Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
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
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
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 ? )
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"
¥ 1-*
P'
= Nf ¥ : ) - E (AHA) dim 4
⑦ 3d mirror
CLEAR't ) aim .
It!balanced,ie not NC - -
- )
. . .
but - . - L to be continued >
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
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
'
¥ : 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