Stable envelopes ,3d mirror symmetry ,bow varieties

Richard RimauyiUNC Chapel Hill UC Davis

April 62021

• joint work with Yiyan Shou

• learned about branes from Lev Rozansky

• related works with

Andrey Smirnov

Alexander Varchenko

Zijun Zhou

Andrzej Weber

Plain :

lcherkisbowvarief.es#Nahajimaguivervarietiey

±i÷÷:i÷÷÷.3dheir ¥z

S non

Nakajima quiver varieties .

Y Vz Vz Vn- l Vh

µiµ¥¥ n. MalW, wz wz Wh-l Wh

quiver Q quiver variety

Ex-

N (II ) = 1-*

Graci"

M In f- #In ... ... ..

4¥11 - EI

Y Vz Vz Vn- l Vh

at N ( eHFIFE.EE"

g) = . . .

W, wz wz Wh-l Wh

• R:= ④ Homicide"" ) ① Hom ( avi, Ini )

i

• µ : R ④ RE → § End"

) m= La , b]- eh

• N ( Q) i' lol"/¥GLn.

WCQ) (typet.IE#aF---IjFqf.

• smooth wi wa wz WA, WH

• holomorphic symplectic• T=(T

"

- T"x .. . xTw*)x¢¥ action

• finitely many fixed pts•"

tautological"

ur,hi

,. . . ,v*- bundles

H¥fNlQH= ?e-*Grid' take't

IX. ① HEH )n ie

PT- fixW W IF IT f/ Etty . - -anti]

If 4/24 I230$

344

1000Localization map , Loc H '×,

'

¥ µ↳ 13 3*4 14

imlloc )= ? Mr N XE m itconstraints among the ftp."

components

1/1/12

For example ,this •o3IO

6- tuple is anelement of H¥FGkQ")

-

we"

j n

(4-23)/2-4-7)Goette)fz ,-z,+tfzz••344 a•Z4 O

'*,

'

¥-O

€4- Zdftu- Zz)fz - z + t) #m#B 344 0014

3 2 § mk

fit ¥"

V

44-Zilla, -Hft - z tht -0002 3 12

Warning .

•T*GqQ" was special ("

GKM")

• In general the constraints

among components aremore restrictive -

-

-

- -

u÷¥¥t. . .

Towards Stabp E HE ( NC Q1 )Morus fixed point )

• fix ④ ITZ te (z , E , Z's . . . , E ,

I )•

FENCQT Leaf ( p) = ( x ENIQ) i zhino Hz) x =p }•

p'tp if heaftp) 7- p

'

• slope ( p) : = U Leaf ( p' )

p'

Ep

un unwat slope

I I

def stab,E HE ( NC Q1) is the unique class

[MO]

• support axiom :

supported on slope ( p )• normalization axiom:

Stabplp = e ffslopep) )OB boundary axiom :

stab pfq divisible by hi for ptg

stably

slope('")

@ 12 , 13 , 23= 0

@ 14 = ( Z,- 2-a) ( Z , - Zzth)

@ is dggigabyte .- Eth ) )

@ zu, divisible by ft ,- za)divisible by hi

@ zs divisible by tr

@ 34 divisible by ft ,- za)divisible by A

@ 35 divisible by hi

@ 45 divisible by H -za- t )divisible by hi

REMARK (2 SLIDES ) ON GEOM .REPR

. THEORY :

stab 's define geometric R- matrices & quantum group actions .[ MO]

• 6=14272-3 . . . . En, 11 HII NIQT ) H*

( NC Q1 )T

1-*1-3 Slab

, p

• other 1-parameter subgroups also define stab 's

stabs

HEC WHET→ HM wcoii ) ④ flat)stab , r

TA

-I• stab

,o stabs , = :

"

geometric R- matrix"

N -

-

= 1-*Grid U T

#Gr

,Q'

w T#

Grz

Stab , 6=(2,2-2,1)HEIN'

)# HEW) E- Giz,

i )state T

1-*

Grid I ↳ I'

Grid 1,0 1-3 (Zz- Z, , O ) ( Zz- Z,th

,te )

to , t ( hi , Z,- Zz th ) ( O j Z,

- Zz)F-Grade I 1-7 l l

swio.mil ! :÷÷÷÷: :÷÷÷÷. ! )

So far-

:

stab,e HE ( WI Q1)

+T- fixed point of NCQ)

• defined axiomatically• remark : main ingredients of

defining quantum groupactions on HE ( NCQ1) .

THE COINCIDENCE ! ! !

q254

4

1-*

Grad- N(}E÷d ) [ rsvz2020]

/ fifteenth! ship gdim =p

Stable Envelopes dim =L,

# fix pts -_ 6# fix pts =

'6

T4xE¥ action TIKI action

8- #Graci"

2-7 Nf }E÷d ) 4crsvz]

r Grad - N('

TIFF'

) 4

64 1-*

Fano .TN/.io.4!qY;. ) "

sKumo' *←m¥•

Ers]

VI.* ¥ i¥/s→Nf¥i¥at⇒z¥m¥ , ) '°

1-*

GIB c- f-*GYBL [RW 2020]

" 1-*

Fast c- N ( § )d

① What exactly is the relationship between

stable Envelopes of 3d mirror dual

spaces ?

② How to find the 3d mirror dual ?

ie what is the combinatorics that

( connects

Iq,with

K¥717'

? )( what is the mirror of 1-

*

Fasa ? )

Cher Kis bowVareitiestype

-A Naka

,CC . . . )

<

ima

quiver

varieties*an MEITEI )

¥tip ,3i4

Branediagramsqftttgg.irApted! ÷ •

,

* * • *

Bao Bao Ga

8¥ "'

YV3V4VsfggggtMfgggg't "4 us

viii. Kenzie:!7f÷:b:anime.

tautological bundlesone for each D3 brake

t.tt/brane

diagram →E ( D )

@yD5branesDJfcYra.h. ybow-Cherkis :

modulispace of Nakajima - Takayama Roz anstey :- the

unitary instantons Hamiltonian reduction "symplecticon multi -Taub - NUT of representations '

spaces ~intersection

'

(key : Nahm'sof certain quivers of generalized

equation ) with relations Lagrangevarieties

dim ( CFD) ) = §,

((dat 1) du - t ( du+ + 1) dat- J+ ¥ss2 dvtdr - - 2¥

,

di

examples

Tim ( C (MHRA ) ) = 2 - I t 2 - I t 2 - I t 2 - I t 2 - I t z - I

LL t 2 - o - I t 2 - I - O - 2 ( l't l't it i)1-* P2

= 4

How are N ( quiver ) special cases ?

- ai;÷ . - -

no B#k=

Examples 1- *p'-NII 's )=E( HAHA )

1-*Grid'=N( ¥1. )=EfK¥ )# Finn ' VK.fi/=efpkp-B#PA )

vi.'

til (AWA )Observes kik "

cobalanced brane diagram"

3D mirror symmetry for bow varieties :

it'll

⑦ 3d mirror

l't' ' I

¥ 1-*

P'

= Nf ¥ : ) - E (AHA) dim 4

⑦ 3d mirror

CLEAR't ) aim .

It!balanced,ie not NC - -

- )

. . .

but - . - L to be continued >

Hanany - Witten transition on brane diagrams .

¥°i. + i. anti

t.fi :c:÷÷÷.

The CID ) = E ( HND ) )

¥ T*p2=Ni¥, )=C(KWH)

⑦ 3d mirror

CHEA )

CIA't't't ) #claims )

n''

HEE.)

⇒ Fp'

t→ NC ,¥¥ , )

def brane charge

charge (N S5 brane ) : = l - k t #$5- branes left of it }a

charge (D5 brane ) : = k - l t # f NS 5- branes right of it }#

f T. I. I Es as as

za egg egg of Bg

charges of D5 branes• & B ⑥ &

I 8

-5 a

is a=

it *

Them The two charge &333

vectors is a ofcomplete invariant B

of HW class

Thin ( up to HW transitions )

T*G/p C N(quiver ) C E( brane diagram )I B

' ' l ' l - - - ' l l weakly decreasing arbitrary

ifI ' tI

E E

closed for transpose !

REMARK ON GEOM . REPR .TH

.CONTINUED ( I SLIDE)

Expectation-:

(known forquivers ) ②

←which representation

arbitrary①

which③ E which

FYI:fu÷÷¥Jiu :q÷;

The first data we need for• HE ( CCD))• stable envelopes :

torus fixed points-9

( a.k.a .

"exact vacuums"

)

BEoM0EBgM

fixed points tie diagrams

•

one of the?ag'Teams )•• a tie must connect 5- branes of different kinds

• each 133 brane to be cored as manytimes as its multiplicity

fixed points of T*

Grad"

"

¥¥fHEA htt Eatrn uftp.#Et- umeet

RHI

'

""

" 3d mirror i-

""" "

#Honfixedpoiuts

---- - horizontal

'

'

=. .

-

i' '

'i'

reflection

f BB.I-3¥ BE. • •

Egg tags , Toes ft tf

ljinary contingency tablesone of the 123 BCTS

BCT : O - I -matrixB Z. I ⑥ B

with row &column sums

I

the charge vectors A

AThin- I

fix pts L BCT 's B

B.

Grace"

• µ, Eng s ! ! ! !-

D { 1,33 AFRA 2 ! If I2 l l

D 4,43 fT¥ a :O'

o

'

o !Ju 2 l l

1433 f# z

'

o ,

'

i'

o

le 2 I O O l

B 1443 FRIT z

' '

i

' iY z Fo i o

Dt { 314 } fF# a

'

o

'

o

'

, iv 2 l l O

fixed points moment

invariant curves → graph →stab,

(with weight )

TEE

Tip'

3d mirror ( Ttp')

•

1-*p-

= N ( I ;)= e (HH)

Fat . -

-Is:÷÷÷: .:A →r w•

vi. fish =CHEH

'

¥ : 3⇒ t.at#.+Ea=..+.t..tEatiiC

f rational limit ( sine - x )Xitxztxz-0 , Yityety3=0 ⇒ 1€,cotfxicotlxdtwtlxzlcotkdtwtkzlatkl-cotfy.tw/ydtwHy4at- lyzltwtlyzcotfy ,)

f trigonometric limit (q → it

fit:}; }⇒ dknyddka.tn/tdk.yddf.yI)tfk..y.ldkfyd--oEll-C

Thank you!