[PPT] - Quantization Kohler structures and generalized Work Bischoff PowerPoint Presentation

SLIDE 1

Quantization

and

generalized

Kohler

structures Work

in

progress

with

Francis

Bischoff

SLIDE 2 . . .

Based

n
ur

paper

:

arXiv

:

1804.05412

SLIDE 3

19-79

Zunino

:

The

2nd

r

model

{ csih )

Is

Nig ) }

(

Lorentz

( Riemann

Inherits

N -6,2 ) Susy

from

Koike

I

!

TMO

II

I

19-84

Gates

Hull
Rock

: same

is

true

for

generalized

Kahler

D

( It

, I . )

g

compatible

complex

structures

St

.

d¥=o

,

dd'±w±#

wt

, w

Hermitian

forms

SLIDE 4

Katherine

g%=z¥Zz

( U

,

#

, ,

,

Zn ))

complex

chart

,

K

E

(

U

, IR )

Zunino :

Use

complex

str

.

to

extend

9

:S

→

M to

a

super field

I

:

E

→

M

Then

action

is

{ oI*K

droke

Q1ianalogofkforgen.tiihler.IT

SLIDE 5

Quantization

for

I koihler

,

w=gI

may

Pre

Quantize

to

(

L

, H

H

,

17 )

(

He !!

at:3 with

FCP )

iw

line

bundle

⇒

It

=

HMM

, L )

using

Je

D

"

⑦,E①L①

I

graded algebra

Q2iAnalogofkandt-forGenka.hu?

SLIDE 6

G

Poi

try

:

Hitchin

2006

,

MiG

.

2007,2010

( Itt

I

) g-

' .

Insist :*

.

" " " " "

#T=I±QtiQI±hpphu+e

QCT

't)

In

the

usual

Kiihler

case ,

QA

= w "

and

T±

=

QB

= O

Deform

a

Kahler

structure

to

a Gk

with a.

to

SLIDE 7

Construction

f

GK

:

( M

, I . ) =

a p2

T E

Ho I

MT )

=

HYP?

013 ) )

← ←

at

91

,

{

Wo

=

Fubini

study

Kainer ←

at

it

i

lenient

'

→

, '

Ii

tr in

it

⇒

I

ve

COL TM )

sit

.

JV

=o(

Wo )

Hit

{ Cv ,r]=o

I

+ =

4 ! ( I

.)

f

=

w (

Itt

a =

So

"

145 It

w .

ds

fg.It.I-lgeuerahzedkiihleroncp2.se

SLIDE 8

Summary

:

want

Gen

.

Kahler

potential

K

want

a

Pre )

Quantization

He ,

A

key

geometrical input

:

holomorphic

Poisson geometry

E

SLIDE 9

TheKoihlerpotenti#

according

to (

Donaldson

' 01 ) w =

i 25

Ka

=

I t.in#Ka)=5Aa,AaEr9Ua )

Aap

=

Aa

Ap

holomorphic

[ Aap]

e

Ht #-)

Glue

Tie

,

to

Tq

,

/←ahkrc

,

using

translation

by

Aap

i⇐÷÷÷÷÷

'

I

SLIDE 10

Result

Z

=

II

TEKE

Est Aap )

(

r )

÷÷÷÷::÷:÷ii .tt#

holomorphic

symplectic

with

rly

=

d (

i 2kg )

= w

Real

.

L

is

[

Lagrangian for

Imr

Symplectic

for

Re R

" I!! Effie!:# entia.LisanA-braneforlmr.LT

SLIDE 11

Ex

( P

' ,

Wes )

Z

=

affine

bundle

with

class [ w ]

e

H' I

r

' )

=¥÷÷: :

inte

i

r

=

dxidy

Lz

L

=

{

real

locus }

SLIDE 12

Moritaca-te.org

Weinstein

:

study

poisson

( X

, r )

via

its symplectic

realizations

it

i÷:

" i

"

tD¥¥

t IT

( Kerth Ttr

lnvolutire

quotient

is

Poisson ( X

, 't )

( X

, ol

( x

' , r ' )

SLIDE 13

Morita

2- category ( Weinstein)

Hr )

anti

.

sp

¥ip

Mort I

x , X ' ) =

( xp ,

I x

' , r 't

sit

.

Ker t*

I

Kent '*

i.

e. {4,0×1}=0

Mor

' I 2- it '

)

=

local

isos

ft ,r )

→

LZ

' ,

r

')

Mor Pic ( X

, r ) = {

§!!! ) µ

Picard

group

.

Identity

in

Pic

is

a

distinguished

self

equivalence

:

Weinstein grouper'd

.

SLIDE 14

⇒

: :::::÷.÷ :* :*

"

Z

=

T*g

,

r

=

scan

a

.

( X

a

' ,

=

x Indy )

2-

=

a'

x

et

a

I

x ,y , a , b )

Z

sf

It

sf It

ix. y)

Ceax , ytxb )

X

r=

dandlytxb

)

dbndx

SLIDE 15

theorem ( Bischoff

,

M.G. , Zab zine ) (Assuming

QI

' =

F

exists)

A

Generalized

Kahler

structure

Cg ,

It

, I

)

is

equivalent

to

a

Hot

. symplectic

Morita

equivalence

CZ , r ) :(

x.

rt

→ txt ,q)

together

with

a

nondegenerate C

's

brane bisection

#t

.

'

. .

÷¥

.

is

z

It

( X

.

f- )

( Xt , rt )

SLIDE 16 . Differ

X

.

ELE

Xt

÷÷£

. s ,

t

foliations

induce

It

, I

n

L

.

Is

Z

It

My

=

F

real

and

#

F'

' ' ' ' +

It

=

F

' " "

I

. =

g

( X

.

f- )

( Xt , rt )

unique

symmetric

tensor

S 'T

't

require

Riemannian

.

SLIDE 17

Potential

function

" ii.

art

dry

=

rly

is

read 1mHz )

=

dk

Ke

LU , R )

⇒

n=2i2K

⇒

dz=r/y=-2i22⇒

g

determined by

real

smooth

function

K

.

SLIDE 18

E

( X

.

a

' ,

=

x Indy )

( Z ,r )

:L X. DO

2-

=

a'

x

eh

a

(

x ,y , a , b )

{

. if

it :*

. ,

r=

damn dlyntnxeb )

tdlszlrdx

I

Pi

9

,

Pz

192

Darboux

chat

.

{

K'

=

19212

Liz (
19212 )

⇒

complete

Gen

.

Kohler

metric

n

EZ

SLIDE 19

Quantization

SLIDE 20

Gutowitfen

:

embed

( M

, w )

into

( Z

,

r

)

sit

.

n/µ=

w

#

r )

ie

.

mis : :;s÷:* :*:L:3

⇒

two

branes

M

,

Zee

(

L

Space filling

brane Lagrangian

brave

⇒

H=HomlM,Z#T

SLIDE 21

proposalforquautization-LC.CZ

,

SL )

:

two

branes

in

A

model

brane

Morita

f

( £

,

Imr )

bisection

equivalence

.

H=Hom(L,Zcc€

SLIDE 22

Construction

f

GK

:

( M

, I . ) =

a p2

T E

Ho I

MT )

=

HYP?

013 ) )

← ←

at

91

,

{

Wo

=

Fubini

study

Kainer ←

at

it

i

lenient

'

→

, '

Ii

tr in

it

⇒

I

ve

COL TM )

sit

.

JV

=o(

Wo )

Hit

{ Cv ,r]=o

I

+ =

4 ! ( I

.)

f

=

w (

Itt

a =

So

"

145 It

w .

ds

fg.It.I-lgeuerahzedkiihleroncp2.se

SLIDE 23

Iterate

the

construction

,

⇐

, :

( X.

DO

7oz
.

a

2oz

Z

, z

e

.

Zoe

£12

723

Ly

d

y I

a ,

d

a,

t

n ,

X

z

X

,

Xo

X

,

Xz

t-qtO.HomCLoqZ.IT

algebra

lndep

.

f

Hamiltonian

deforms

f Lok

.

SLIDE 24

for

IP

' ,

Mor

Pic

( P

' ) =

PG ↳ ( a )

x

Pic C P

' )

( 4

,

Ocn ) )

If

we

compute

in

this

case

4=

Id

A

=

to Ho ( IP

' ,

Ockn ) )

\

k >

let

Id

Ay

Vandenbergh

noncommutative

deformation

f

A

,

SLIDE 25