1. The Kreuzer-Skarke Axiverse
Liam McAllister
Cornell
String Phenomenology 2018, Warsaw, July 2, 2018

2. Based on:
Mehmet Demirtas, Cody Long, L.M., and Mike Stillman,
“The Kreuzer-Skarke Axiverse,” to appear.
See also:
Talk by Demirtas in parallel session
Braun, Long, L.M., Stillman, Sung,
“The Hodge Numbers of Divisors of Calabi-Yau Hypersurfaces,”
Long, L.M., Stout,
“Systematics of Axion Inflation in Calabi-Yau Hypersurfaces,”
Long, L.M., McGuirk,
“Heavy Tails in Calabi-Yau Moduli Spaces”,

3. Main idea We explore the landscape of CY3 hypersurfaces at large .
We can now triangulate and obtain topological data of CY3 anywhere in the Kreuzer-Skarke list, in seconds per geometry.
At large we find narrow Kähler cones, large volumes, ultralight axions, and small field ranges.

4. Plan Motivation Computation: Kreuzer-Skarke at large
Results: large volumes, ultralight axions
Outlook

5. Motivation
To learn about quantum gravity, can study ensembles of explicit compactifications of string theory.
Use to refine, refute, or support no-go conjectures
about de Sitter solutions, super-Planckian field ranges, etc.
Complementary to broad-brush landscape statistics
suggests new statistical models
Can inform questions that hinge on topological data
e.g., axion alignment from special axion charges of instantons
May find new mechanisms for moduli stabilization
and reveal relative prevalence of known mechanisms
Do not know what we will find, so we should look!

6. Motivation
How can we generate and characterize a large ensemble of explicit solutions?
Setting: type IIB orientifold compactifications on Calabi-Yau threefolds.
Take CY3 hypersurfaces X in 4d toric varieties V.
Such V correspond to triangulations of 4d reflexive polytopes, all 473,800,776 of which Kreuzer and Skarke have enumerated.

7. Goal
Generate and study a large ensemble of 4d effective quantum gravity theories by analyzing (subsets of) KS.
Requires adaptations of existing packages, as well as new algorithms and new mathematical results.
Realistically, begin with separate studies of different aspects of EFT (e.g., instantons, kinetic terms, D-brane configurations), but hope to merge someday.
What can we do to build such an ensemble?

8. Computational task Triangulate polytope
Examine fine regular star triangulation to obtain
Toric variety V, Mori cone of V.
Intersection numbers of
This gives the metric on Kähler moduli space, at leading order in and .
A. Braun
Oda, Park; Berglund, Katz, Klemm
To compute ED3 superpotential, need topological data of divisors D of X.
Witten
cf. scenarios:
Kachru, Kallosh, Linde, Trivedi
Balasubramanian, Berglund, Conlon, Quevedo
Bobkov, Braun, Kumar, Raby
Blumenhagen, Jurke, Rahn, Roschy
Bies, Mayrhofer, Weigand
Braun, Long, L.M., Stillman, Sung

9. Computational task Triangulate polytope
Examine fine regular star triangulation to obtain
Toric variety V, Mori cone of V.
Intersection numbers of
Number of relevant points = (‘favorable’ case)
Number of intersection numbers ~
A. Braun
Oda, Park; Berglund, Katz, Klemm
Number of triangulations/polytope grows exponentially.

10. accessible now 473,800,776 4d polytopes h1,1 h2,1 many, systematic
hardest
accessible now
cf. Huang and Taylor
h1,1
many, systematic
few, limited
h2,1
Altman, He, Gray, Jejjala, Nelson
Cicoli, Ciupke, Mayrhofer, Shukla
Altman, He, Jejjala, Nelson
Braun, Lukas, Sun
Long, L.M., McGuirk
Taylor, Wang
Gao, Shukla
Cicoli, Krippendorf, Mayrhofer, Quevedo, Valandro
+ many more
473,800,776 4d polytopes
easiest

11. Computational task Triangulate polytope
Examine triangulation to obtain
Mori cone of V
Intersection numbers of X
Used and
FRST method: A. Braun
Intersection numbers in :
1 CY3 at in 30 min
New intersection number algorithm:
1 CY3 at in 30 sec
All 653,057 CY3 with in 1 day
1 FRST/Δ for in 2 hr

12. Strategy: go big
Enumerating geometries anywhere in KS is feasible. Obtaining all distinct geometries is not.
Idea: develop statistical models of particularly interesting properties of EFTs.
Obtain these by enumerating millions of geometries at
Find empirical scaling laws with
Use as small parameter in model (e.g., RMT).
Aim: understand broad properties of EFTs in bulk of KS, before a purely mechanical ingestion of all of KS.
Possible that most vacua are in this realm.
Long, L.M., McGuirk
Bachlechner, Dias, Frazer, L.M.
Bachlechner, Long, L.M.
Long, L.M., Stout
Heidenreich, Long, L.M., Reece, Rudelius, Stout
cf. fourfold case: Taylor, Wang
cf. String Axiverse, Arvanitaki et al.

13. Random matrix theory = Supergravity Hessian: Wigner+Wishart+Wishart
Marsh, L.M., Wrase
Alignment from eigenvector delocalization
Heavy tails in Calabi-Yau spectra
Long, L.M., McGuirk
Bachlechner, Dias, Frazer, L.M.

14. Limitation: Weak Coupling
Topological data contained in KS determines (many aspects of) EFT only in the regime of weak coupling and large volume.
Hope to learn about EFTs beyond this regime of perturbative control, but we have to start somewhere.
This talk: implications of requiring control of curvature expansion.

15. Kähler and Mori cones

16. Kähler and Mori cones
Kähler cone: = set of cohomology classes of Kähler forms.
Mori cone: = cone of effective curves.
i.e. classes realized by 1d subvarieties.
are dual cones:
generators of
Stretched Kähler cone:

17. Stretched Kähler cone Kähler cone generator Mori cone generator

18. Stretched Kähler cone
Logic: hard to compute perturbative corrections as functions of moduli.
But when an effective curve is small, can have unsuppressed WSI.
So use as a proxy for control of expansion.

19. Key observation When , the condition
implies that some curves and divisors are very large.
What is the origin of this phenomenon?
How large is the effect, in real Calabi-Yau threefolds?
What are its consequences?

20. Key observation
Suppose:
Solution:

21. Stretched Kähler cone Kähler cone generator Mori cone generator

22. Stretched Kähler cone

23. Example
Prime toric divisor volumes:

24. Approximating the cones
For a CY3 hypersurface , we cannot actually compute Kähler cone of X when
Can compute ; “it is often assumed that this is a good approximation to ”
Can also compute cone generated by curves of the form
Call this cone
Knapp and Kreuzer

25. Approximating the cones
cf. Cicoli, Ciupke, Mayrhofer, Shukla

26. Results:
Narrow cones,
which beget
large cycles

27. Kähler cone walls are nearly parallel

28. Stretched Kähler cone is far from origin

29. Intersection numbers are sparse

30. …and of order one

31. Calabi-Yau is LARGE

32. Divisors are large

33. …so some axions are ultralight

34. Ultralight axion caveats
Holds in region of control of curvature expansion.
If all divisors are large, all ED3 wrapping divisors give small terms in superpotential.
What about axion masses from ED3 wrapping non-holomorphic cycles?
That is a whole talk. Bottom line: not known, quite interesting.

35. Axion fundamental domain is small
in the most-restrictive region

36. …but can be large
in the least-restrictive region

37. Implications
PICK ONE: corrections, massless axions, small
Stabilization by nonperturbative effects appears difficult at large , even if we could choose which divisors are rigid.
Many couplings controlled by Can search our data for outliers with anomalously small volume, and Planckian axion decay constants.
cf. Douglas Denef Florea Grassi Kachru
Dimopoulos Kachru McGreevy Wacker
Rudelius
Bachlechner, Long, L.M.

38. Cosmological constraints
Kreuzer-Skarke axiverse contains many axions, some massless, with small periodicities.
Possible constraints from couplings to SM (helioscope, holoscope, red giants, supernovae, CMB, X-rays)
But first have to specify realization of SM!
Possible constraints from moduli decays to dark radiation, or from overproduction of axion DM.
But first have to specify (non-)thermal history.
Possible constraints from black hole superradiance.
Dedicated analysis required, worthwhile.
Could impact/inspire axion inflation model-building.
Sikivie
M.C.D. Marsh et al.
Cicoli, Conlon, Quevedo
Higaki, Takahashi
Arvanitaki et al.
Stott and D.J.E. Marsh
cf. Physics Reports by D.J.E. Marsh

39. Conclusions Constructed and studied a landscape of CY3 hypersurfaces:
2 million CY3 with
including one FRST of each polytope with
This is a step toward the goal of constructing a large ensemble of solutions of string theory, by stabilizing moduli in these geometries.
Can use to test ideas about quantum gravity EFTs.
When curvature expansion is under control, many cycles are very large, with , and some axions are massless.

40. Outlook
In a few months on a cluster, could get one CY3 for each of the 473,776,800 polytopes in KS list.
Number of triangulations per polytope not known, but likely vast for
What is generic in the KS ensemble?
Could estimate , and weight polytopes by this factor.
Obtaining raw topological data (intersection numbers, Mori cone of V) was the first step.
Manipulating this data to stabilize moduli and study vacua will be challenging at
Extracting physical lessons from this dataset is a natural target for machine learning.

41. Thanks!

42. Non-holomorphic instantons

43. Non-holomorphic instantons
We’ve shown that requiring that no holomorphic curve is small implies that some holomorphic curves and divisors are large.
Corresponding instantons in W are highly suppressed.
Are Kähler potential instantons correspondingly suppressed?
i.e., are non-holomorphic instanton contributions under control whenever one is well inside the Kähler cone?
Not asking that they be negligible, just not parametrically larger than instantons in W.

44. Minimal volumes Suppose X is a compact Kähler manifold, and
If is a holomorphic representative, it is calibrated by J, and absolutely volume minimizing in its class.
Specifying a point inside the Kähler cone determines the size of corresponding superpotential term.
What if has no holomorphic representative? (e.g., classes outside the cones of effective curves or effective divisors).
Can we constrain the size of contributions from D-brane instantons wrapping ?

45. Toy example
Suppose with volumes generate the cone of effective divisors.
Take
W cannot contain terms
But can there be terms ?
General question: given a point in the Kähler cone, can we lower-bound the actions of D-brane instantons wrapping classes outside the effective cone?

46. Minimal volumes
Question: let X be a compact Kähler manifold, and consider such that has no holomorphic representative. Denote by the volume of the minimum-volume representative of Is there a lower bound on ?

47. Plateau’s problem
Given a smooth curve , what is the least-area surface with boundary ?
Minimizing within the class of smooth submanifolds does not always converge to a limit within this class.
i.e., singular configurations can have smaller volume than any smooth submanifold.

48. Plateau’s problem
Given a smooth curve , what is the least-area surface with boundary ?
Minimizing within the class of smooth submanifolds does not always converge to a limit within this class.
i.e., singular configurations can have smaller volume than any smooth submanifold.
Federer and Fleming (1960): defined integral currents
Formal sums of submanifolds except for measure-0 singularities.

49. Geometric measure theory
Defines minimization of area in class of integral currents.
Fundamental theorem: a minimizer always exists!
Singularities have codimension at least 2! (Almgren)
Holomorphic chain: formal sum with irreducible subvarieties.
Positive holomorphic chain:
Positive holomorphic chains minimize area. Is a general minimizer a general holomorphic chain?

50. Are minimizers holomorphic?
Suppose X is a compact CY, and are effective.
Minimizers for are holomorphic curves , with volumes .
Q: is minimizer of given by ? In this case,
A: (Micallef and Wolfson, 2005) No.
In a K3 orbifold, they exhibit a volume-minimizer with
Holomorphic and antiholomorphic curves fuse and recombine to reduce their energy (volume).

52. Bounding Kähler instantons?
If energy reduction can be arbitrarily effective, then in principle we could find unsuppressed non-holomorphic instantons anywhere in the Kähler cone.
I have found no example of large energy reduction in a CYn. (Micallef-Wolfson case: reduction is ~ volume of blown-down curve.)
But can one establish a bound?

53. Hyperplanes:
Fundamental domain, of radius