degenerations of K3 surface * A. Braun (Oxford) and T. Watari - - PowerPoint PPT Presentation

Heterotic—IIA duality and degenerations of K3 surface

• A. Braun (Oxford) and T. Watari (Kavli IPMU)

April 23, ’16, Southeast Regional Meeting

*

*

away until Aug. ‘16

based on 1604.xxxxx (appeared yesterday)

• Duality Het IIA @6D
• 6D eff. theories w/ (1,1) SUSY

fibred adiabatically over 4D N=2 SUSY.

4 Narain

T

K3

(4,20; ) O (4) (20) O O 

4,20

Isom(II )

3

IIA K3-fib.CY M 

2

Het /" "K3 T 

1

Kachru Vafa ’95 Klemm Lerche Mayr ’95 Ferrara et.al. ’95, Vafa Witten ’95, ……. Seiberg ’88, Aspinwall Morrison ’94, Vafa Witten ’94 …

• fibre adiabatically over

– first step: specify a lattice polarization of K3 (IIA). – second: two aspects to study

• further discrete choices in fibration.
• degeneration of fibre. not adiabatic.

1

[ ]

S

U   



T

 

4,20

II  

8 9

( ) k ik 

6 7

( ) k ik 

( )[ 3] B iJ K 

(K3) 

“fixed” over vary over

1

1

Part I Part II

Part I: Duality Dictionary of Discrete Data

• Multiple choices of lattice-pol. K3 fibration

3 K

 

S

 

4  2 

1,1

II U 

1   1   2.  

3 3

2

K 

 

Choose any one from

3 CY

 

For blue points only.

1,1(

) 1, h M   

Candelas Font ‘96

toric data (polytopes)

• Multiple choices of lattice-pol. K3 fibration

3 K

 

S

 

4  2 

1,1

II U 

1   1   2.  

3 3

2

K 

 

Choose any one from

3 CY

 

For blue points only.

1,1(

) 1, h M   

Candelas Font ‘96

M =ell.fibr. over

n

F

2 2. n    

toric data (polytopes)

• Multiple choices of lattice-pol. K3 fibration

3 K

 

S

 

4  2 

1,1

II U 

1   1   2.  

3 3

2

K 

 

Choose any one from

3 CY

 

For blue points only.

1,1(

) 1, h M   

Candelas Font ‘96 Klemm et.al. ‘04

• Multiple choices of lattice-pol. K3 fibration

3 K

 

S

 

2 

1  

3 CY

 

Type IIA on CY3 Het on “T2 x” K3 instanton 4+10+10

Kachru Vafa ‘95

1,1

1 #( ) 3 h vect   

2,1

1 #( ) 129. h hypr    which one is dual? GW-inv of vert. classes Het 1-loop + <div.div.div.> intersection threshold

Kaplunovsky et.al., Antoniadis et.al. ’95 Klemm et.al. ‘04

• Multiple choices of lattice-pol. K3 fibration

– 4319 choices as toric hypersurface – 3117 of them admit -K3 fibration – 1983 of them come with multiple choices,

• sometimes the same , sometimes not.

3 CY

 

( , )

S T

 

Kreuzer Skarke ‘98

1,1(

) 1, h M   

S



with

…… in Type IIA language

2,1

h

• A. Braun

• In the case of deg.2-K3 fibration

3 K

 

S

 

2 

1  

3 CY

 

Klemm et.al. ‘04

Type IIA on CY3 Het on “T2 x” K3 instanton 4+10+10

Kachru Vafa ‘95

1,1

1 #( ) 3 h vect   

2,1

1 #( ) 129. h hypr    exploit detailed info. of hyper –mult. moduli space

Braun TW ‘16

2 (6) 2 3 4

( , , ). y F X X X  each coefficient  polynomial on right DOFs for or not?

1

8 8

E E 

which one is dual?

3 K

 

3 K

 

Braun TW ‘16

( ) ( )

r d K 

 

( )

B B

r dK  

• coeff. w/ scaling

section of

• In the case of deg.2-K3 fibration

3 K

 

S

 

2 

1  

3 CY

 

Klemm et.al. ‘04

Type IIA on CY3 Het on “T2 x” K3 instanton 4+10+10

1,1

1 #( ) 3 h vect   

2,1

1 #( ) 129. h hypr    exploit detailed info. of hyper –mult. moduli space

Braun TW ‘16

2 (6) 2 3 4

( , , ). y F X X X  each coefficient  polynomial on right DOFs for or not?

1

8 8

E E 

Others do not allow free 4+10+10 instanton interpret’n.

Part II: degenerations of K3 and solitons

{( , , ) | } X x y t xy t  

An example of degeneration

t 

• Type IIA / M = deg-2 K3 fibr. over

1

Add point(s) from

3 3

2

K 

 

1 8 2 2

Bl ( ) S  

10 2 7

Bl ( ). [ ]

C

F S dP    

Friedman ’84, …., Davis et.al. ’13, Braun TW ‘16 ruled surface

• ver ell. curve C

• Generalization
• f IIA /

1 3

ell.K3fibr. over with degeneration ell.fibr.over Bl ( )

k n

CY F   (-1) (-2) (-2) (-1) (-1) (-2) (-2) (-1)

2 1 1

RES R (T ) ES

k

S



    Dual to Het / T2 x K3 with k NS 5-branes

Morrison Vafa ‘96 Ganor Hanany ‘96

( )

k n

Bl F 

• Generalization

IIA /

1 3

• K3fibr. over

with degenerati pol.

• n

S

CY   (-1) (-2) (-2) (-1)

2 1 1

RES R (T ) ES

k

S



   

Type II degeneration of lattice-pol. K3 surface

generic fibr. degen. to

1 1 k k

S V V V V



  

rational surfaces monodromy

: ( ) ( )

T t T t

S T S   

2

:exp[ ], 0. T N N  

t

S

1

• fibr
• ver ell. curve

Clemens—Schmid exact sequence

Type II degeneration of lattice-pol. K3 surface

generic fibr. degen. to

1 1 k k

S V V V V



  

rational surfaces monodromy

: ( ) ( )

T t T t

S T S   

2

:exp[ ], 0. T N N  

t

S

Kulikov , Persson, Pinkham, Friedman, Morrison, Looijenga, Saha, Scattone, ….

2 2

1 1 , N a a                                    

1.    [rank 4] [transc. lattice] R  

T



Het dual: soliton, monodromy in Narain moduli

• fibr
• ver ell. curve

1

Clemens—Schmid exact sequence

• back to examples. (deg-2 K3 fibre)
• Het interpretation: defects in = corridor branches

– NS 5-brane: – 1st eg. above:

1 8 2 2

Bl ( ) S  

7 10 2

[ ] Bl ( ).

C

S F dP    

• degen. to
• degen. to

7 10 2

( ); , R E D  

17 3

; . R A 

all fall into 4 classes for deg2 K3 Type II degen.

8 8

, ,

S

U R E E    

1

7 10 2

2 , ( ); .

S

R E D      

Braun TW ‘16

• degen. to

8 2(

13 [ ) ]

C

d V S P      

2 8 1

R E A



 

[rank 4]

T

R   

• More varieties in degeneration of K3 surface

– Type III: dual graph = triangulation of sphere

• monodromy
• construction: Davis et.al. ’13
• more hyper-moduli -tuned solitons.

– non semi-stable: reducible fibre with

• turned into semi-stable, after base change of order k
• would-be Type II or III.
• Lattice polarization: which pair of solitons can

be BPS together.

1. m 

3

exp[ ], 0. T N N  

3

exp[ ], 0.

k

T N N  