Fwbenius Manifolds

Fwbenius Manifolds

1

. \

March , 2nd , 2018 .

1

•

History of

^Fwbenius- ^Manifolds i

~ 1980 , Saito 's

singularity ^Theory

~ 1990 , Vafa 's Quantum Cohomology ^{( A - model 1}

"

Moduli ^" of C-Y Holds ⁽ B. model )

(1) (19941 Dubrovin ^{: of ZD}

Geof

^{TFT ( textbook 1}

I

PDE ^:

eqns

WDVV

(2) ( 1998) Manin ^: Fwbenius Manifolds _{, Quantum} Cohomology ^and Moduli Space

( Gwmov ^{- Witten}

Theory

⁾

(3) ( 2004 )

Hartling

^:

^Fwbenius

^{Manifold & Moduli Space fur}

Singularity

^.

MainQuestio=

^:

Analytic

Continuation

of Fwbenius

Mfds

?

Compact

Example

^{? (}

Compaocitication

^{? ? )}

Definition

Fwbenius

of

^{Manifold :}

@

-

M ^: C ^{° .} mfd g^{: TMXTM -} IR ^{" metric "} ( not necessarily positive def , only require ^{non .}

deg

^.)

commutative

A

multiplication

^{Str : o : TM×TM - TM .}

In

any

^{local word Ix', ... , X"l , the basis { Jj}

:=÷xgl

^{on TM}

^lu

,

We denote The Str crust ,

Jjo

^{2k = C}

^Fk

^{2m .}

^{They satisfy}

^:

(1)

g

^{13 flat i.e,}

Riemlg

^{) to . Therefore , 7- flat word , Lt'' ... it " 1}

sit ,

9abi= ^g(2a

^{, 2b ) are constants}

(2) Ze ^: M ^→ TM : identity ¹ unity ^{section . st .} eoX=X . HXETM

T

global

^section

(3)

GIXOY

^{, Z ) = SIX , YOZ ) Define cc X. Y ,Z ) :=9( XOY, Z ) :}

totally symmetric

totally Lo_, } ) ^- tensor

→ 2;

o2j

⁼

glkcije

^{Jk Cijl : symmetric wire . iijik .}

(4) V :L ^- C connection

Nxc

Wirth g ⁾ ( Yiz , W )

EPYC

^{) (X, Ziwl}

Regard

^{PC :C 0,4 ) - tensor Then above}

identity

^:

^totally symmetric

^!

→ This is the

associatively

^{of 0}

(5) 7e=o

Rink^: Later , we will introduce the Euler

vf

^. E

In flat ^{word ,}

Hillel

we may ^{assume that e=}

2¥

,

=

(4) becomes ^:

|2aCb_cd=2b_Cqcd_

^{Calculus ( ! )} implies ⁱ I ( Ioc,)

fcn

^f- It ^', ... , -01 st . Cabo

=g?[§µ-yqt

→ The function F determines everything ^{on The Fwb. mfd i} Metric ⁱ gl ^{. ,. 1 -}

gleo

^{. , . ) = c (e, . , . ) ⇒ Gab =}

Iffy

Str const . of

multiplicative

^{Str : Ckij =}

gklg.fi#yq

^{→ Cikj : function}

^of

^{F .}

Associativity ^of

^{0 : ( Ji}

^Jj

^{) ° Jk =}

;

^{Jio ( 2jo 21<1}

11

Kfj

^{2eo2k )} 2i ^. (

Cjtkael

"

"

£

Cijcek

^{as 2}

ciecglk

^as

⁽

^{Witten -}

^Dijgraaf

^{- Verhhde - Verlihde )}

lis lis

Edicts

^{- ( i) WDVV eqn}

III

"

IIIIEimm.int#es

^{Cotati : FindWDVVa wordeqn ,}

^{Issue only}

^{- free timeexpressionin flatfor wwd .}

WDVVEI ^ICE-f.fr

^The

^o=÷eb| ftp.cpab.cdpbcpae/+Cpae(

^existence

^Cpdacpef

^{of -}

^{potential Cpdfcpae}

^{)F - ⇒}

^tff

^"

^3k¥

^.

^{2k¥ )}

+ c ^.

^total

^.

^epaulet :flY!a÷

^.

^Ethanol ;t÷Y÷al

^-

^campsite

-

fare

2Eb

)

⇒

(

^cpda

¥5

⁺

^{cpdiscfeta )}

^-

⁽ cdp.tt#+cdpt2claeb- tangled )

⁺

^{( deg} ¥ ;

^-

Claim

^{: 137 holds in}

general

^wwdi

system

^{(Xi) ( } )}

Exert

^: (a)

Verity

^{the claim}

( b)

Heveling

(^{. Manin ) . (3) is the word .}

^{expression of}

^The

^following

^: KY .Z ^e PCTM ),

hx

^. , ^, LZI⁼

Xohylzl

^{+ Lx l ZIOY}

Still form another

^{of WDVV :}

y

^{X'T extends to}

^rf

^.

Li TM ^→ TM llil ) ^- Tensor ie , a vector ^- valued 1- form ^.

Torsion

of ^L

^:

TLCXIY

^{) := [ LX, LY ] - L 1 [ LXIY ]) - L 1[ x. LY}

^]|+L2[

^{KY ]}

Hantjes

_tensor ^:

_HLCKYI

:-.

TLILXILYI

^-

^LTL

^{( LKY ) - LTLCX .LY ) +}

,

for

^{XIY : vectors L ' TLLKY )}

Now , for XETM ,

Lxl

'll ⁼

XOY

^⇒

Lx

^{: TM → TM X: u}

^f.

Exerts

^:

HL×

^{to ←→ WDVV eqn .}

Example ^from Singularity ^Theory

^{I Saito}

⁾

^:

Surface

Singularity

^:

flxiyiz

^{) =}

xIy2tp(

^{2-1=0 in 1 3}

pA ^{) =}

Z5ta=3Z3t A±Z2ta±Zt a±

^:

deformation

of A4 ^-_Sing^.

-

ry tmodwloiratpaaornaumueptero

^{An sing :}

^xtytznt

^'=o_As-

sing^.

←dim

,cA=4

^{iso .}

sing

^.

A ⁼

QKXIYIZD ^/

^(complete

^fx

^,

^{fy ,fz}

intersection^{) = 1 [Z}

^]/(

^{I 'cz)) I 'lZ1=5Z4t 3A} Z2t2azZt a,} PLZ⁾, qcz^{) EA . plz ) 0912-1} =p ^{( z) .} qizl ^{( mod 'LZ}

I

))

The ^is

^given METI ^by

Gwthendieuk 's residue ^:

RmIi ^{0 - complete} intersection

dimt

A= KKZI ^.. .iZmD/l fi ^,.. ,fm1 finite ^{-di n} 't k ^- Us.

( say ^{k -} ¢ )

I ^: A ^→

tree

K map

ydfin ^. ..

ndtm ₎

c- &.

YEA _, IC 41 ^:=

Resfm=o Resfm

,=o

- - ' ''' '

Resfeo ( FTIFN

It B ^a trace ^{i. c .}

gcaib

^{) := Icab )} - ^isPldj^non)9141^.

degi Exert

^: (a)

Show

^That

glpizhqizsl

⁼

IT II

(b ) The metric is flat in

R4

take K ^- IR ) , with

ds2=

daodaz

^{+ daidaz -}

3pA

^}

^daadas

^-

st azdaj

to = Ao ^-

¥

^{Azaz Write down :}

The flat Wd '

|

^{, ,= a,}

^.÷aj £u°¥s=CFj÷tk

tz ^{= Az}

tge

^A} 11

March , 9th , 2018 .

• WDVV with Euler vector field ⁱ

n

f- It ) ⁼ Flt ^',....

, tnl ^: potential Capr ⁼

2£ prf

^{T= [}

tie

_;

2=1

1) 7 xp ^:=

Cup

^{is a constant , non .}

deg

^{. matrix ( metric )}

C

Ip

^{= 7 "}

Ceap

^:

"

Str const .

"

21

Associatively

^{: ea. ep =}

C%Hler

^{B a} _ring ^{At , with} _unity ^e,

Then _led.ep ) ^. er ⁼ ex ^. (ep^{. erl ⇒}

§

^Cases

^Csf

⁼

§ ^Calscpsr

^{→ WDVV eqn}

3) Additional ^"

quasi

^-

homogeneity property

^{" :}

require ^:

deglt

^{) = I ,}

deg

^{I e. 1=0 ( so,}

deglt

^{' 1=1 )} t=t' eit that ^{... .} + then

Notation ^: da ^:=

deglt

^{'1 9ai=}

degledj

^{= I - da}

Recall ^{; F:}

weighted

homogeneous

of degf

^{=D # means :}

F (cd 't ,_,

cd2tz

, ^{. . . .}, cdntnl ⁼

Cdffct

, _, fce ^{1 *}

|

§=£dlf¥

, Eh , ⁼

↳ § ^datba

^{, 2am}

IT

_,

)

The case ^:

we only ^care about Bnp_, F

, we require only ^:

e=¥

^.

^EHl=daT£2a

|£

^{,"for" some}

^Lef

^=wrist

^dff

^.

^df

^{+ ,}

^Aapt

^{Adp, Ba}

^'tf

^{, C+ , Batwith 't CEuler -}

^vf

^{*, '}

^{) £EeI[ genteel}

EH

)

=§g

⁽⁹

ftp.rd/2as.T

^.

Lee

^{= - d. e -}

(

_Quasi

-

homogeneity

property

-nLemmal_

:

KEY _)&p

^{= 1}

df

^{- di}

)9xp

^{i. e. E is an} infinitesimal

conformal tmnsfi

Wint . The flat metric < i ) = (Map⁾

Rinkh ^: conformalwuformal

transfi

is

generated >/3i

^{in Euclidean}

by

^: isomethy_space

;

^{: , h=2dilationconformal, inversion}

^Transf

^{. (, Lionville}

^{iff holomaphiz}

^,

cf

^,

Dubrovm ^, Fomenko , Nouikov ^UOLI )

pfi Apply 2p2a2

, ^{to C *}I and do the wmmutator with E.

2p2a2EF= df ^2p2x2f

⁼

dfY&p

observe ^: 2. EF ⁼ [21, E] ^Ft EZF ⁼ did , ft E 2. F ^⇒

2p2£2EF=di7apt2p2aE2iF

Jp2aE2

, f= 2ps ( ^{[ 2£}. E) ^{+ Eda ) 2iF =} 2p⁽

99

2p2^, f) ⁺ 2pE2£2iF

⇒

dfrlap

⁼

diyxp +997Pa

^{+ Efs 7Pa}

Cd ^: If 4^,, ⁼

Mleiilil

^{= 0 ,} and ^Q= ( 9fs⁾ has simple

eigenualues

^{, then}

by

^{a linear}

change of

^word, , we may ^{assume :}

Map

^{) =} ₍

I

^..

.f )

^{, and then}

Flt ^{) =}

Itt 'I2tntIt§nt

^{'t at " " + fltz , ... .tw )}

Also , for d ,==l , 91=0 , qn ^{:=d ,}

df

^{= ztdn = ltdat dnti - a =3 - d}

→ %a + 9 one ,, ^-a ⁼ 2- ( da ^{+ dcnti , . a} ) ⁼ 2- (2 ^- d ) =D .

pf

^{: Lei ill > =o ⇒ Can choose}

eigenvector

^en

of

^Q sit. ( e_, .eu ^> =1

Now , on span

Seiienlt

_, can use Q ^-

eigenvector

^to get ^{(A) . ( Check I}

Rink ^{: In}

general

^{, if Q is}

^diagonal

^{izable , Then}

^EHI

^{can be}

^transf

^{. (}

by

^a

linear

transf

^{.1 to :}

Ect ) ⁼

§dat&2&

⁺

^[7

^r22a

12162=0}

.

Example

^:

1 , n=2 _, WDVV B empty ^,

Scaling

^{conditions ⇒}

I) Flt , ,Tz ) ⁼

lztitz

⁺

^tzk

^,

k=[d1

^{dtl , 2,3 , D=}

^deglezs

^.

iii Fltiitz ) ⁼

ftp.tzttz2 _logtz

^{, D= - I}

iiil Fltntz ₎ ⁼

Itptz

⁺

lugtz

^{, d =3}

iv ) FH , ,tu ) ⁼

Ititz

⁺

e£

^" _, ^{D= lirto}

V) Flt , ,Tz) ⁼

ftp.tz

^{, del , r=o .}

2, n=3 , Fit ) ⁼

It its

⁺

It

^,

^{tit fltzits}

⁾

WDVV ^⇒

flxiy

^{) satisfies} f×2×y ⁼

fyyy

^t

fxxxfxyy

pf

^: multiplicative ^{table : 4=l , ez , e3 (}Yap⁾

=/

, ,

1

)

e

}

⁼ Fzz, 7 ^{" e}} ⁺ fzss

)

"

e_, +

Fussy

^{" e}, ⁼ es ⁺

f×××

ez +

fxxye

^,

es^. ez= Fzs_, ez ⁺ Fsss ^{e2 +}

Fs33e|

⁼

fxxyez

⁺

fxyyel e5= ^Fzz

, ezt ^F 33202 ^t

Fzzzei

^a

fxyyezt fyyyei

Associating

^{( WDVV ) ez:(. ez) .e } = ezlez . e} )}

⇒

ejtfxxxez

^.

eztftxyez

⁼

ftxyeftfxyyea

⇒ ^{z +}

^fyyye

^{, +}

fxtxfhxyezt Westside fxx\yez tfxyf ^fttxftyyei Rft

^{= t (}

^)ez+t2××ye

^{# ,}

( Check₎

Scaling

^{conditions : (al ( I -}

f) ^xfxt

^{(I- d)}

Yfy

^{= ( 3- d)}

^f

^of

#

^{,z , } .}

Cb) d=l ^:

lzxfxtrfy ^=2f

( c) d=2 ^:

rfx

^-

yfy

⁼

^f

(d) d ^{=3 :}

lzxfx

⁺

zyfy

^{= const .}

idea_i

Use the

first integral ^of

^{EHI , will do case ( d ) :}

Question

^:

Find

a

good

^{, word ,} system ^,

vf

^{. on the I Ky) -} plane ^. (

zx

^, 2y ) i. ^e,

{

^X

's

I×

y'=

ay

⇒

|×=c,ez±

Y= ^{Cze "}

Define 5=7×4

B a const ,

along any integral

^{curve .} We set

5=4×-4

^t=x

gx=t _y=sE4

Change of

Variable ^:

ftfcxiy

⁾

=3¥ .TT +2¥

^.

}Iu=f×+4t3sfy

=

I

⁽

Ix2×t2y2y ^{) f}

^⇒

staff

^{= ( ⇒}

ft=2÷

^⇒

fezclogtiocs

⁾

⇒

flxiy )=2cl°gx+

^011×14

⁾

^{- L * * 1}

Thm_ iflky

^{) satisfies}

flixy

⁼

fyyytfxxxfxyy

^{can be} Transformed into ODE

of ¢e¢lZ

^{) via L**1}

4^{" ' =}

400412+32 c¢"

⁺

11202-0114 "t 7842-2

^{@" 2+16 czd}

"f16oz2¢ 'd

^"'t

1922301

^,

"4

^"

This B a

special

^{case .}

^of

^{Pain level It ( as well as all other n -3 cases )}

• Card ^.-

free form of

_Euler v.

f.

^{: EEPLTMII}

- THE ) ⁼⁰ ( This make sense since Map ^{B flat )}

- Q ^- OE is a covariant ^const .

operator

^⇒

Eigen

^values

^of

^{Q are}

const ,

fans

on M , sit

,

Meta Lee :c

^{= - e}

^:

herfap

=D Map

for

some unst . D.

March , lbth 2018 ^.

•

Symmetry

^{of WDVV :}

Type

^{I :}

Legendre

^{- Type} Transformation

Sa

^{17<=1 ,. . .in ) : Courd ,}

change th £

^' sit , Lal

Ir

⁼

2r2xFHi

(b )

Fpiwtt

^{) =}

FµrH

⁾

( ^c)

ftp.

⁼

Yap

( ⁱ )

2a=s÷I=2j[÷a÷÷p

=gBr2aa< =7 ^{jrfctsttfp Farr}

^"

^Jp

⁼

^{Ffa Jp}

Cd

^:

If

2x ^. is invertible ^. then

?a=2x

^.

^In

^{. In}

^particular

^,

2<=2×2^2

^{⇒ 2x=e .}

pf

^{: =}

^FLA 22.2¥ _Ip

^' 2k ^. Let

Gx

^{be the inverse matrix for 2z . :}

GIs

,

FI

, 2x ⁼ 24

"

GXI

,

Fla Ip

^{- 2x =}

Jai

2k .

RMII

^: _Sx 's commute fur

different

^{K 's. S}, ⁼

identity

^{. So, at the end ,}

we need eo

renumbering

^{the indices to switch K c- 1 .}

(2)

2£Fµv=7PrFaxr2^p ^{Fair =9Pr}

^Faxr

fair

⁼

^Ffa Fair

11

F£µv

⇒ Under ^{t ' →}

It

,

Far

⁼

Ffa f§µnj

13 )

Preserving

^{WDVV eqn :}

Iggy fi×kfµg,g type

^{= F " "}

^Feig

^⇒

^WDVV

ⁱⁿ

^original

^{word . system.}

II ; IIR ^{IIR EE 'a}

Examples

^{: ln=2} , del , r=2 ) F ⁼

Itt

^'

^)2t2+ et _Yap =/ T! /

7<=2 ,

Sa

^: I ^'=

ta

⁼ Faz ^{= et '} _{check that} ;

ELE

^',

.i2j

⁼

IF

^'

^# It _'s

^LE

^'t

f2= I

, ⁼ F ,z= t ^'

Uogf

^{' -}

^E)

, Then switch 1← ^' 2 .

Elf'iE2j= 's FZIF

^{'s' e}

ICE ⁹²

^(1.

get

³¹²¹

→ The case D= ^{-1 ,}

Rink

^:

If

^9x_, ^{= .. ".}

=9xs

^{, then}

^fc=Ld

^{. . . . ., Cst . can consider}

Sc

^:

I

. ⁼

§

,

ci2&Jx _; FHI ⁼⁾ So is ^a transf ^' if Ci2x _,^{. B invertible} ,

Type

^{2 :}

^The

^Inversion

^I

I

^{' =}

's two Ed=¥I

^{for a # 1. n}

^In

.tn/ffYpIyEyHlf4

^{' -}

^{that 'l=einif+ ÷ # tie}

It B

conformal

^:

okpdfddff

⁼

Tqn 'T

^72ps

^dttdets

Effect

^{on Euler} vuf ^{. ( Dubnvm Lemma B. 1).}

Exercise

^: 542 ^;¢ ) acts on solh

of

WDVV eqns ^{with d=l .}

t ^{' -} t 't

I ok _§

,

tat ^. t ^{" →}

t%tn+d

) ^x±lin

qn ^{- ( At}

"tbY(

ctntd ⁾

• 1- dimll

affine

connection ^'- real

lcpx

D#i

^{This B}

given by

^{a function ret ) =}

title

) ^{sit . on}

k

^- form

R•£

^:

fdtke Rok

^,

Of

^{dtkt '}

:=(f¥

^{- Kmt ,}

^{f) dekt}

^{' - ( *I}

In E ^:

fltsdtk

⁼

( ^fit

⁾ (

dt1dF1k1dFk-i.fTF1dEk-HIe.triEiF1dEkH-ldfedalfTEKEIlkl-krrhfTFllfElY-dHfEtfiEtktEyaEElktlEjktis_feYdekalFelYtii4dE1dI-TransformatinruleirLEl-YfI-C@d2Igfddg-CtxiMiie.g.E

.EE#adI=Eibd=e*aediE=-KTtdj3

⇒

FLE

^{) =}

⁽ Cttdprlt

) ^+2C ( Cttd )

No local inv ^{. :}

Locally

^,

_May

^{solve :}

^w=¢df

^{sit . 0w=o i. e.}

^¢

'

-

84=0

Then

^{we set : w=¢dt=} :d× × ^: flat parameter

Notice that ^LH) reads as ^:

of dtkt

^{' =}

dkadqlfofkjdzktt

Def ^Projective :( ^{Structure )}

1- dim

't attire

connection

wl Mobins Transformation

^as

symmetries

B called ^a

projective

^{Str ,}

Prof

^{: 4)}

^Rdt

^{' ..}

^quadrate differential

_,

R=£T

^-

^IT

^{is an inv.}

under Mobius

transf

^. ( cheek )

(2) _{t can} be reduced to ⁰

by Mobius transf

^{, #}

^R=o

pf ^of

^{( 21 : ( ⇒ ) trivial C ⇐ ) reo ⇒}

^day

-

^I

^{I r2=o}

2dy÷=

dt ⇒ 2

fdtf

^{= T - To ⇒ - 2}

f-

^{= T - To ⇒ r=}

⇒

.

Now

_, set

I

^:=

#

,

⇒

FLEI

⁼ -21T ^- To ) ^{+ LLT - To} )=0

a

Rmk_i Recall That

moduli

of ^flat

connection ^{< →}

{

_rep ^.

of

^{Ti ( Xix )}

^t

⇒

No local

MV^.

EXamp1e_ ^{: L = - doxt}, + Ulx ) on D= S^' or ICP^'

Ly ,=Lyz=o

^{Set T=}

¥11

^-

^projective

^{Str . !}

( Ioc .) as our new local word ,

The non ^.

trivial attire

connection is

defined

sit , X=

flat

word ,

⇒ Rdt ^{' =} zudx ^. l

Check

₎

g fixed ^k

Exercise

ⁱ

plr.rs

^{. . ..} ) ^:

poly

^{. sit ,}

pdtk

^{is in. under Mobius}

^transf

^frr

any

^{attire to , then} pe ^{QI Ri VR ,} ON ii. ^{.., 1} , where Q B a

grd

homogeneous

0k

has degree

CHAZY Eqn

^{: 3d Fwbemus Chart with D=}

93=1

^{, r=o . E = t}

'2

_, +

It

^{222 to '}

Ask for ^Solis of

^WDVV ,

periodic

^{in t3 . period =L ,}

analytir ^Hit

^it

³⁾

^{= 10,0 , in}

F hit

'it4

⁼

}EY2t3t _I ^eilltl

^{' .}

¥5 ^'rlt3

^{) ,}

_rtl=n§ anq

^"

.q=e2

^{'T 't}

WDVV eqn

^{becomes : 8}

"

=

brr

^{' -}

9

^{r'' 1}

of

^{. 1.13 Cbi )}

Exert

^{: rule}

Ijl ⁽

^{1 -}

249-7292-9693

^{. . . .}

Trettin ^e.)

d=l ^,

Sllzilc

₎ acts on the ODE ! ⇒

Fits

into the

theory of affine

connections

Eqns ^from

^{QIR , OR , ...}

.0kR1=o _R= drlde

^-

IV

set

u=I #

^2,

R=dr/dI

^-

IV

^{where w=dx}

Set U=

's ¥2

^{, where W=d× ,0w=o 7w=o}

L= ^-

ftp.ucxl

^{yilx⇒), yzlx ) normalize Yi , Yz}

^by 1¥ Yy

^,

}|

^=/

t.br/y.~sr=dYh=#l

^{' since}

^he

^.

www..gl

^.

Leo

, R=o

k=1

^{, 71=0 ⇒ r ".}

^3rr'

⁺

^r3=o

k=2 ,

82ft cN=o

for some C

IF

y

'"

- brr

"t9r

^{' 2 +}

(

^{c . 121}

(

^{r' -}

It "F=o

^CHZ _gives

_Chazy

_eqn ^!

In terms

of

^{u, the} eqn ^{d u}

"t2cu2=o

Observation

^: _p

'2=4p3

^-

gzp

^-

^g

^{} ⇒} 2ps

"p

^{' =L}

^'2p2p

^{' -}

^gap

^'

=) p ^" =6p ^{' -}

92/2

Thus _, 92=0 ^,

93=1 U=±oPol×

⁾

Poieqni

^{- an - harmonize elliptic}

function

^.

←

malize

ODI

Lame

^' _{eqn y}^{'' t (} Apt ^B) p=o

y

^{" +}

I

^polx

)y=o

^{Let t=l -}

ppcxi

^{, we}

_get

^:

tcttlddtte +1ft

^-

f) ^date *

^=o

March ^{. 2}

3rd

, 2018 .

ZD

Topological

^field

Theory

⁽

Atiyah

^{'s axioms )}

1 , Space of ^local

physical

^{States A i. e. dine A =n < is}

2. ^{T[ , J{ I :-) Vee A ( s ,}

St

^{as , := ¢ ,}

^if

^{22=4 , [ : connected}

cpt^, oriented surface

|

^, if ^Ci has orientation

,

#

^{Ai ,}

Ai=|

^{Aa* induced from E .}

if ^{J{ acid .}_Ci^{... I}_as^Ck' ^{i 'F not ,}

⇒ K¥1 ,¥ a- ^Alqyz

^{YE, as 1 e , = A*a}

^A*oA=Hom

^{( ADA i A)}

Also

, we require ^The

assignment

^{is a Top. in . i. e .}

They

^{are the same}

under Aomeo.

÷

₎ normalization

it

^:

DID

^{, id}

^eA*

^{@ A}

2)

multiplication

^{: Vs}, # sz ⁼ vs_, ¹ Vsz

} ,

factorization

^:

j(¥#y Ying ¥¥"

^S

T

Vs^{, C-} A •

A*o

^{. . ". Then Vs =} contraction

of

_vs^, _along ^The

#

^•

_{Genus g}

, s ^- pt ^{wrrelator : cut .}

vgis

^:=

HQ.PE#M*genus=g

symmetric function^{@ sat}

rnof

• ⇐ v

of Da

genus

a) a-

^{0 e.}

A*•A*oA=HomlAoA

_, At

gives

^the

^alge

^{. Str.}

•

y

^{:= V}

of III.

^e

^A*bA*

^{i. e,}

^y=

^Von

^symmetnl

^pairing

. e ^:= v

of

^a cap ( _{disk )}

g§n

^{e A}

Thai lciyie

⁾

gives

^{A a}

Fwbenius age

^{, Str}

( primary

^chiral

fields of

^the

theory

⁾

pf

^:

Associating

^{:(L*p)*r =L * ( p*r ) x.} p^{, re} A.

a

Ml

^a

^1¥

^€

.

.tn#.r

^-

^e¥n↳r

g=o

Unity

^{: Cut and}

glue

^{a disk = do}

^nothing

Non ^.

degeneracy of

_y ^:

Let

F

^{:= v}

of (f=§

^{Then observe that}

^Dy¥=%

⇒

n5=id AI¥ ¥1

⇒ y ^{; invertible}

Fwbenius Property

^:

ykxpirl

^{= Vo,}}

^Hip

^{, r ) via}

g$[YpIp ^{4mL .hr ;}

^.

Rink

^: This use

only

^The

genus

^{0 part !}

Actual Physics

consideration ^:

QFT ^on _D ^{din 't}

-

^{mfd [}

Data ^{: •} A

family

^{of local fields ¢& (x ) XEE}

• A

Lagrangian

^{L =L ( 4,10 :$ ,}

"

... .

)

→

S[

^¢

^]

⁼

|zL( ^4,41

^{, ... . ) :}_~ ^action _{involves metric}

_Gijcx

)

.

Quantization

via

Faymann

^'s

path integrals

^:

partition ^function

^{: 2 £ =}

f

^{[ d¢}

^]

^{e- {(¢]}

f ^{:-. The spare}

of

^t

( an fields ) ^" _measure ^"

- sc ¢ ] Correlation functions ^: ( 4dL^x),

¢piy

^{), ... .}

7g

^:=

)

^[

^d¢]4&(

^{x )}

^dpcy

^{) ... .. E}

s

The classical

they

^is

conformal if

^{fS=o , for}

_any 8Sij

⁼

Egij

topological if

85=0

ffgij

TQFT ^: The correlation function depends only ^{on the} topo ^{. on} { , not on

it XIY, . . .

< da

,¢p

^{, "}

_-7g

_←

genus

Now,

for

^a TFT

arising

^{from TAFT ( or a TCFT .}

if integrable

^{over all wnf.}

May

^consider

deformations

_preserving ^the _top^{. ihv. Li → [ + Ettha} .

classes of

⇒ Moduli space ^{L local 1 E)}

Method

^{: QFT with}

nilpotent

_symmetry ^{Q : H - H ←}

Some

^{Hilbert space}

^of

chtarge 62=0 States

Observables ^{'' =} operator ^on

H

which commutes with Q i. e.

{

^Q,

41=0

5

Q ^- cohomology ^on all operators ^{: 22 - bracket}

By

^{22 -}

graded

^Tawbi

identity

^{: | Q , Said}

{10,011+-101.541}=0 }1±|Q

^,

⇒

{

^{a. s} a.

0111=0

^°

Define ^: A ⁼

kerclfimce ⁽

^The

_primary

^States fields )

' : Q is a

symmetry

^⇒

( {

0,4 }, 4, .dz, ^{... .}

1=0

LAI

If

for

any primary

^{field ,}

¢a=$£l×

) _, we may ^{solve :}