then,smceXn.IamilatelDn,theremainig

^{term.fr eachn.is}

e m

nmz.IM )

^,

kifi.li/lX-nZ2lEml!Enm.ltZ'MMx-im,liilli-mlEml!En

1 ^{nm ml}

=

ema.mil [ m-anne-x.nl iǚlǜ )

^{- m… … (}

⼗

三

emem^!

ǛT ii-mlm-n-lm-ann.at

Kkz

(⼆)

^{" 。 Kkz} (

Ēǐ

xletm-l-jn-z.vn_Z,

= e

kzziixite-mzizlk-jnjj.io

[ ^{⼼' !}

(

^En ioljn^)!

(

En

Theu

, we

和

✗

ie

"""_""

iie

"""""

: 1^{0) =}

expl-kiz-lmzi.it _0.Em.in

^e"""

^è

^戒

Kkz

(⼆)

^{" 。 Kkz} (

Ēǐ

^x

T吲

eim

^Zi

kznjni.ci

^z

^liixitē

^{" 百} j.io^三 ⼼^{' !}

⁽

^rn

^āil

^⼼。

^{𢇃 川}

è

^""

(

^{lk.io以☒1 10不}

)

Thus.to

fmd ^<

oli Ǜ

^"""_iie^""""" : 1⁰⁾

wenedji-oandktkz-os.nu {

¹⁰⁾ ,

xi

^⼼ ,

成

⼼

}

,

aeorthogonal.Hence.wegetk.kz

_

之 ^{" Ekkzt,}

恬

(

^王

川 rexphīzn

^{i e}

Texp 1点

^{- u z, .SCk.tk) ,}

Thnanuergesto expff 1点 ^{i 1 到}

^"

_⼀点 ^{# 㢦}

⁺²

吵

expl-fllogh-Z.lt/og(E-E)DkiKz=K-Zz)E-E)J2-znS(ktkz).f(Z.-Zz)(E-)J

=

^t

^,

*

Geometryand Topdogīcd

^Fīeld

Thary

^F

^09221011

莊

秉 熏 ⼒

Week5 Hmework

Exeia_lett-TFTz.IS

^{.5)= (} " "' "比

icoordnnateoftoruswithz.IT

⁷ ZT ) 了 ^三 3^{+1 三}3⁺

t.Assumethatf4.lt

^{, S)=}

^Eia

^{4.lt, S+2 下) =} eiibtlttzāiz , S+2不下

)

出 ^{(t. S)= e …ā}

4ft

^{, S+2 下) = e"⾼ 出 lttzāiz,}

stz.TT/CheckthatperodicDiracFermonscoupledtoflatconnectnn

A^{" = TT}

( b-tajd5.AE

^{「" (5 -}

īājds

Tz

T.ms

^m 9

andsdvethegenerdsdutnn.pro f.Wewanttofndthetwistwlt.si

suchthattlt.se^""' B

"

perodo.ie

^{, 4}

^!

^{(七,512刊 ⼆ 4}

^!

^{(t.SK 4}

^!

^{(七⼗2𥑮 , stzāti ) , 4!(t.si}

4!(t.stz.FI) ^{= 4. (t.se}zne

""'""'

= 4_(

t.sje-aewlt.si

⁾

火 (^七⼗2TTz , StZTG) ^{= 4(}

ttznez.stziityewlttZTTz.se

^{2 TT}

= 4⁽t.SI ezābe^{W(七⼗2TTz.se2 TT,1}

Let

WU.sl-uttVS.Theu.wegtfv-taia-OU.lzitz.lt

^v.(^2TT

)

^{tztib = O} .

1

(

^ibtiti

a)

Sdve and

和 {

^{" 在}

,

V=

iaina-blt-iasisperodic.tn

^,

比

t.si ⁼ 4. (t.se ^在

Now

wegeti.LT

,

changnguariablesi.IT

^-_b)

⁽³⁺⁵⁾

^{, t= 不}

^(3-5)

^{i ,}

t.no

Tatbs

^_

^āatb

5

e ^T

T.z.ie

^T -

Then

,

t.heflatcomectnnisgi.eu by

7 ē^{""" =} _dt _

āūbdstāatbdī

Tz Ez ^-

ns.lsmilarargumentshowsthat

^{7 ē"}

^"

^{" = dt}

^TǛ

^{ㄘ-5以下 -5瓦}

^dī

^,

能

Now,

wesduethegeneralsdutnto4E.S.nu

⁴

^Íltsl

_B

_pantc

_,

wehave4llt.sk _蝱

_tae^"

{ ^比

^{(t.si = 三}

^Ectlè

"

{

^nez

出

(t.sk ^三

Emé

^…

4flt.si

^{= 三}

^Titlě

^"

nez

nEZ.ms

^{_ 出"b)} ttias

Then, cect.si ⁼

e.eu

[_EZ 出(七)

_

ihablt

⼆ 三

t.lt/-eT2.eErsrEZtaiiit-icis4+(t.s)=IFn.eiS.etrnEZiwt-

5)

t.is

⼆ 三

f)

^. e ^{5 .} e

FEZ^tǎ

I.

(^{t.si =} [

Ene

^{" .}

ě

^"""

t-iastznc-zik.cn

^-_b)

t

⼆ 三

Elt

^{) - e 5 .}

^eirs

rEZ^-a

_

⼼

的

_ttiās

4₊(t.si ⁼

ztn.e-ins.eu

_EZ

_

iliiā-5)

t.e.is

= 三

Ǖtie

⁵

FEZ^_ǎ ☒

Geometryand Topdogīcd

^Fīeld

Thary

^F

^09221011

莊

秉 熏 ⼒

Weekb Hmework

Homewokl ShowthatEBoftheform-ollyltdt.ly Chird superfidd

^{): +00}

^I

Flg^{⼆ 0⼟,)} , ⁼

_

iōō

.

Fermīm

functonlfarefunctnsmxti.co

^4=0'0

^ōē ) prof

^:

^Suppose

^that

在 ⁼

ftōf

+0ft

Ǖftōf

tōftēf

⁺

Üfētōftōft Ǖf

+

Üf

^{_ +0⼗⼀}

f

_ +0^⼗⼀

f

^{+== +0-} _f^__ +04f

- _ -

fti Òftòftt

⁺⁰

f.it Óf

⁼⁼

tiōlōstftōzftō

⽟

E)

^-0"

ft-t-0-ft-t-0-f-tidlo-jf_tfjf.it

^0-2+f-

) +074

+I.

042

_tfit

5

正 ^{= -} f ^{+ i}

02ft Òfe

⁺⁰

f

^-_-

Üf

_

ti 0

( Òzftōzf

⁺⁰²

E)

^-0"

fētòft

_ +0^" f^_=⼲ + i 0

(

^{0⼗⼆2ft⼆ ⼗}

^Üaftī

⁺⁰

5_f-7-0-f4-i.042_fthen.wehave.fi

^{f⼀千 ⼆ f⼆千 ⼆ f__ = 0} _, ftttii-fi.co ftt ^⼆⽇+f^_

, f⼗^⼆千 ^⼆ I2tf.tt^⼗⽇+5^{== 0} fwm

可

在 ^{⼆ 0} and f⁼⁼ f+ ==

fī

⁼ f^{_ = 0} _, f^{_= t E 2f = 0} ,

ft-ti.2-f-O.fr

⼲ ⁺ I 2

f

^{= 0} ,

fti.2_ftt-ofwm.EE

^{⼆ 0 .}

Now_, 在 becomes ft

Òftōf

^+0"

fttòfttōf

_

tot-f_tot-ft-ittf4.Tofmdt.tn

^,

F.weneedtheflowigk.ru

^ma;

Lemma For

pdynmid ^functnn

glz.nl/wehaveglyt,j)=g(xTxI-i2tg0ti-i2.g0-22+g04.subpf: arehearmg.soitsufiestopwethecasewhengBmmmid.ie

^{Bnh sides}

gh.zzleti.gl ^yt.gl

^{= ( i-}

^{ii) (iii)}

^{b = (☒Eai}

^{(iii) (ix)}

^{"- bi(i)"⼼}

⁾

= (州(

x-P-ailxtiixPU-biliilx-ME-abliilx-M04-gci.il-iigE-is.gr

^_ sstg 04 ^☐

Now

, we Iet

4

^⼆

flyt

^,

51,4

^{⼗ ⼆}

ftlytj

⁾,

4_

= f.ly

Tj

^{) ,}

F-ft-lgt.y7and4.it

^{⼆ 0 ,}

wherewevieuf.ft.f.f-asfunctnmoftandsubstituteey.tn

, wehave

4 (i) ^+0'4 +151

^+04_

(i)

^{+00 Fl 1}

lemma

f-izuf0-i2.fi

^-2.2+1

-04

+

Ò

f-ijf0-i2.f0-2_2tfdlt0lf-izuf.U-i2_f_0-2_2-if.tl

⼗ Òō

lft-ihfott-iaft-0-22.tt ^-04 )

= ftf.it

Òtf

^_= 0^⼆ ⼗ f

04

tft

Òtft

^{-0⼗⼆ ⼗}

fōtftī

^0⼗⼀ tft^-0⼗

⼀

⼆

,

* Homework² Fmd

theonseruedcurrentsofsskmtSwviaNoetherpwcedre.pro fwefrstfndthevariationofchrdsuperfe.US

^.

Write

t.ly#00F(yf),S=E+Q-E-Q+-EE+EE.Tuafunctnng(y),wehaue

Clnal

superfeld

^{mto E= 4(}

^扪

⁺⁰⁴⁺

⁽ⁱ⁾

⁺⁰

2

aitiōjtiōyog

Qg

^⼆

otiō

^⼦_我^{± g = 29} _对

₎

^{⼆ 0 .}

Eg

^{⼆ ⼀}

Jō

^-

^iō

²_{j g} ⁼ _ai^"

⁽ⁱⁱ⁾

^-

^iō (

_,²

州

^{9 ⼆ -2}

^iòzg

^,

Ncte

t.hatSEisdsochirdsmcesauti-commuteswithI.ie

^{, IĪSI-5旺正}

0.Then.SE

⁼

S4tlSoT4tt0TS4dt@74_t0lS4_1tSl00lFt00lSF7.iziE 02-4-2 ^IEÒ

²

₊₄

^-9.4+ ⼗

Òlzi

^02-4+ -2iēō ²₊₄

+1

t Et ^4_ +0

lzi

^02.4_

-2

IEÒTH

+

(

^{- E}+0^{⼗ ⼀ E.}

01 Ftòōti

^{+02F -2}

IEÒIF ⁾

_。

⼆ (44^{- E}-4

+1 tōlzi

^{E 2}

₊₄

⁺⁴

F)

⁺⁰

(

^{-2i E 2}4 ^{+ E-}

F)

+0

0-zigat-z.EE

²+4

⼀)

Henu,

thecorrespmdngvaviatnon4.YE.Fisgn.eu ^by

{ ^{SYÉIZǗHEI 84}

^{⼆ 千4-_-4+}

Alsabytakngampley ^gugate

^{, wegt :}

^SIÉTIIIEĒ

^SFTEET

SF ^{- _ -}Ziā

2_4i-z.EE

²+4^_ _

SĒ

^{- _ -}

ZIG2-E-ziE.si

,

Now ,

wearereadytoapplytheNoetherpwcedure.to

fndtheansenedcharge.OuractonBS-Sk.int

^{Sw = fǎx L}

=

fi 12412 -12,412

^{⼀ ⼼4112+}

^EE

^的。+2,14_

^t.EU

^。-2,14⼗

⼀ ⼼₍4¹ 4比 ^⼀ ⼼⼼

EE

+1

Ftūhi

^-

The _equatnn

of

^{mtn 乃 F -_ - Ū'}

^(I).

Forsmpliāty wefocusontheEttemsmSL.Themeanngfultermsofst.SYE.SF.SE

^, are S

4

^{= E+4_} ,

84_⼗ ^⼆ Et

F.SE

^{⼆ 0}

SI-o.SE

^⼆

O.SE

^{⼆ E+22_}

t.SI

^⼆ )_。 (^E+4

⼀)

²。

0T

^-2, (₄₄⼀)²,中 ^{- W"(}4^{) (E+4}

⼀)

^Ū

年

⁾ ① candout

ti

(

G^,2⽇ ) ⁽²。+2,14_ ti

E

^的。 ①

- 2

.

bythequatnnof_i.net

) (_EF^- z)

- W^"(4^{) (GF}1 4

-.-

^Ū

"

(I)

(

G.ziz.tl

⁴₊₂ ^{mtīm F -_-ŪID}

⼆ 的_。G) ^{4_ (2}。47 ^⼗ _Et(2_。七)(汗)_: ^{( 可}414^{_ (}

汗

^{) - G (2.七) (2,47}

4

-4

的

^。-2,1

吼

^的。+2,1

4-J.EE 中

^。-2,1

_EIŪ

^'(E)

iiij-i.it

⼆ 的_。G)^{4_ (}汗) ^- ( 可4) ^4_(

汗

) ^{- E}+ (汗) (2,4⼀) +4(汗^{) (2。4-1}

ī ,^⼀

⼆

-

EEN

^-2,1

_幻

^Ū

年

⁾

mtegratnnbypatslThen.fdxL-fdxltako-2.lt

^{(可 Et)}

_[

(^不到

Iti

^{4_ ti}

^EŪŪIE EÜŪ

)

1

^{+ E+12+ 1的。可:D}

⁴⁷⁴

^4__

|

Canceli

Fmdly

^{, smce Et}

Barbitrary.weknowthattheconseweredcuweutsaregnenbyG.to audthecorrespondugcihitt.EE

^-2,1Itti

^EŪŪ

^⼼

啊

¹

^]

^{== 2222}

^-

^{⼀年44-_⼗-ii}

^EŪID EŪŪ

⁾

consewedchargesarefdiciandfdiGISmilaramputatimc.by termsmSL.wewilgetaltheconseweredcuweutsGE.GE focusngontheE.EE

_,

EE.Candthe.ir conespondng supercharges

^,

Next,

wecomputetheconsewedchargeofaxialntat.vn:4 ^-4,4

^{⼟ ⼼ 以牡 ,}

Thevariatonofaxialwtatmisg.ve

ⁿ

by

^{的4 ⼆ 0 ,}

^SAI

^{0 ,}

⾃

⁴==^,

2

ēiecy icE.SE

⼆

FECYE.SI

^{= ⼟}

EO

Then

, we have

SAS-fdic.FI

^{2。+2,14_ _}

^I

^{( ⼦。+2, )}

^{( c.4-1-c.EC}

^2。-2,14+

⁺⁵

^{的。-可 ) 《-4+1}

-

wititic

⁴

_⼗)

^{4_ _ w"}₍₄₎4₊ (iii)

till-icEIE-niniiiii.io

^"_。

⼆

似的

^。

⁴¹

^4_

^{+54 +1}

⁺¹

2.cl

^{- Et}

^我

⁴

+1

corvespondug

^{currents :}

iii

^A supercharges^:

FAY ^指

di.Fmally.wecomputetheconsewedchargeofvectorntat.vn

^:

⁴

^{- e""⼼}

⁴

underthecond.tn ^{Wl到 ⼆}

CÉ

.

4±^- e

'"""^"

4_{_}

Thevariatonofaxialwtatmisgnenbylz.lk

)īea

ye

MKHKEGE

S

v4

^{= 2 e 4 - =} iia

^中

,

我在 ^{⼆ ⼆} (ii) ^{ia 4}±

2 E Eo 2E

Eo

Svf

^{- _ -}

iiaōl

,

Sv4 ^-_- (

E

^{- 1}

)

iaEThen.wehavesvs-fdh.li

^ia

^{4 12 年 +44 )} 北 iiat-2.li

^ia

^{4 12 点 (}

^可

⁴⁾

2.ti.at

ii-i.io ^{- _ --_-}

tcklk-DIE.afl.ckftckf-cklk-ill-i.at^"

¹ iotli-laF.li

^{。+2,14_ - E}

⁽

E ^-1

⁾

^{(2。+2.1 (a t )}

2 + (

i

^{- 1}

⁾ 無

^的。-2,14+

⁽

k

-1

)

^(2。-aiiāfǐ

Z ià^{___ =O}

- cklkt )(⼼(

kia 中⼼ 24

+4^{_ ⼗} cklkt )(⼼(

kiaf FE

。^{⼆ ⼀} cklk-114

"

(

E ^-1

⁾

^{ia 4+4_}

tcklk-nfli-ljiaEE-cklk.lt

^{" 4}₊

(

E ^-1

⁾

ia4_tcklk-HTI.li ^-1

⁾

iaE-fixboallilt-i-I.it

^{) -}

⁽

^{2k - 1 )}

^{( I4.tt}

⁺⁴

₊₁₇

i

Jǒ

+ (

可以 li

^{-4 可}

^II.

^2,

^中

^{) +}

⁽

^{2 - 1 )}

^{( 4.tt}

⁺⁴

+17

consewedcurreuts.kandthesupercha.ge

^{B Fv ⼆}

^{f Jědx}

^{' .}

☒

Geometryand Topdogīcd

^Fīeld

Thary

^F

^09221011

莊

秉 熏 ⼒

Week7 Hmework

Homewak

gij

^⼆

Jijk

^{任,Īl and (}

gijl

^{> 0 ,}

Lkifǎok

^任,主

⁾

^{= -}

gjiiiǖtigij

^{4 旺}

^咩

^t

Rājkētitiǖǖ

tg.jlF-Ii.tt

⁾

^{( ÉIǗFIǕ )}

^,

Show abae

equdity

^!!

profi

^Fist

wehauetheexpressmi-oliM-iotii-ioi.ci

^{, -04.2.2+4}

^年 04年

^-

^iōiii

+04 iō

忙

⁺⁰

E.

É

iii)

⁺

iòiitioiǖtsiiōǖ

^-

^iō

^球

-

ōǖtiōiǖtō

UseyrexpansnnonklI.IT

^, weget

K任 , Ē

)

^{= K (}4 ^,

I)

⁺²

iklt.4T-iotii-i02.ci

-0422+

i

⁺⁰

É

+0

+4 年

^{⼀ iō 4}

¥+04 EÒTI

+

2jKH.FI tiōzǖtiōiǖ ^-0422

+4

ōi

-

ōǖ

^- iōiiōǖtiōii

tisik C) ^(⼀)

⁺²

ijkc.lt

⁾

tispj

^{(⼀)(⼀) ⼗ ⼀ .}

Smcewearemtgraloverthemeasured40.weonlyfocusonthe.co efficient of

^{04 m KIIĪ ) :}

K任^,Ī )

。

⼆

2iklt.4T-2.su i)

⁺

2jKH.FI

^-22+

Ǖ

+2

ijklt ^E)

^-2+

iti

⁺²

_ijklt ^E)

^-2+

Ǖ 球

+ 我

jk

^{( 中,中)- 2+}

ⁱ

^-2-

Ǘ

⁺²

.ci

²+

Ǖt ÉÉ

titiaǖ

^-

iiiǖ tii-I-ittt.FI

⼗四⽇jklttt-ijf.4-I-iii.4ii-4.fi

^É

tziijkl4.tl.is#Tii+i2-f4iI+Tiit2e2E2ijK(4.D'4f ^ǗYIǗ

^_

Note that

nnderthe

^{Kàhler metric}

dEgcjdE@di.we

^{have 不 9}ij ^⼆

gsj Iki

_,

我

9jg.is 唷

^{, and ⽟我}

gjilgsj ^Iei

^⼆

gsīǘj ^{Ǜit Rijeī}

^,

mtegratnmbypats

c. L

Then_,

Lkm-2iklt.FI

^-22+

i)

⁺

2jKH.FI

^-22+

Ǖ

+2

ijklt ^E)

^-2+

iti

⁺²

_ijklt ^E)

^-2+

Ǖ 球

+

gij-2-ii-2.ci

⁺²

^.ci

²⁺

^{Ǖt ÉÉ}

titiaǖ

^-

iiiǖ tii-I-ittt.FI

tgsj 嵫

^。

iii.

⁴

I-iii.ci

ǖiiiǖ

g-ij.is#Tii+i2-f4iI+Tiitgsp-IIjIei-RijeE'4f

+

^ǗÜIĪ

⼆

2jikkh.at?2+4it9ij2I.2+cit2i2jkl4b2.i.2+It9ijii.ii

+

2ij.KH.D-iiiitsijk.lt ^I)

^-2+

^{Ǖ 球}

与 ²

+

gij

²⁺

Ü

^-2_

Ǖ

⁺²_

Ü

²+4

tig

^⼀

球

_-⁺²

ǛǛǗ ^.it Ǖtitsiiii

⼀

球

^+2_

ǘǜiǘtitīiiizktipǘ

+

Rijkēliuhii

tgji

^É

^- 唬

^4:49 ÉIǕǗǗF

⁺

^Ieǖihiǘǘ

=

itzgj-lD.tn ig.jo

^。+2,1

^t.la

^{) 。}

^Ǖ

^{-2,1 Ǖ++12}

^É

^。-2,1

⁽

^吐

^t.la

^{D. 4。}

^上

^{+2,1(D。- D. )}

^{4i-4.it ti (}

^{D。- D, 4}

+

Rijkēliuhii

tgij ^É

^-

唬 ^{4:49 ) (} ÉIǕǗǗ

g.jhizi-ii.si/=-gijiniIutegratnn'ti9ij4i(DotD,

⼆

Yitigijti ⁽

^{D。- D, 4 +}

_Rjkēti

^ǛIǕǛ

bypautgji-Iii.tt ^{) (} ÉIǕǗǗ

^.

#