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 based on 1604.xxxxx (appeared yesterday) away until Aug. ‘16 *

• Duality Het IIA @6D 4 Narain K3 T (4,20; ) O  Isom(II ) (4) (20) O O 4,20 Seiberg ’88, Aspinwall Morrison ’94, Vafa Witten ’94 … • 6D eff. theories w/ (1,1) SUSY 1 fibred adiabatically over 4D N=2 SUSY. IIA K3-fib. CY T  2 Het /" "K3  M 3 Kachru Vafa ’95 Klemm Lerche Mayr ’95 Ferrara et.al. ’95, Vafa Witten ’95, …….

1 • fibre adiabatically over – first step: specify a lattice polarization of K3 (IIA).  U        [ ] II S T 4,20  1  ( )[ 3] 8 9 “fixed” over B iJ K ( ) k ik   1 6 7 vary over (K3) ( ) k ik – second: two aspects to study • further discrete choices in fibration. Part I • degeneration of fibre. not adiabatic. Part II

Part I: Duality Dictionary of Discrete Data

• Multiple choices of lattice-pol. K3 fibration toric data (polytopes)   3 K U  II     1,1 4 2       1 1 S 2.   Choose any one from 3 CY    3 2 3 K    1,1 ( For ) 1, h M blue points only. Candelas Font ‘96

• Multiple choices of lattice-pol. K3 fibration toric data (polytopes)   3 K U  II     1,1 4 2       1 1 S 2.   Choose any one from 3 CY    3 2 3 K    1,1 ( For ) 1, h M blue points only. F M =ell.fibr. over n     Candelas Font ‘96 2 2. n

• Multiple choices of lattice-pol. K3 fibration   3 K U  II     1,1 4 2       1 1 S 2.   Choose any one from 3 CY    3 2 3 K    1,1 ( For ) 1, h M blue points only. Candelas Font ‘96 Klemm et.al. ‘04

• Multiple choices of lattice-pol. K3 fibration    1,1 1 #( ) 3 h vect   Kachru Vafa ‘95    2,1 1 #( ) 129. h hypr 3 K Het on “T2 x” K3    Type IIA on CY3 2 instanton 4+10+10 S   1 GW-inv of vert. classes Het 1-loop + <div.div.div.> intersection threshold   3 CY Kaplunovsky et.al., Antoniadis et.al. ’95 Klemm et.al. ‘04 which one is dual?

• Multiple choices of lattice-pol. K3 fibration   – 4319 choices as toric hypersurface ( , ) S T Kreuzer Skarke ‘98  – 3117 of them admit -K3 fibration S    1,1 ( with ) 1, h M – 1983 of them come with multiple choices, • sometimes the same , sometimes not. 2,1 h A. Braun   …… in Type IIA language 3 CY

• In the case of deg.2-K3 fibration Braun TW ‘16 exploit detailed info. of hyper – mult. moduli space   3  K 2 (6) ( , , ). y F X X X 2 3 4 1 each coefficient  polynomial on     right DOFs for or not? E E 2   8 8 1 S Type IIA on CY3 Het on “T2 x” K3 instanton 4+10+10      3 1,1 CY 1 #( ) 3 h vect    2,1 1 #( ) 129. h hypr Kachru Vafa ‘95 which one is dual? Klemm et.al. ‘04

    3 K 3 K coeff. w/ scaling   r d ( ) ( )  K section o f   ( ) r dK B B Braun TW ‘16

• In the case of deg.2-K3 fibration Braun TW ‘16 exploit detailed info. of hyper – mult. moduli space   3  K 2 (6) ( , , ). y F X X X 2 3 4 1 each coefficient  polynomial on     right DOFs for or not? E E 2   8 8 1 S Type IIA on CY3 Het on “T2 x” K3 instanton 4+10+10      3 1,1 CY 1 #( ) 3 h vect    2,1 1 #( ) 129. h hypr Others do not allow free 4+10+10 Klemm et.al. ‘04 instanton interpret’n .

Part II: degenerations of K3 and solitons An example of degeneration   {( , , ) | } X x y t xy t t 

Friedman ’84, …., Davis et.al. ’13, Braun TW ‘16 • Type IIA / M = deg-2 K3 fibr. over 1 Add point(s) from    3 2 3 K S   2 1 8 2 Bl ( ) 0 ruled surface over ell. curve C     10 [ ] Bl ( ). S dP F 0 7 2 C

(-1) • Generalization (-2)  k ( ) Bl F (-2) n  1 ell.K3fibr. over of IIA / CY 3 (-1) with degeneration  k ell.fibr.over Bl ( ) F n      2 1 1 k R ES (T ) RES S 0 Dual to Het / T2 x K3 with k NS 5-branes Ganor Hanany ‘96 (-1) (-2) (-2) (-1) Morrison Vafa ‘96

• Generalization Type II degeneration of lattice-pol. K3 surface CY   1 pol. K3fibr. over IIA / 3 S with degenerati on generic fibr. degen. to S t    S V V V V  0 0 1 1 k k      2 1 1 k R ES (T ) RES S 0 1 rational surfaces -fibr over ell. curve Clemens — Schmid exact sequence monodromy    : ( ) ( ) T S S T t T t   2 :exp[ ], 0. T N N (-1) (-2) (-2) (-1)

Kulikov , Persson, Pinkham, Friedman, Type II degeneration of Morrison, Looijenga, Saha, Scattone , …. lattice-pol. K3 surface generic fibr. degen. to S t    S V V V V   0 0 1 1 k k T   [rank 4] [transc. lattice] R 1 rational surfaces -fibr over ell. curve Clemens — Schmid     1 1     exact sequence              , N 1.         2 2 a a monodromy              : ( ) ( ) T S S T t T t Het dual: soliton,   2 :exp[ ], 0. T N N monodromy in Narain moduli

Braun TW ‘16 • back to examples. (deg-2 K3 fibre)     10   degen. to [ ] Bl ( ). S dP F ( ); , R E D 0 7 2 C 7 10 2  S   2 1 8 2 ; . R A degen. to Bl ( ) 17 3 0          2 [ ] 2 ( 13 ) degen. to S d P V R E A 0 8 C 8 1    all fall into 4 classes for deg2 K3 Type II degen. [rank 4] R T • Het interpretation: defects in = corridor branches 1 – NS 5-brane:     , , U R E E S 8 8 – 1 st eg. above:       2 , ( ); . R E D 7 10 2 S

• More varieties in degeneration of K3 surface – Type III: dual graph = triangulation of sphere   • monodromy 3 exp[ ], 0. T N N • construction: Davis et.al. ’13 • more hyper-moduli -tuned solitons. m  – non semi-stable: reducible fibre with 1. • turned into semi-stable, after base change of order k   3 • would-be Type II or III. k exp[ ], 0. T N N • Lattice polarization: which pair of solitons can be BPS together.