The Monotone Wrapped Fukaya Category and the Open - Closed string map,

. preparedby Leon Li

IIIMotivation

2 Preliminaries : WCX ).

SHE X )

3 String mapsand their compatibilities

4 Generation Criterion and applications

Main Reference (Foundational)

I Ritter - Smith'

The Monotone wrapped Fuka ya Categoryand the open - closed string map

,version 7

Additional Reference I Applications )

I Ritter'

14 ] Floor theory for negativeline bundles

via Gromov - Witten Invariants

I Ritter'

15 ] Circle - actions, quantum cohomology

and the Fukaya Category of Fano Toric varieties

MI Motivation

D2Motivation

Goal Extend Abou zaid 's generation criterion for Wrapped Fuka

ya Category W I X )

to monotone convex symplectic setting with desirable applications .

Motivating Qu Formulate and proveHMS for

"

local symplmfds"

I Ex :X = To

-44¥- Ks → pm)

,Kk em )

Cmonotonetoric ) negative line bdls

Formulation Issue At ( X ,w ) : non opt sympl . ;convex at infty i NOTexact( Ex : Ipm ↳ IX. w ) sympl .subnet )

:.

" Wks"

,

"

FIX )"

,

"

SH't IXI

"

have no definitions a priori .

Proving HMS e ed to find split generators for Wl Opt -H)

A

Gta ① Formulate Generation criterion for WIX ) for X :convex sympl .

② Apply to Wl Op

.tn-

H)

.

Historical remarks on negative line bdls

. ( Coates - Given toil'

07 ; Y.

P .be

'

01 ) :

GwtwisalE → ↳ B ) via GW 1B) and all )

.

Convex Sm pug

-

( Quantum Lefschetz Hyperplane 7hm )

. I Oancea'

08 ) SH't

( Q → L → (B. WB ) ) = O,

via Leroy - Some Spectral Sequence for Stfconvex

WpslTHB ,

= O

. I I Smith 'll ) ( Tot lot - is → IP'

),

w ) T'

: essential torus

SILICO,H

't

IfTy)

←o

. ( Ritter'

14 ) ✓ SH't

( Ot - k ) → pm ),

tf If KE E : via

Qtfloc- KD SH 'T Oc - ki )

orKZHM

( Viterbo 's ) canonical map as localisation

("

Sympl.

.Lefschetz Hyperplane Thin

"

)

. l Ritter'

15 ) ✓ SH't

COL - k ) → pm ),

tf IfkEm : via [ Ritter - Smith ]

• SH 'T E → X → LB , we ) ) : determined by QH*f× )t

"

non -

Cpt Seidel representation"

monotone toric hey . monotone closed

line bdl Sympl . tonic

Historical remarks on generation criterion

( Viterbo'

96 ) SH*

( X ) =him HF*lH ) as Ik - alg . for X : LiouvilleHi linen at infty

( Habitation)

I Above aid - Seidel'

lo ) .

SHEX s tfman .

HI

"

.li ) for X : Liouville

"

homotopy direct limit"

. WLX ) wrapped Fukaya Category , following ideas of Efukaya- Seidel - Smith ]

( Aboozaid'

to ) {08 : Htt

* CWCXD → stfu ,

•

GO : Stf ( X ) → Htfcwcxystring maps .

for Xi Liouville

• generation criterion

( Ritter - Smith ) . ( -- I -)

for X : monotone convex

Over th : - dik Novikov field

. C - as - ) For Q HMX ),

FIX ) :

opt Fuka

ya Cat .

• Above are compatible ,

and descend to"

GITX) - eigen summand,

"

Th m ( Ritter - Smith ) ① ⇒ FIX )# WIX ) Acceleration functor as A

asfunctor at .

HH.IT#DHH*HIIiHH*lWlXDasQtFCX

) - mod08h I too ,⇒ acceleration diagrams

QHECX) ⇒ SHHH

a.)ya ,

tf I x

Goat17

180. ring

HHTFIXDHtt'T

WIN② tf X E Spee ( a ITH * t I : QH't

Him a) Er,

above diagramsdescend to

"

I - eigen simmonds"

when . Wxlx ) E WH ) w/ oblwxlxl ) a L satisfying Moll ) = Xfull sub cat .

. F,

CX ) e FIX ) ( - l . - )

. SH ESH 'T X ) generalised I - eigen space ,as A - subalg .

.

QHE.

QHTX) ( -

i . -- )

③ ( Generation Criterion ) If Im

108×1) contains invertible of Stil resp .

Qtf )&

for S 11 sub cat. of Wx IX ) ( resp . Fat X ) )then S split generate .

Wxlx ) ( resp . FIX ) ).

Until 2M

Applications Toti-7 X (

B.WB ) ) w/ 4kt - www.7-kekso

.

Monotone toric neg . cpxlb .

Closed

monotonesympl . toric

Thmttzeaitlwx

),

7- Lz

:'-(LET"

E IX. w ) s . -1 .

HF

'The.

#=HW*kz.

Ato

monotone lag Tons fiber t t {

0840:

HEH.

ha→ SHH )

4bar E DX ! Moment polytopew/

hot.

determined by Z

bury centre lindep . of Z )

Cor ① Stfu ) # O ② WIX ) : nontrivial

Conj WCX ) :

split -

gon .by/lL.Fe.DforsuitabkholonomiesQ ..

For X. w ) = Hotta - k ) → 01pm ),

Wow,) for Isk Em :

7hm ①'

[ Ritter'

is ] Stil O Hd ) =JaclwoHT"

A ) N¥" '

ZE.wkz.w.n.z.mw.am

Sit.

Efc ,

TOC- ki ) ) ← Work ,

⇐ " "

.

Zmiw) ↳Zit . . .tZm+¥¥n+w

,

Cor Speck't

) = Critvlwoi . " ,

② [ Ritter - Smith

]f¥.sn#ff.loc-kD.W/OC-H

) : split- generated by

hz.FZECritlwa.nlo

2

Key new ingredient : ✓ QHTOFK ) ) ( ⇒ t.ro . Qtflotksh.es ) sit .Won = X.

2 Preliminaries : WIX ),

SHTX )

Convex Sympl .mfd

Def I X,

w ) . monotone convex sympl .if

① a ITX )=X×[ w ] EHYXIK ).

77×30

② I Xin EX opt domain sit.

I XII" n

, whaxin ) I yx-LI.IT?dlRH).IlY.d ) contact I ⇒ ( Xl w = do ) exact sympe.)

Ex ⑧ Liouville mfd. E.g . ( X. w )= # Q

,Wan )

¥iFh=- kw

. .I Kh )

-

gHerm . metric

② I Ritter'

143 Monotone

negativeline bdl

:① → X ( B

,WB ) sit .

. CKX ) =

- KEWBI.tk > 0

closed monotone . XB - k > O

us ( X,

W =TFWBTIT.dk?O)) Rao w/ a I TX ) = HB- k ) - Ew ] ⇒Ex .ws : monotoneUI ¥10 O : angular I - form 11 11Xin

{ 11×11 ,⇐ I }TFC,

ITB ) -14*6,1×1)

Ex E → Ol - 1)→ (IP'

, Wes ) :. hlz ) = lzftl . GITIP

'

) =2[ was ]

• G ( OL - D) = - [ cops ) . C, ( Toth ) = [ w ]

Lagrangian

②L

"

ECXYW) cylindrical monotone ori . Lag

•LAY= :L

" "

. 0-15-0 on XWin

and near Y ( us L = hx El - Eos ) EYx-LI-e.es ) )

② LE ( X. w ) closed. monotone .

oni Lag

Ex ② # Q, Wan )2TEQ :

exact Lag .②E → OH ) → IP'

Ul

§ → 510th ) → IP'

Ul Ul

§ → L → Tai = { 1701=1713

Monotone Lag Tons

Rmk In fact L = Lysa.

for Ysu E OOH )

barycenter

kmk Later,

Ob I WIX ) ) = { LE IX. w ) Lag of type ① or ② )

Wrapped Floor cohomology as direct limit

Setup X IR C'

- small ,Morse

,

linear at infty w/ slope C min { Reeb periods of Y )

Ul

{ L :3 !

;① or ② us Elli

.

L;

i WH ) = { integer Ham.chords of weightWE

Buy}

( generic Hms finite ) 10,1 ]

-8> X w/ 8-

=Xwµ)

Def

.CI#LLi.L;..wH) :-.

⑦ A. Y ,A- I FI ait

"I ai elk

.ni EIR

.

'i'

'

Fani-

-as } o → L ;

YEOL Li,

Li : WH ) X Novikov fieldw/

I I → LjI Oil ]

.

'

I

:*-

THE. ,

#

one. I . 't

E D= B .

HRsBu - J Lau -WXHHO

Fact { Li ) : Monotone Lag w/ same mo,

F- top

" ) - E = I § 11 du -

wxhxodtlldsdt=

§U*wtwHly. ) -

WHly,then 22=0

.

Def . HFT Li.

L; ;wH ) :=HMCECLi.li : WH )

,

's )

. HW 'T Li, Lj ) life HELL

i. L ;: WH ) w . r . t .

K : H # hi,

Li.

. WH ) →HET Li.

L ;i cwt DH )

continuationmap

Pnp For opt Lags ,

HELL , L ; I is well - defined w/HFMLi.li#HW*CLi.Ld

Wrapped Floor complex via telescopic construction

Def

CWHL:

.

'

i = IQ

CFTLi.liWHILE ]

, deg g=-

l ; geo

Telescope construction of the wrapped complex as

Clara] - chain cpx

2 2

w/ N' = Cone ( K - Id : IQ CELLI,

Lj : wth D ) A a

CHCH ) CF 'T2HlTH KM TzdKy .

. - -

ie . µ'

:

Xxg. Y

tsl- n'

"

axil - It

" "

today + KY - Y )gCF*fH ) g.CI#C2HI

Else ] -

linear G or2 2

Rmk Meaning of'

k¥4; .li :

WH ) ⑦

CFTLi.li;wHlg)

T

yield representatives of

⑤HEIL.

.

, Lj : WH ) )For Yul 21=0 , Identify y

and Ky via lily .gl

⇒ I y ] = [ Ky ] in H

#Li

, Lj )

Prop I Abovzaid - Seidel ) H*KWTLi.

L ;),

M'

I EHWTLi.li )

-

-

Wrapped

FukayaCategory

Def ( X , w ) convex ms WIX ) ÷sWdX ),

wrapped Fukaya Category as lunar red ) As - cat .

Ob ( Walk ) ) = { LE IX. w ) Lag of type ① or ② satisfying Moll ) = X )

Mor : L%,

. Hom who .L , ) Chill . .

4) OU'

Acs structures : Given ( Lo,

..

.

. Ld ) E Ob I WIN,

d 72

µdin

:CFt Kd. ,

,

Ld i Watt ) #.

.. ④ CELL

. ,L

, ;w.HN#CFTLo.LdiWoHH9I

9:T.E.IS#siiE:ti*eetxol-iac*

z

E " REto

plus = E

I do - Xi ,④ of

'

's o where & : determined by 2g-

eqivariancy of µd 't't.

.

2g ( Mdgid . ④ . . . ④ gi : = It, thudgidxdxo . . -

④ aglgikx , . ) ④ -. . ④ gi :X!µd=¥wµd%

: cuff La. , .

La ) ④ .. . ④ CWHL

. .L . ) → CW

't

Ko. Ld )

Prop I Abou zaid - Seidel ) ( WIX ),Md ) : Aos -

category .

Key ingredient of proof : Degeneration of domains t÷

Exµ '(µ4Gx,t µ4µ4gxt,x,)tµ49x.M'HI ) = O :

⇒

.

¥¥÷÷

SymplecticCohomology

.

as direct limit

Setup X# IR C

'

- small ,Morse

,

linear at infty w/ slope L min { Reeb periods of Y }

us 81 WH ) = { integer Ham.

orbit of weightWEB ) F IRI

#X w/ I

=Xwµ )

( generic Hms finite )

Def

.CI#LwH):=t0A.Y-h=fIEoaitnilaielk.nietR.tIEni--cs3

YEA wth

"

Novikov fieldXtalk.

'

I

:*-

THE. .

#

f!IitEu ]=BHRs

Bu - J Lau -WXHHOFact 22=0

.

Etopw ) - E = I § 11 du -

wxhxodtldsd't =§U*wtwHly

. ) -

WHly+,

Def . HFT WH ) :=HMCFTWHI.se )

. SH't

IX) life HF 'T WH ) w.r.tn .

K : HFTWH ) →HET artist )

continuationmap

Symplectic

admin complex via telescopic construction

Def

SIX) : -

IQ.CI#twHlIoD.degg---i;ai=0Telescope construction of the wrapped complex as

Clara] - chain cpx

2 2

w/ N' = Cone ( K - Id : EI.CI#LwHlD ) a a

CECH ) CF 'T2HlTH KM TzdKy .

. - -

ie . µ'

:

Xtg. Y

tsl- n'

"

ax -11 - 1)

" "

today + KY - Y )gCf*fH ) qCF*C2H )

Glad -

linearon or

Rmk Meaning of

.IQ/CF*twHltOCFTwHlq

)a a

T

yield representatives of HFTWH ) )For Yul 21=0 , Identify y

and Ky via lily .gl

⇒ ly ] = Cky ] in

Stix)

Prop It 'T

X.MY =

SHHH

Acs structures

µ d. Fin

: Cftc watt ) #.

.. ④CFTW.HN#CFTwoHH9I

[ 0.13

Is #

exeehaha

E EIR

Eeopw )=E

I do - Xi ,④ of

'

'=o where & : determined by 2g-

eqivariancy of µd' Fin.

.

2g gid . ④ . . . ④ gi : #= It

, thudgidxdxo . . -

④ aglgikx , . ) ④ -. . ④ gi :X!

µd=¥aµd%: #X ) ④ .

. . ④SIX) →SETHProp I Ritter - Smith I (SHIH,Md ) : Aos -

category .

g. µKey ingredient of proof : Degeneration of domains !

3 String mapsand their compatibilities

Summary of"

classical results"

7hm I 80 : SHH X ) → HH IWM ) closed -

open map as -1 - alg . mor.

Cor HH't

I Wha ), HH * IWM ) : SH TX ) - mod .

② I 08 : HH* LWIXD → SH

't

IX ) open- dosed map as Stik )

- mod. mor

.

New features :

compact analogues

Def . FIX ) : = ¥!s.

FIX ) kept ) Fukaya Category as As cat

by considering only Cpt Lags . and Hom ! Li

.Li ) : = CET Li

.

L; ) as in closed

case .

. Qtftxl :=H*lQC*Lx ),

a ) quantum cohomology as A- alg .

Thru Above results hold . replacing WH ). SHIH by FIX )

,Q HMX ) respectively .

Cor HH 'T 't KD, HH * IFM ) :QH TX ) - mod .

Upshot The above Thins are compatible .

Acceleration Functor

Goal Define HF : FIX ) → WIX ) as As functor

Naively : Ob : ↳ Las cpt Lags

Mor : CFT Lo, 4)

"

as "CW*( Lo,

L, ) as

"neo piece

"

Issue ¥ w = o piece in CWM Lo,

4) = ¥ CFT Lo,

L , :

WHITE]

So EExtend CW't

l Lo, 41 to include CELLO , L , ) for Lo ,

L,

: cpt

Def ① Wok ) extended wrapped Fukaya Cat. of X as As cat

.

•

Obi = Ob C WIX ) ) Hom w

'll,

L,

),

Lo or L,

: cylindrical

° Mor : Homwollo.li )f

C # ko, 45193 to ¥ CELLO

. hi wth -193,

Lo,

Li optI ,

d !!•

Me: essentially the same as Minix , CW Lo ,

Lil

② FIX ) E Wok ) i full sub cat . of opt Lags.

Homological

Lemma ①

Sw: WIX ) → Walk : A

. -

gis' IEEE-5W

'

: Wow → Wky )

Ob : t.

↳ LI g

asi - equivalence )

Mor : -5W: Hanako , 4) ↳ Homw! Lol,) ;.tw :-O

② I : FIX ) →

Tak ) : As -

q 's similarly defined asabove

Def acceleration functor FIX ) - FENIX )

fats on's aisslfirTak ) a Wish )

HHTIHH

't "

( only

hnetorialw.at?gflsAosubcatjHH*tHxhtH*ITHH*lpWHHHtt.tt#Iitftiii'HH*fqrlXll1511Ill I

HHxttolXD-HHIWHXDHHITHXD-HHIWa.IM

Canonical map

Def ①

SEOK) extended sympl . coohaincpx of X as As alg .

SCI CX ) QETX to CFT wth -193

d. Mda: essentially the same as

Msc't

Cor ① SEIN -

SEIN) : A

. gisE

SH't

IX ) ⇒SHIK

) : A -

alg isom.

QECX ) -

SEIN) : chain map of alg .

E

it : Qtftx) →

SHIK) :

A -

alg map5. t

.¥ Tio

I Stfu )

( Viterbo ) canonical map: "

non opt PSSmap

"

PropI Ritter ] QTFIX) toSH't

IX ) Aab . mapin > SHH x ) : Q HMX ) - mod

G O preserving c,

- action

( or C,

:=

@iTXHtMmslElCilTXD.t) ) = :c

,

Def Qtftxh,

SHHH,

generalised I - eigen space . XE Speck ,) i generalised eigenvalues

Compatibility

Thin ( Ritter - Smith ) I acceleration diagrams .

Axes

HH*

IF

,fxDH¥HE, HH *

I

WHNo:÷¥i÷÷÷÷÷:j.at#xi..modx x ) as ring

HHIFI

xDTia ;

HH lWHY

as

, -

-

-

- -

--

-

⇒

Key ideasof proof . FIX ) a, Wo I X ) N W CX )

. QIN ) as SEOK ) a SEC x )'

--

- - .

-

- --

--

"

E• Of

, 80 extends to 0.

- FB

Upshot The above diagramsdescend to I - eigen simmonds

Thru ( -a s - )

Key ideas of proof . Horizontal arrows : tautological .

. Vertical arrows : 2 key lemmas :

Lemma I I Auwuxi Ritter -

Smith ) V LE ob C Walk ) l resp .FIX ) )

,

GOO : Q CX ) SEH ) CWHL.

L ) satisfies

[ X ] 1- E whom. unit

a L TX ) I -1.

eat U'

la)

Lemma 2a I TX ) -

2.20HH

*I

WHnil potently :

titty

H WE Htt * I M ),

7- N sit .

Cc . -

a4Now = O

.

Proving OG : HH * lwxlxl ) → SHIH,

:

①U

W

⇒ C c,

-

x.

I w -

- o n°8, ( c

, - x. I. 081W ) -0

,o

titty .

Sti - nod §H*µ,

Proving 80,

HMX )HHHWnkxb.tn# '

Y 1-1%01×1=0.

⇒ Cci - a × ⇒ i

¥¥ie04¥¥%=unite ,

'

governing .es : isom. ⇒ 80K¥

§H*tX) ring nor. Hfftfwcx , ) HWTL.LI As -

nonsense

4 Generation Criterion and applications

Det

¥

.ae?SfunsgpliatjgeneratesAi-V-Ao-obA

.

⇒ twisted complex built from

obsw/ A as a summand

.

Th m ① I Abou zaid ] X : Liouville med i W I X ) Z S

→ HH *

IS) → HH * I Who ) SHEN

'

-- -

-

-

- -

- s

08 Is

If Im 108 Is) contains I,

then S split generate .WH )

Rink By Shi -

equivalency of 081s

.

it suffices that Im 108 Is) contains Stf - invertible.

② I Ritter - Smith ] X :

convex ;

WIX)

IS→ HH *

IS) → HH * I Who ) SHEN'

-- -

-e- -

- s x

08 Is

If Im1081g

) contains invertible .then S split generate .

WIX)

Rmk ② still holdsI also for X : dead ), replacing WIX )

, SHH ) by FIX ),

Q HMX ) respectively .

Idea of proof

Notation W W.

#S

.

IT: test object

E.g

L ! -homglk. - I:SCh

Rg! - honest, Ks:S→ ch

Yoneda modules

. D :S → L ④ R copwduct as S - bimodmap

8

HHHOIHHx-CSHHHIS.HR#H*lRxosL )

s-

Dth

Key diagram-

HHIS.lt#QH*tRxgHassume/0eolstII tamp

.si#n-f)AEJVuStflXh.eoo-sHW*lK.k

)

Final step I I

Applications

⑥ I Abou zaid ] T ZTIQ split generates WITH

)zS:={T5Q}

" " " " " " " " " "

*" " " " " " " " ^

)key diagram

÷¥¥ILae:¥a÷÷.is#:m.mapHH*lC-*kaQDH*cggsHn-*kQ

)u

Lemma s 7- b Goodwill ie mapI L ] L

✓unit

.

..

7- a s . e. Ofc =L ⇒rrt . ✓Rmk Actually TIQ generates

WITH t .

'

7- F )

②"

Practical generation criterion" for X : monotone convex

Goal Showing L split generates WIX ).

X - mom

IT Gen .Crit .

080 : HWML.LI → SHHH,

hits invertible

for L : closed IT accel. diag .

a " "'

' " ¥

:±±¥÷.

"i÷i÷i÷÷fThin Of : HFML.4-sqtflxh.esIt for TEO

O

checkable for X = Tot I E → Ot - k )4pm,west)

Reasons ① [ Ritter 't5) ✓ QH 'T Oth ) ( for * o.

QTFLOHYEA )

② [ Ritter - Smith ) Hato,

choose L= Lz

Cmonotone) Lag . Tows fiber ⇒ .HF*=Hw**o%°QH*U u

7- Z E Crit ( Won ,)§

"

-7247,

.

EP "

1-s-kx.tiwr.sth.at. # O

¢ → Of -

ki-0 pm Z Tai

sit . Won.fZ) = Xt out:

' t tOOH ,01pm ay ;!!

②Monotone closed toric sympl .

¥,

" "

meFEET

setup-

in.co.ma#o.eirnEx

' '

Y'e ' R

:*.

Ul 40

Lag µ Into ,

10.1 )

U U

TI Ly → y& =L ,

→ y

Z t Iz. W :( rid

"

→the LG super potential Wise → A

Ul Ul U U

ZE critlw ) → Crit Vfw ) '

II Speak , fqpg ,I ⇐ It#HI 2T

'tI i GO

QHTIPYE-szwotkwzw.ie ,

I Ivatt W%, c→ zuI t " '

then 12W).tw#t4=t2T.ywttE)

I C- ( 0,1 ) s

II. Ybaf Intlbarycenter e

g ⇐2T't ) i distinct ;multi

.

II u

Lz :'-( Ly,-5 Yz E H' I ↳

idol) w/ Moltz ) = z

HH = ( Tai.

It C- H 't to ) )holonomy

ThmI-sspeccaoQHTBDidistinctofmutti.LCor-ai.tl

) split generates F±. ÷

UP'

).

then Lz split generates FIB ).

Ps ;080 : H #↳↳ →QHMBh.ES

¥3 I → 08%4*1*0