P RCS G 2022

Disk Potential Function S for Poly
( Joint with SirChanggon

Lou & X i ao Zh enginSpaceProgress)

York Kim( ResanNational Union )Polygc.int
FX E( re=SO13) :=,b,◦•◦.hn +31 €SO (3 : R ) or R3RT ( The length of Edge )2linearPoly gon

Mr :.

y,Z)

ER3 : 가구Space={ i. (vi. V2.

ooo.vn +3 )ft Z2

ET 5 ( ri)=V2 }M3:i= 1졹 Vi=0•^

V3(V5T•.

( r) : = { (x'LvSO (3) 2•V1>••

V3SO (3) (5V4R34diagonalact.io )nSome facto•••Mf ≠ ¢Mfis≈

ifricr.to( CP M3called'◦

tri.it rit.to 00th +3◦

PGL ( 2.0 )ric If In o E

•

ric

gene
vii. iskcihler.dimc.ME N

2o o o,( E.io.

,En +3 ) E

仕叩".

,

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s.t.IE ri
ii= 1'

Mi is sm00th ⇔ r is.

projectivevarie.ly=gene( i= 1=O.

18

2

Diskpotentidfut( FuKay a•

L:aOh

Lagrangiantorusoflx.co) ( www.riO.dim.RL-dimp.X 2)
,

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,IPI
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en :>

.

=,

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htriviallinebundle.ITe.

=d)>

F is a flat

-Connection onL}~

Series aComplex Mirror chart.

18

Disk•s

L = T ! Close Oriental

100ps 0.02.000 On St

피 (ㄴ) ≈ 고거 ≈ 조 ( Q 02.

{ F F isflata

Definit.io n ( Diske.g(X-,On )Connection on

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,L

=(.

:-( 卵尸≈~(아)
np.si?hk:一亐石川 ㄴ) ≈ 고2 ≈ ㅍTake O• • •,.

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potential function )EP ! Ois )=,"W. ( x)>

3g

potential function (cost)

2P and set x)

equation>(,5 p):-ii,1lbfAtl-.

i1hol.CO ) E At××ll!

!

5-PL
W.CNNpx
ns.px-•+

1.x +'hx

generally.mayhavemutplehagrangiantor.in Multiple (107chan다
Chart
gluedtoproduceoibe.tt
More>"areMirror,.

4

Ben ding Systems

18

Want to construct( kapov.ch•

Lagrangiantori.nu poly gas Space)(Millson )Me has a Hamiltoniantorusactionldefinedonitsdenseopensubse.tl
For E ( reb, ooo.hn +3 ),ERMwetakeatriangulat.ionof (M3)-gon

I (bendingalongthechosendiagonals.li
Lagtorusf.toration 토Have a:

Mt

">

R

givenbythelengthoftheohosendiag.in d)(isapolytopegivenbytriangleinequalities.e.gs•

TheImageof 또E = ( 1.1.1.1.1 )n - 2.

•V12n

3=( 輻輻 )IV5d.,•E•

daA'EN••v3V445€ Mf in R3=:

f = Im IinR가2

SYZmirrorsofpolygo5.se
Theorem ( LouZheng))
Assumethatiisequilateralandgeneric.X-MFCPolygonspo.eeK.

Then an SYZ Mirror of X is a LG Model ( ☒ W ),•※=Gr ( 2.1CM3 ) 7 { 加二 P23="=°

1개+2m3=

P1계3 }( Record Gr ( 2.1CM3 ) ≤ 1P ( N1대3 ) viathePliickerembeddingpi.gl•

W:I>

W(印忙堂 阮汨i. - l•.

이'2•2邏

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53 •

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Disk

potential

6

sforcaterpillarbendingsyste.ms)
Assumethatiisequilateralandgeneric.X-MFCPolygonspo.ee•function18( 휴3 휴4.000 Int2 )thecarterpillarbendingsyste.in
L-themonotonehagrangiantorusfiberofI.locatedatt.be CenterEE:

2WLe.g

P13=> N= 2,hk

妨叩V1Mt==p,'lP볘

拓的卄

2

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.

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邏휪ᵗ,'5yen2

邏

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5V4V2

3

Rjd P5R3.itP쮀
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W=2=•

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4

Remark ( F000 & Pascal3

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繼千

3

뛻

繼千

eff-Tonkonoglcomputethed.sk potential of GP5.

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CIustervarietie.si/iaSYZmirrorsymmetry__( Fominzelevinskyl.Mutat.ions.seedmklt.B) ( 必別 (k=X =( 1G 가2,7세 재+1.000000,,,,1Cm )-dustervar.froze.in var
-b. .

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Uiglueviathebirationaltransformat.io, *Foragivendustervarietyx.findasymp.ms/dandaLag.torus
fibrationwhoseFloertheoreticalmirror.isG.

X.

8
Polygonspacesdustervariet.ie/8

Symplecticmreauevcyofaontpett
PolyI
ComplexSpaceon,Without toZen variablebeing Systemse.g

dBInitialsee.dk?G=pi3.7bip.4).B).SetB=(9i)andB'=N= 2>

E = ( 1.1.1.1.1 )

,

Mt

,

=

•

•

7C2 )( 1••( 네 가2) B)

•幾.

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=

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p.4

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2=P24

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•

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P35'

P25R5135

P35'P13BB5

9Competition of d.sk potential functionS18( To ric Case)Oh F000 Wood Ward Chan

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etheFanotoriccasete.gs⇒

Gelfandzeit.li'

Disk potential function
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sandtoricdegeneratic.rs

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11

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18( Hausmannknutson.Noharalle.cla)

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Caterpillarbendingsyste.in

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✗

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D(prg.

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pi.it:☒3¢

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Gr ( 2.CI?t3).L:torusorbit
•.

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↓ "TUcm3
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isStrategy•

Tocalculatethed.sk potential functions forvariousbendingsystems.it sufices.to Computer
Oneof them.

!
•Global ⇒ Local ⇒ New Global.

Global Part

ialflagvarietiesofvarioustyp.es( Ch oK.

Oh )咸晁廳韆一ALocal # 남 ( 1에 .tk ( R7 TU (n) TSU에 H50세 ㆊ,New Global!

Each

Polygonspac.es,,,.

diskcorrespondingtoateminwiscontainedinalocalmodd.ci共)14

singularfibersofbendingsyste.ms

18

The fiber ofUnder I is anisotropicsubmflddiffeomorphic.to( Bad) If F isric ,gene5013계

x

any Point( 5Px TM3

Need localmodels.ES 013) F5,e.g>卞二 ( 1 1N= 8( dB 30이

8^"L2>i↳y4v3

1=,,in9앵

=1,d, >

=0de,=1 ,d.gs 0,dso>ㅇ)give.rs/R3

The fiber is ≈ SO (3)~>EdlfvxxP0SO13)

10Lr,2-12"0=>※1 ^9)

di 5,i56

SO13)

0.0,.

5e.g>

N= 2E = ( 1.1,

•Mt

=d P55•에

LLV4V23,V5암•1.1.1 )

lV12,

••

v345

€

dlf.in R3

↓

S'•f

↓

.
•E.

rpt

E

•

=Im IinR2

15

Competition of d.sk potential functionS for OGG ¢4 ) and 0G (2 ¢5 )
,0G ( I.

5013)

0G ( 2, ¢5 )

5)

新

홉?

htii.hr∴(X) = 기자 14기2t

간x,

Wt (X)+ A=,8,각+++

개각각각"기2+x,+b'

가3강No matter What a b ( E ㅍ) are L admitsFs.to ( L0 )isanonzeroobj.ee•,( Sheri dan AF000 ),의,(TO G) U~GH90G)

Fukx ( OGG ¢4 )) isnon-trivialwhe.in 入二 4.0,-.

4,

Fuk 10G (2.05 )) isnontrivialwhe.in 入二 4. -4.4 i. -4 i (⇔ OGG.IE )> A2 &=In sumb=''( ⇔ OGG ¢4 ) = EP XGP )=(P3 )

0,W. ( X) = 기자 14기2t.

간+x,2x2

WK시

=객+.

.
++

개각

각각

"x,간t강

16

C.com/sutationofdiskpotentialfunctionsforpolygon_spacesM_F

Foragenericriuiissymp.Fanote.fi

IX)

ENdetermineeffectivehomotopydass.esof

⇒

•

•

Area = 1 +Master Index 2

18.2.2121

(x)

Local

models.com/suteopenGWi
詞垢 nvariantse.gsn=2•g+棟邯 앚k II]

P5000여 勃注P3

鄙埼'to ric deGenerationmm

엦쐊q

坮馹

P4페나湟邯
P3P4.LI#Q=EnEtttEsiiolXo.Ldxoa=a52①OGU.at/D2≈

一一UM ( L 鄙坮 坑珩 )kisof.

.

degree

2.

가.

IT
18

Competition of d.sk potential functionsforpolygonspaces.ME ( cost )Fore.gInhigherdimensionalcas.es>

M= 4F.

=( 1.1.

◦ ◦e,I).

710에 5.

thiscase.needtocarethefollowinghagrangianfibe.rs
Two

SO (3)

Two

5

✗

T

8th U4'

6th U3

T2×

一一

For in Stand 5X Focus at ( 암,,암,암

4th U2암 ) = (0.1.1.1 )

,

一一

2U.

'✗

뼙

AnbdUaround5xFcanbetrivializedasUT2x@tPviatoric.deg
Then have an

orientati#onpreservingdi
ffeom.
tlp
(U<>.

.

<<a2)

3VP

7f

•

>

.

,,<<>>n,•^

OrrSv.

^아>

M ( UHF T2 P ) x T2'>.

J

Choice of F
We areconoernedwiththeequilateralco.se.

ifwechange.ir?Canhavedif erentalgebraicvarieties
differentfibrationse.gs
許弘拈

G. WhatF.

.

( 1.1.3.3.3 )Mi≈,'EP x EP'W.

=Y +.

y.tjtyatjle.gr
핢tjty.yiye.gs
許拈十 京淡e.g>

ri;i)

=

( 1.3.3.3=( 1.1.3.22).

千一 (2224.4)

G,

M 占二邨

MF,

≈'x EP'¢5 # 21CM

ME, ≈ ( F # 30P2

△ f.

W2-W3

=Y +.

WE Y +.

ID

△ f,

얘

Aslongasdlfismonotoneonecansimilarlycompute.hr.

ThankYou 鐫