×

zbMATH — the first resource for mathematics

PALP: a package for analysing lattice polytopes with applications to toric geometry. (English) Zbl 1196.14007
Summary: We describe our package PALP of C programs for calculations with lattice polytopes and applications to toric geometry, which is freely available on the internet. It contains routines for vertex and facet enumeration, computation of incidences and symmetries, as well as completion of the set of lattice points in the convex hull of a given set of points. In addition, there are procedures specialized to reflexive polytopes such as the enumeration of reflexive subpolytopes, and applications to toric geometry and string theory, like the computation of Hodge data and fibration structures for toric Calabi-Yau varieties. The package is well tested and optimized in speed as it was used for time consuming tasks such as the classification of reflexive polyhedra in 4 dimensions and the creation and manipulation of very large lists of 5-dimensional polyhedra. While originally intended for low-dimensional applications, the algorithms work in any dimension and our key routine for vertex and facet enumeration compares well with existing packages.

MSC:
14-04 Software, source code, etc. for problems pertaining to algebraic geometry
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
Software:
cdd; lrs; PALP; PORTA; Qhull
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Kreuzer, M.; Skarke, H.
[6] Avis, D.; Bremner, D.; Seidel, R., How good are convex hull algorithms?, Comput. geom., 7, 265, (1997) · Zbl 0877.68119
[7] Kreuzer, M.; Skarke, H., On the classification of reflexive polyhedra, Commun. math. phys., 185, 495, (1997), References like [hep-th/yymmnnn] refer to http://arXiv.org/ where articles can be obtained in postscript or pdf format. · Zbl 0894.14026
[8] Kreuzer, M.; Skarke, H., Reflexive polyhedra, weights and toric calabi – yau fibrations, Rev. math. phys., 14, 343, (2002) · Zbl 1079.14534
[9] Avram, A.; Kreuzer, M.; Mandelberg, M.; Skarke, H., Searching for K3 fibrations, Nucl. phys. B, 494, 567, (1997) · Zbl 0951.81060
[10] Kreuzer, M.; Skarke, H., Calabi – yau fourfolds and toric fibrations, J. geom. phys., 466, 1, (1997)
[11] Kreuzer, M.; Skarke, H., Classification of reflexive polyhedra in three dimensions, Adv. theor. math. phys., 2, 847, (1998) · Zbl 0934.52006
[12] Skarke, H., Weight systems for toric calabi – yau varieties and reflexivity of Newton polyhedra, Mod. phys. lett. A, 11, 1637, (1996) · Zbl 1020.14503
[13] Lynker, M.; Schimmrigk, R.; Wisskirchen, A., Landau – ginzburg vacua of string, M- and F-theory at c=12, Nucl. phys. B, 550, 123, (1999) · Zbl 1063.14504
[14] Batyrev, V.V.; Borisov, L.A., Dual cones and mirror symmetry for generalized Calabi-Yau manifolds, in: S.-T. Yau (Ed.), Mirror Symmetry II · Zbl 0927.14019
[15] Batyrev, V.V.; Borisov, L.A., On calabi – yau complete intersections in toric varieties, () · Zbl 0908.14015
[16] Batyrev, V.V.; Borisov, L.A., Mirror duality and string theoretic Hodge numbers · Zbl 0872.14035
[17] Kreuzer, M.; Riegler, E.; Sahakyan, D., Toric complete intersections and weighted projective space · Zbl 1061.14037
[18] Kreuzer, M., Strings on calabi – yau spaces and toric geometry, Nucl. phys. proc. (suppl.), 102, 87, (2001) · Zbl 1006.83062
[19] Candelas, P.; Lynker, M.; Schimmrigk, R., Calabi – yau manifolds in weighted \(P\^{}\{4\}\), Nucl. phys. B, 341, 383, (1990) · Zbl 0962.14029
[20] Kreuzer, M.; Skarke, H., On the classification of quasihomogeneous functions, Commun. math. phys., 150, 137, (1992) · Zbl 0767.57019
[21] Kreuzer, M.; Skarke, H., No mirror symmetry in landau – ginzburg spectra!, Nucl. phys. B, 388, 113, (1992)
[22] Klemm, A.; Schimmrigk, R., Landau – ginzburg string vacua, Nucl. phys. B, 411, 559, (1994) · Zbl 1049.81601
[23] Lerche, W.; Vafa, C.; Warner, N., Chiral rings in N=2 superconformal theories, Nucl. phys. B, 324, 427, (1989)
[24] Vafa, C., String vacua and orbifoldized LG models, Mod. phys. lett. A, 4A, 1169, (1989), Superstring Vacua, HUTP-89/A057 preprint
[25] Intriligator, K.; Vafa, C., Landau – ginzburg orbifolds, Nucl. phys. B, 339, 95, (1990)
[26] Kreuzer, M.; Schweigert, C., On the extended Poincaré polynomial, Phys. lett. B, 352, 276, (1995)
[27] Kreuzer, M.; Skarke, H., All abelian symmetries of landau – ginzburg potentials, Nucl. phys. B, 405, 305, (1993) · Zbl 0990.81635
[28] Kreuzer, M.; Skarke, H.; Kreuzer, M.; Skarke, H., Landau – ginzburg orbifolds with discrete torsion, Phys. lett. B, Mod. phys. lett. A, 10, 1073, (1995) · Zbl 1022.81694
[29] Batyrev, V.V., Dual polyhedra and mirror symmetry for calabi – yau hypersurfaces in toric varieties, J. alg. geom., 3, 493, (1994) · Zbl 0829.14023
[30] Fulton, W., Introduction to toric varieties, (1993), Princeton Univ. Press Princeton
[31] Cox, D., Recent developments in toric geometry · Zbl 0899.14025
[32] Cox, D.; Katz, S., Algebraic geometry and mirror symmetry, Math. surveys monogr., vol. 68, (1999), Amer. Math. Society Providence · Zbl 0951.14026
[33] Klemm, A.; Lerche, W.; Mayr, P., K3 fibrations and heterotic type II string duality, Phys. lett. B, 357, 313, (1995)
[34] Aspinwall, P.S.; Louis, J., On the ubiquity of K3 fibrations in string duality, Phys. lett. B, 369, 233, (1996)
[35] Candelas, P.; Font, A., Duality between the webs of heterotic and type II vacua, Nucl. phys. B, 511, 295, (1998) · Zbl 0947.81054
[36] Vafa, C., Evidence for F-theory, Nucl. phys. B, 469, 403, (1996) · Zbl 1003.81531
[37] Hu, Y.; Liu, C.-H.; Yau, S.-T., Toric morphisms and fibrations of toric calabi – yau hypersurfaces · Zbl 1033.81069
[38] A. Klemm, M. Kreuzer, E. Riegler, E. Scheidegger, work in progress
[39] Kreuzer, M.; Skarke, H., Complete classification of reflexive polyhedra in four dimensions · Zbl 1017.52007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.