Theories Based

Cohomological

Field

Theories _Based

from _jointon ^work

with ^{B. Kim}

Gauged _& ^with

Linear ^{g. can-Fontanne -}

G- uéré ^-Kim

Sigma ^{- shoemaker}

Models

I ^: GLSMS

II ^{: COHFTS}

III ^: History

II ^! Construction

I ^{: GLSMS ( Roughly )}

A GLSM ^{is a G- IT} quotient

of affine space with ^a global

function ^.

c- → ¢

where

• 11/10-6 ^{is a smooth}

DM stack

• Zcaw) ^≤ V%G ^{is proper}

Data ^{: ( V} _, ^G, O _, ^W )

• V ^{Ctvector space}

• G- ≤ Glad

• ⊖ ^{: G- → character}

• wE§ymVYG ^is_function^{a G-inv}

G-ITQvo-t.int

V70 ^{) : =}

• f-_* Symv /

{^REV / ^{•• fly}_{f- (v) -1-0}^0-491

VK.ci^{. =} @ ^"%. ]

( quotient ^stack)

Een ^[1/10]

Examples ^{of affine Gltq what}

• Projective ^{space .}

• Grassmann ^{/ ans}

• semi projective ^{toric varieties}

• Quiver varieties

By ^{varying ⊖ c- ( V. GO,w)}

GLSMS specialize ^to

• Complete intersections ⁱⁿ

all ^{of the above}

• Quantum singularities ^(Efate)

w :[^{%-) → ¢}

To specialize ^{to complete}

intersections in G-

Choose

• W ^{a G-} representation

• s ^←Gym ^IN w)G

Gives

• → w Not

• <^s _, ^{-> :} totw^"→ v16 ^G-

Form the ^GLSM

(v⊕W!G ^{, 0 , < si>)}

2- (^s ) ^{≤ G}

GLSMS ^{( Precise]}

Data :( 11,721, wir)

• 1) ^{¢ - vector} space

• P≤ ^{Gen) linear red.} _gp^.

• Xt M _→ surjective ^character

• V a Q1 ^- character ^of

Define fr ^extends

• G- :=kerX ^G- by ^"R-cha.ge^")

• D- ^{: =} VIG

Require

• Vˢˢ(^{v1 -_} 11%-2.us/o-1fYgo:ki-t_off ^")

• 2- (dw) ^is _proper

Goat ^{: Produce an}

enumerative theory

for ^G-↳ Ms

• specializes ^to

Gromov ^- Witten theory

for complete intersections

in G-

• specializes ^{to FJRW}

theory ^for quantum singularities

(Ende) [^{VIG] -4¢}

• Varying ⁰ interpolates

between the above

2¥ ^{①✗ @ an +2}

= Spec ^{Q [Xo , . . . , ✗n ,} _p ^]

weights ^{: 1 , . _ ^ , 1 , - d}

a) 0--1 ^{: ①×}

b) f- ^: a.^✗ is

-2 ↳ _t ^{- '}

• f-^(x) ^≈ homogeneous polynomial ^{of degree of}

• w ^{≈ =} Pf ^(X)

a) ^{ex = tot} Opntd ^¢

Gives ^G-W theory ^{of 2-(f) ≤ 1pm} b) ^an-%¢✗=[Aⁿ%d^{] ¢}

Gives FJRW theory

TT-coht-TSDefn-AC-homologica-fieldtheory-i.si^.

• H ^a graded ^{a-vector space}

• 4-_,^-7 a supercommutate pairing

• 7- EH ^lunit)

• Rg^,:D: H←→H*(MgT⁾

• satisfying ^{natural axioms}

• permutation ^covariance

• tree

• loop

• forgetting ^tails

• metric

EI ^{1GW theory of Z)}

Let ^{2- be a smooth} variety Mg_,!aZ ^{) = Moduli of maps}

e. ^{→ z}

• 741=1-1*12-7 ^{(stale space)}

• LY _,V2^{> _' =} SV ^{, u V2}

2- n[m jir

ev¥H¥Mg,r,dlZY→H*Mg%_↓^for*

1-1*1%7

/^{IS PD}

1*-1-1*174 ^- _{- if}

?ᵈ µ*Ñsid

EI ^CFJRW theory^]

• H ^{: =}_get^{④ Jae (}Wtvg ^{) dwg}

14 ^* ? HHMJ.ir ⁾

Observe ^:

* ⁼

g-Q-o-J-aclwtpgldwg-g-Q.o.HN?Hldw1ygY ⁾

= 1-1*1=11%-7*1^duty

In general ^{, given * -90}

there ^{is a twisted} Hodge complex^:

@*%r:i-% ^{. . -7k¥ ]}

if W ^{= 0}

[ 0*-3 ^{R'✗ → . . . -5s}:)

1-1-1*1541,01=+0+1*4*1

If ^{W 13 an} isolated singularity ^{in 11th}

Idw ^is regular

& ^{(R•* , Idw) = * (dw)"-5Jackdaw}

H*( _Rj ^{, rdw) = Jacko)dw}

In general ^{, given A, Gaw)}

define

H-tt.HR_,=%o-• Mdw⁾

14%-11-1*19_Kiimeth ?-µqᵈw* ⁾

Formula

Observation

If ^{✗ c-} HIM

.si#,dw)Bc-H1*lRj-,dv)=arBc-1H*CA*,d(w+vD-dS

%^{= -?^dcw+vandWAB -1 ✗) ABAD )}

Idea ^{: Let}

Marie __M_gridᵗᵗlV,FQw^{) be the}

moduli _space ^{of LG} quasi ^maps

to he ^{→ 2-law})G

• Embedd Mg,r,d→Ug,r;d¥¥¥^?

↓ ^{en er '}

1%6-7 _¢^{¥> Most}

• Construct a virtual cycle ⁱⁿ

twisted Hodge ^{cohomology of Vg,r,d}

[ ^M]_,, ^{c- It}/_Mgird^*(RugBe ^.me _,rdlÑ◦w⁾)

Supported ^{on Mg,rid}

( ^{which is} _proper ^{by FJR 2018})

G- LSM Invariants

1[mjrir₉₁₄₄

Hf*Ngµd,AdlÑ◦w☒Y^{) -711-1*14}_.ro/Mgird

↑ñ* ↓f%*

11-1*6^.

ygy.gr ±%EMᵈw☒%^?-4%1-41%1;)

# History ^d- enumerative

invariantsofGLS.MS

• Fan^{-Jarvis-Ruan 12013) G- finite}

• Polish^{chub-Vainttib (20/6) Purely} algebraic

version

usingfactorizations^matrix

• Kiem ^-Li ^{12018) using cosection} localization

• Fan^{-Jarvis -Ruan 12018}) ^"_Narrow^sectors"

for general

GLSMS

• G-^•can-Fontanne^{-F- Gvéré - convex} hybrid

Kim ^-Shoemaker ^{120181 models}

• F- ^- Kim ¹²⁰²⁰⁾ General

case

Thin ^{( CF} -1=-0--1-1--5,2018⁾

Enumerative ^{invariants for}

convex hybrid ^{models specialize}

to FJRW theory

and ^{to GW '} theory

as defined _using ^{the co section}

localized ^virtual cycle ^.

Thm (^Kim-04,2018^{) The cosection}

localized ^virtual cycle ^{agrees with}

the Behrend^- Fantechi virtual cycle^.

Thm ( ^{F- Kim, 2020) The}general

GLSM invariants ^{form a CohFT.}

Construction

• Embedd Myrie ^{in a tm°°ᵗh}

Space

Mg,r,d ^-7 Ug,r,d

\

, .

. _

- -

"

I

, .FI

↓ →n÷

I%-% ^-

• Find ^{a "}virtual ^" matrix factorization

1kg,r,d ^on ( _Ug ,r,d,Ñ°w☒r)

supported

• [^m _]; ^{a :=} tdlbg.r.ae/chlKgisd)FeEetined

Construction ^of 1kg _,r,d

LG quasimap ^data :

• C _{genus g} ^curve

• q-tgy-n.ir^{) marked points}

• É _principal ^{f- bindle}

• K ^: px

, Cly ^→ WEE

• u :c ^→ PIV

vector

87% ^bundle_:

↓

L ^universal

curve

↓^'

{ ^{( C , q , F , K)} }

d

RiT*&×n%^{) = [ A → B)}

RIT_* (8%10)=[1--413]

her ^{D= til} 88%-7^{← extra}

want_construct^toa toÉp*Bᵈ*a

c.se#w ↓ ↑ ^'

using

≤Ugarit ^c-

"

'tot A

2- (B)

" 24%8^, Huh ↓p

= M ^" LUGG⁾

{ ^{( C , q}_, 8,15 ⁾ }

01

zlt^{) =}Mig,;D

odd ²

A ≈[Ñf*B←→Ap*B)₂

2 =) _{B -11} ^{✗ Supp}_{= Mg}^CK)-2-1%7

,rid

Problems ^{: In general ✗}

only ^exists locally ^{on Ugigd}

a-EIHYUg.ve/KlBH

Want^{: To find a matrix} factorization ^{which is}

locally ^{the Ko szul} factorization

%¥T%Factorizations ^come

from ^{sheaves of CDGAS}

:

• * ^{DM stalk}

• (A_, d) ^{sheaf of CDGAS over}

• we MCX _, 0×3

• a ^C- M( ^✗ A-^{,) s.t.dk)=w}

Then _even -2_, ^odd

A ⇐ A

2 =D ^{-1 • a}

graded ^{leibnitz rule 22__ w}

- '

. this ^{is a matrix} factorization

- Idea ^:

To realize

✗ ^e-HI ^' ( tot A)^¥1B))

we need ^to replace ^75lb)

by ^{a d-} acyclic ^complex

but retain the ^CDGA structure

Thom^-Sullivan

✗ ^(B) É%%• ^# B) ^→Th%%B

e- acyclic P5h%¥

coslhplicial

sheaf ^of ¥817S

CDGAS

Then

• ✗ ^c- Tti ÉKIBL _,

• and alla) ^{= w}

1kg,r,d:=&h•G%TBÑ¥Hh%*Ñ

locally ^{looks like}

" p*Ñ→←iᵈᵈp* _]