FTheoryTools - Goals for 2024 and beyond

[HereAround]

To be discussed

• [ ] Agenda for FTheoryTools paper.
• [ ] Use https://docs.oscar-system.org/stable/Groups/action/ for the action of the discrete quotient on the center of the Lie group factors. E.g., the gauge group SU(6)/Z_3 would be stored as the Lie algebra sl(6), then a group action of the quotient factor Z_3 on the center Z_6 of SU(6).
  • Add these to the existing .json files in place of the current specification of the gauge quotients
• [ ] Make computation of Tate sections lazy? Would speed-up the constructor of the big model A LOT!
• Andrew:
  • [ ] Kodaira type of SU(3)s in toric hypersurface models coming back as nonsplit?? Fix and extend tests for the Kodaira type of the toric hypersurface models.
  • [ ] Documentation of HypersurfaceModels/Constructors.jl and alike needs updating. For instance D1 etc. are probably to be removed/explained better. Also, the above changes need to be worked in/should be reflected.
  • [ ] Add comments in code that fully explain defining_classes, explicit_model_sections, etc.
  • [ ] Fix tutorial (Martin: What does this mean? What is broken?)
  • [ ] Add Tate table functionality (Please elaborate. What does that mean?)
  • [ ] Improve/enhance family of spaces.
    • Martin: ~~I discussed this at length with Matthias today (Feb. 14). Based on this, I see only one way forward. Family of bases serves as placeholder and placeholder only. That is, it must always be replaced with an explicit geometry (be it a toric variety, scheme, ...) if any non-trivial computations are to be conducted, such as blowups/resolutions of singularities.~~ Based on recent conversations with @wdecker , I believe we should discuss if abstract varieties (cf. https://docs.oscar-system.org/dev/Experimental/IntersectionTheory/AbstractVarieties/) are to replace family of spaces. (Related old PR: https://github.com/oscar-system/Oscar.jl/pull/3544.)
    • Related question: Can we have defining_classes, explicit_model_sections and alike for those abstract varieties? But maybe we indeed can...
• Mikelis:
  • [ ] Update attribute that associates a Weierstrass model to a hypersurface/Tate model, so that it can construct the WS model associated to a given model from the database when another database file exists for it.
• Martin:
  • [ ] Update serialization and first steps towards database 2.0 (cf. https://github.com/oscar-system/Oscar.jl/pull/3932)

QSM focus

• [ ] Expect matter at curves between different U(2) factors, that is not charged under U(1). Add technology for these computations into FTheoryTools. (Andrew)
• [ ] Present the blow_ups (in particular if conducted on affine charts) in a way that is readable for physicists. Also interpret complicated blowups. To be discussed with Anne. (Andrew)
• [ ] Some level of support for chiral spectra. (Martin)

Final goals:

• [ ] Study singular limit of QSMs (c.f. root bundles papers) and try to resolve the singular geometry.
• [ ] Keep track of U(1) sections. (Yes, their resolution is at another level, but we should be able to keep track of them.)

New models for data base (for @emikelsons)

• [ ] Put in the model https://arxiv.org/pdf/2210.09473.pdf, to test Weierstrass model resolution. Equ. 3.10 resolves in Weierstrass form. Equ. 3.11 involves a messy locus and thus valuable test to gauge our computational state of affairs.
• [ ] Nikhil & Andrew model and Mordell-Weil -> Future avenue and to be supported in the FTheoryTools. (And paper to be released eventually).
• [ ] CKPT (https://arxiv.org/abs/1507.05954)
• [ ] Raghuram charge 3 and 4 (https://arxiv.org/abs/1711.03210) (cf. #4155)
• [ ] Exotic Matter (https://arxiv.org/abs/1706.08194)
  • [ ] Matter in Transition (https://arxiv.org/abs/1512.05791)
  • [ ] These models may require new technology for us
• [ ] Generic SM models (https://arxiv.org/abs/1912.10991)
  • [ ] Related to QSMs
  • [ ] Also (https://arxiv.org/abs/2210.09473), which is also above
• [ ] Tate with MW rank 1 (https://arxiv.org/abs/1406.5174)
• [ ] Matter and Singularities (https://arxiv.org/abs/1106.3563), maybe
• [ ] Highest MW rank (https://arxiv.org/abs/2105.02863)
• [ ] Include the model by Wati and Kobe (https://arxiv.org/pdf/2401.00040.pdf).
• [ ] Coding task: Support complete intersection models.
• [ ] Coding: Function put_over_concrete_base should work for hypersurface (and complete intersection) models.

Data and serialization (for and with Antony)

• [ ] We could consider serialzing G4-fluxes (a.k.a. cohomology classes on toric spaces). At least for the QSMs, there is each time a specifically chosen G4-fluxes (cf. https://arxiv.org/pdf/2102.10115, equ. 2.25).
• [ ] Based on https://github.com/oscar-system/Oscar.jl/pull/3980, could also serialize Lie algebras of the FTheoryModels. Again, a particular instance includes the QSM models.
• [ ] Add an artifact/database or alike for the model discussed in https://github.com/oscar-system/Oscar.jl/pull/3978. I have the corresponding .mrdi files and will be happy to share them. They are about 1.5TB in total size.)
• [ ] Make use of the attribute saver of @antonydellavecchia (cf. https://github.com/oscar-system/Oscar.jl/pull/4020) to automatically serialize more/all attributes. For instance, even after the above, for instance the known resolutions are not saved for FTheory models.
• [ ] Database 2.0, i.e. use mrdi files.
• [ ] Metadata via Mardi portal.

Milestones

• [ ] Tate autoresolver: Use insights of "Tateform on steroids" to automatically resolve a Tate model as much as possible. (Andrew)
• [ ] Migrate the code from experimental into source.
• [ ] Nagell's algorithm?

Completed

• [x] Book chapter and good names for wrapper functions
• [x] Toric schemes (see https://github.com/oscar-system/Oscar.jl/pull/2700)
• [x] Uniform blow_up code among toric and scheme settings (cf. https://github.com/oscar-system/Oscar.jl/pull/2741).
• [x] Add function blow_up (or resolve) that accepts an F-theory model and blows it up/resolves it. (https://github.com/oscar-system/Oscar.jl/pull/3012)
• [x] Do we need new "types" for partially or fully-resolved models? Answer: No.
• [x] Add function tune, that allows to set parameters to specialized choice (-> add singularity/enlarge gauge group). (Cf. https://github.com/oscar-system/Oscar.jl/pull/3146)
• [x] Complicated centers can be blown up with the schemes technology. However, this is currently only possible for smooth toric varieties. This in turn is the case because we can create ideal sheaves only for such toric varieties. Extended in https://github.com/oscar-system/Oscar.jl/pull/3513.
• [x] Support non-toric blowups and update _construct_literature_model_over_concrete_base and the documentation of put_over_concrete_base to use generic sections. (Cf. https://github.com/oscar-system/Oscar.jl/pull/3519)
• [x] Tuning must work for setting sections to zero. (Cf. https://github.com/oscar-system/Oscar.jl/pull/3726)
• [x] Use (https://docs.oscar-system.org/stable/Experimental/LieAlgebras/lie_algebras/) for the Lie algebras (cf. https://github.com/oscar-system/Oscar.jl/pull/3822, https://github.com/oscar-system/Oscar.jl/pull/3908)
• [x] Extend the tutorial: https://www.oscar-system.org/tutorials/FTheoryTools/
• [x] Use Matthias' proper transformation function when possible/needed.
• [x] Limit coefficients in generic sections (cf. https://github.com/oscar-system/Oscar.jl/pull/3930)
• [x] Provide serialization support for toric divisor classes (cf. https://github.com/oscar-system/Oscar.jl/pull/3931)
• [x] Added F-theory models on all toric hypersurfaces (https://arxiv.org/abs/1408.4808v1): https://github.com/oscar-system/Oscar.jl/pull/3568
• [x] More models from "F-theory on all toric hypersurfaces": https://github.com/oscar-system/Oscar.jl/pull/3642
• [x] Clarify and improve mechanics of literature model constructor: https://github.com/oscar-system/Oscar.jl/pull/3675
• [x] Reparametrized literature models in "F-theory on all toric hypersurfaces": https://github.com/oscar-system/Oscar.jl/pull/3693
• [x] Updates to hypersurface model constructor: https://github.com/oscar-system/Oscar.jl/pull/3699
• [x] Various improvements: https://github.com/oscar-system/Oscar.jl/pull/3712
• [x] https://github.com/oscar-system/Oscar.jl/pull/3424 (see also https://github.com/oscar-system/Oscar.jl/pull/3506)
• [x] Support more than just {w = 0} as gauge divisor for calculation of singular loci (cf. #3718)
• [x] Basic support the QSMs as literature models. (https://github.com/oscar-system/Oscar.jl/pull/3363)
• [x] Refactoring of the QSMs, so that attributes are better available (https://github.com/oscar-system/Oscar.jl/pull/3871)

Abandoned (at least for now)

• Convert a Weierstrass into a Tate model? Can be done locally, but what is the global obstruction? Answer: Let us not (yet) get into this.~~

(cc @apturner)