Weight systems for 5 and more dimensions

• The database of all weight systems for 5d polyhedra with the IP property has a searchable web front-end.
  It is now also available in Parquet format at the Hugging Face Hub.


• Other results:
  
  • All 5d weighted projective spaces with CY 4-fold hypersurfaces defined by quasihomogeneous polynomials satisfying a transversality condidition: 1,100,055 (23MB);
    subset of weight systems defining reflexive polytopes: 252,933 (6MB)
    
  • Weight systems for reflexive 5d polyhedra (CY 4-folds) with degrees 6-100 (2.7MB), 101-150 (12MB), 151-200 (31MB), 201-250 (63MB), 251-260 (18MB), 261-270 (20MB), 271-280 (22MB), 281-290 (24MB), 291-300 (26MB)
    
  • Weight systems for reflexive 6d polyhedra with degrees 7-100 (3.3 MB), 101-150 (20 MB)
    
  • 6d weighted projective spaces with transverse polynomials: degrees 7-200 (6MB)
    
  • 7d weighted projective spaces with transverse polynomials: degrees 8-150 (17MB), 151-200 (26MB)
    
  • All files
    
  • The format of the files is
    w1 ... w6 deg=d M:#p #v N:#p #v H:h11,h12,h13 [chi] . . . (reflexive)
    w1 ... w6 deg=d M:#p #v N:#p #v V:h11,h12,h13 [chi] . . . (reflexive, transverse)
    w1 ... w6 deg=d M:#p #v F:#f V:h11,h12,h13 [chi] . . . (transverse)
    (or with the standard PALP format "d w1 ..." instead of "w1 ... deg=d"), where
    • #p (#v) = number of points (vertices) in the M or N lattice;
      #f = number of facets (for non-reflexive polyhedra);
    • H = toric Hodge data (non-transverse case, Batyrev's formula),
      V = chiral ring data (Vafa's formula; for all weights that define both reflexive polytopes and transverse polynomials, i.e. ... N ... V ..., we checked that both formulas give the same result).
    
    
• History and references:
  • In hep-th/9701023 Klemm, Lian, Roan and Yau computed all Fermat weights and the transverse weights up to degree 400.
  • In hep-th/9701175 Kreuzer and Skarke computed all reflexive weights up to degree 150 and all transverse weights up to degree 4000.
  • In hep-th/9812195 Lynker, Schimmrigk and Wisskirchen constructed all 1,100,055 transverse configurations. We took their weights and added the M- and N-lattice data in our file format.
  • Later Max Kreuzer also computed all reflexive 4-fold weight systems up to degree 300 and all reflexive 5-fold weight systems up to degree 150. These can be used as starting points for complete intersections of codimension 2 and 3, respectively (cf. math.AG/0103214).
  • In arXiv:1808.02422 Schöller and Skarke found all weight systems that determine 5d IP polytopes.
  • The 6d and 7d weighted projective spaces up to degree 200 were first constructed by Hirst and Gherardini in the context of a machine learning project (see arXiv:2311.17146). The present lists, produced by a recently optimized version of PALP, are consistent with their results.