The logarithmic gauged linear

Sigma modele Theory

Joint work with Felix Janda

Yongbin Ruan

Adrien Sauvaget

Input on CWsidex smooth prooj varietyIDMstack e.g Ip

v b Ix e.g V 0h52p't

w c How s t Y Ako CXsmooth of codimYx rkV

W n V a st Y dw o

eex E w action

mm R chargec Po V W EE

Rink in Ipo needs not to be V b2 Ipo can't be proper

stable maps with p fields

nodal curve

no markings forsimplicity

fuglieing ive we owEI

Pox w c ew PII J f map with p field

Igp field

f Cls x PCHTf V we

f stable if h is stable

f RLS M instanton moduliMP f C PE stableU

e INDy

I

UVK.cdw.gg

P o

u NP has G w action

2 It recovers CW of Y w oj

Kien Li cosection localization

MP c A NYS

rely on upoW EEs t Ca XP hey rir

2

b Guido NP rely on up EI

Chang Li Kim oh Chang A L

C J Webb R Picciotto

es MP is non proper

We fix this

The RS CJR

There is a compactification

XP c upsit

1 UP is propyer Dan with 6 action

2 3 vis Cycles

up upred upyup

red

depend ondepend on

poop E w Pkp Ciw W

s.t a MP up red

Cbs upVir

upJed m upbupyred

m order of poles of W along co Flpo

How do we compactify MP

How to construct the reduced theory

FLO use stable log maps ofAbramovich C Gross Sierber

1 Poc Ip 2 QIc 11 as a log varietyW c PCV G co

2 Pc i P hq

j J f IvymapClog curve

s

stability f representable

why jtf f Hk f as o

for Ks o I Hs o

Hi ample class on X

Wc at

Fordprinciplizationof piPlµP

UP f jyp5 I

log Rmaps logRmapswith specialconfiguration

Kiem Li coseetion extends to UPIMPnicely non degenerate

poleof order poleof orderof W alongs

up red