1. Heterotic—IIA duality map of discrete data
July 17, ’17, 　at KEK
Taizan Watari (IPMU)
based on arXiv: with A.P.Braun (Oxford)
work in progress

2. Heterotic—Type II duality
Type IIA / M-theory
mirror
symmetry
Heterotic
Type IIB / F-theory
Yau-Zaslow conjecture ‘95, Katz--Klemm—Vafa conjecture,
Harvey-Moore ‘95, Maulik—Pandharipande ‘07, …. and maybe more.

3. Duality Het IIA @6D K3 (metric, B) 16 SUSY charges preserved.
BPS particle counting on both sides:
K3　(metric, B)
Seiberg ’88, Aspinwall Morrison ’94, Vafa Witten ’94,
Harvey—Strominger ’95, Bershadsky-Sadov-Vafa ’95 …
particles with electric charge
using
counting
in Het
IIA
reasoning
math

4. Duality Het IIA @6D
16 SUSY charges preserved.
6D eff. theories w/ (1,1) SUSY fibred “adiabatically” over D N=2 SUSY.
K3　(metric, B)
Seiberg ’88, Aspinwall Morrison ’94, Vafa Witten ’94,
Harvey—Strominger ’95, Bershadsky-Sadov-Vafa ’95 …
4D N=2  right mover has
Kachru Vafa ’95 Klemm Lerche Mayr ’95
Ferrara et.al. ’95, Vafa Witten ’95, …….
Banks Dixon ‘88

5. fibre adiabatically over
first step: specify a lattice polarization of K3 (IIA).
examples:
quartic K3
elliptic K3
“fixed” over
vary over

6. 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.
“fixed” over
vary over
Part II
Part I

7. Part I: degenerations of K3 and solitons
An example of degeneration

8. Type IIA / M = deg-2 K3 fibr. over
Friedman ’84, …., Davis et.al. ’13, Braun TW ‘16
Type IIA / M = deg-2 K3 fibr. over
Add point(s) from

9. A well-understood case
(-1)
A well-understood case
(-2)
(-2)
IIA /
(-1)
Dual to Het / T2 x K3
with k NS5-branes
(-1)
(-2)
(-2)
(-1)
Ganor Hanany ‘96
Morrison Vafa ‘96

10. Generalization Type II degeneration of lattice-pol. K3 surface IIA /
generic fibr degen. to
rational surfaces
-fibr
over ell. curve
Clemens—Schmid
exact sequence
monodromy
(-1)
(-2)
(-2)
(-1)

11. Type II degeneration of lattice-pol. K3 surface
Kulikov , Persson, Pinkham, Friedman,
Morrison, Looijenga, Saha, Scattone, ….
Type II degeneration of
lattice-pol. K3 surface
generic fibr degen. to
rational surfaces
-fibr
over ell. curve
Clemens—Schmid
exact sequence
monodromy
Het dual: soliton,
monodromy in Narain moduli

12. back to examples. (deg-2 K3 fibre)
Braun TW ‘16
back to examples. (deg-2 K3 fibre)
Het interpretation: defects in = corridor branches
NS 5-brane:
1st eg. above:
degen. to
degen. to
degen. to
all fall into 4 classes for deg2 K3 Type II degeneration.

13. 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
fractional Type II or III soliton.
Lattice polarization: which pair of solitons can be BPS together.

14. Part II: Duality Dictionary of Discrete Data
restrict attention to CY3 w/o Type II or III degen.

15. Multiple choices for a given lattice pol. K3
toric data (polytopes)
Choose any one from
For
blue points only.
M =ell.fibr. over
Candelas Font ‘96

16. Multiple choices of lattice-pol. K3 fibration
Choose any one from
For
blue points only.
Klemm et.al. ‘04
Candelas Font ‘96

17. 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.
Kreuzer Skarke ‘98
with
…… in Type IIA language
Choose any one from
For
blue points only.
Klemm et.al. ‘04
Candelas Font ‘96

18. A coarse classification
labeled by #(BPS charged hyper)
Detailed duality dictionary
has to involve, in IIA language,
classical intersection ring of divisors
also hypermultiplet moduli
how to work out Het vacua systematically?
more tools need to be developed.
NL loci counting
GW inv. of vert. curve classes (IIA)
new SUSY index (Het)
massless charged hyper multiplicity (4D eff. theory)
Harvey-Moore ’95, Klemm Kreuzer Riegler Scheidegger ’04,
Maulik Pandharipande ’07, Haghighat Klemm ’09,..
(deg.2  )

19. Multiple choices for a given lattice pol. K3
toric data (polytopes)
The same #(hyper), and
The same GW inv. for
vertical curve classes,
for all of n = 0,1,2.
The intersection ring distinguishes
n= even and n= odd cases.
(12+n, 12-n) instanton
distrib. in Het
M =ell.fibr. over
Morrison Vafa ‘96

20. In the case of deg.2 K3 in the fibre,
Kachru Vafa ‘95
Het on “T2 x” K3
instanton
Type IIA on CY3
GW-inv of vert. classes Het 1-loop
+ <div.div.div.> intersection threshold
Kaplunovsky et.al.,
Antoniadis et.al. ’95
Klemm et.al. ‘04
which one is dual?

21. In the case of deg.2 K3 in the fibre,
Kachru Vafa ‘95
Het on “T2 x” K3
instanton
Type IIA on CY3
GW-inv of vert. classes Het 1-loop
+ <div.div.div.> intersection threshold
Braun TW ‘16
This one is dual to
the instant. distrib.

22. Summary
There is a long way to go to get to the level of understanding of mirror symmetry.
reflexive polytopes

23. Braun TW ‘16