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Advances in High Energy Physics
Volume 2011, Article ID 152749, 15 pages
http://dx.doi.org/10.1155/2011/152749
Review Article

Computational Tools for Cohomology of Toric Varieties

Max-Planck-Institut für Physik, Föhringer Ring 6, 80805 München, Germany

Received 15 April 2011; Accepted 27 June 2011

Academic Editor: André Lukas

Copyright © 2011 Ralph Blumenhagen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. Fulton, Introduction to Toric Varieties, vol. 131 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, USA, 1993. View at Zentralblatt MATH
  2. M. Kreuzer, “Toric geometry and calabi-yau compactifications,” Ukrainian Journal of Physics, vol. 55, no. 5, pp. 613–625, 2010. View at Google Scholar
  3. S. Reffert, “The geometer's toolkit to string compactifications,” http://arxiv.org/abs/0706.1310.
  4. D. A. Cox, J. B. Little, and H. Schenck, Toric Varieties, American Mathematical Society, 2011, http://www.cs.amherst.edu/~dac/toric.html.
  5. E. Witten, “Phases of N=2 theories in two dimensions,” Nuclear Physics B, vol. 403, no. 1-2, pp. 159–222, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. Distler, B. R. Greene, and D. R. Morrison, “Resolving singularities in (0,2) models,” Nuclear Physics B, vol. 481, no. 1-2, pp. 289–312, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. R. Blumenhagen, “Target space duality for (0,2) compactifications,” Nuclear Physics. B, vol. 513, no. 3, pp. 573–590, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy, “Cohomology of line bundles: a computational algorithm,” Journal of Mathematical Physics, vol. 51, no. 10, Article ID 103525, 15 pages, 2010. View at Publisher · View at Google Scholar
  9. S. Jow, “Cohomology of toric line bundles via simplicial Alexander duality,” Journal of Mathematical Physics, vol. 52, no. 3, Article ID 033506, 2011. View at Publisher · View at Google Scholar
  10. H. Roschy and T. Rahn, “Cohomology of line bundles: proof of the algorithm,” Journal of Mathematical Physics, vol. 51, no. 10, Article ID 103520, 11 pages, 2010. View at Publisher · View at Google Scholar
  11. cohomCalg package, “High-performance line bundle cohomology computation based on [8],” 2010, http://wwwth.mpp.mpg.de/members/bjurke/cohomcalg/.
  12. M. Cvetič, I. García-Etxebarria, and J. Halverson, “On the computation of non-perturbative effective potentials in the string theory landscape - IIB/F-theory perspective -,” Fortschritte der Physik, vol. 59, no. 3-4, pp. 243–283, 2011. View at Publisher · View at Google Scholar
  13. R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy, “Cohomology of line bundles: applications,” http://arxiv.org/abs/1010.3717.
  14. R. Blumenhagen, A. Collinucci, and B. Jurke, “On instanton effects in F-theory,” Journal of High Energy Physics, vol. 2010, no. 8, article 079, 2010. View at Publisher · View at Google Scholar
  15. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons, New York, NY, USA, 1994.
  16. E. Witten, “New issues in manifolds of SU(3) holonomy,” Nuclear Physics B, vol. 268, no. 1, pp. 79–112, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  17. R. Donagi, Y.-H. He, B. A. Ovrut, and R. Reinbacher, “The particle spectrum of heterotic compactifications,” Journal of High Energy Physics, no. 12, article 054, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  18. J. Distler and S. Kachru, “Duality of (0,2) string vacua,” Nuclear Physics B, vol. 442, no. 1-2, pp. 64–74, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. T.-M. Chiang, J. Distler, and B. R. Greene, “Some features of (0,2) moduli space,” Nuclear Physics B, vol. 496, no. 3, pp. 590–616, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. R. Blumenhagen and T. Rahn, “Landscape study of target space duality of (0,2) heterotic string models,” http://arxiv.org/abs/1106.4998.
  21. M. Kreuzer and H. Skarke, “PALP: a package for analysing lattice polytopes with applications to toric geometry,” Computer Physics Communications, vol. 157, no. 1, pp. 87–106, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. L. Borisov, “Towards the mirror symmetry for Calabi-Yau complete intersections in Gorenstein toric Fano varieties,” http://arxiv.org/abs/alg-geom/9310001.
  23. V. V. Batyrev and L. A. Borisov, “Dual cones and mirror symmetry for generalized Calabi-Yau manifolds,” in Mirror Symmetry, II, vol. 1 of AMS/IP Studies in Advanced Mathematics, pp. 71–86, American Mathematical Society, Providence, RI, USA, 1997. View at Google Scholar · View at Zentralblatt MATH
  24. V. V. Batyrev and L. A. Borisov, “On Calabi-Yau complete intersections in toric varieties,” in Higher-Dimensional Complex Varieties (Trento, 1994), pp. 39–65, de Gruyter, Berlin, Germany, 1996. View at Google Scholar · View at Zentralblatt MATH
  25. J. Rambau, “TOPCOM: triangulations of point configurations and oriented matroids,” in Mathematical Software—ICMS 2002, A. M. Cohen, X.-S. Gao, and N. Takayama, Eds., pp. 330–340, World Scientific, River Edge, NJ, USA, 2002. View at Google Scholar · View at Zentralblatt MATH
  26. D. R. Grayson and M. E. Stillman, “Macaulay2, a software system for research in algebraic geometry,” http://www.math.uiuc.edu/Macaulay2/.
  27. W. Stein et al., Sage Mathematics Software, The Sage Development Team, 2010, http://www.sagemath.org/.
  28. R. Birkner, N. O. Ilten, and L. Petersen, “Computations with equivariant toric vector bundles,” Journal of Software for Algebra and Geometry, vol. 2, pp. 11–14, 2010. View at Google Scholar
  29. L. Borisov and Z. Hua, “On the conjecture of King for smooth toric Deligne-Mumford stacks,” Advances in Mathematics, vol. 221, no. 1, pp. 277–301, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH