Computational tools for cohomology of toric varieties. (English) Zbl 1234.81107
Summary: Novel nonstandard techniques for the computation of cohomology classes on toric varieties are summarized. After an introduction of the basic definitions and properties of toric geometry, we discuss a specific computational algorithm for the determination of the dimension of line-bundle-valued cohomology groups on toric varieties. Applications to the computation of chiral massless matter spectra in string compactifications are discussed, and using the software package cohomCalg, its utility is highlighted on a new target space dual pair of \((0,2)\) heterotic string models.
MSC:
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 81V25 | Other elementary particle theory in quantum theory |
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 81T80 | Simulation and numerical modelling (quantum field theory) (MSC2010) |
| 81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |
References:
| [1] | W. Fulton, Introduction to Toric Varieties, vol. 131 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, USA, 1993. · Zbl 0890.65087 · doi:10.1145/155743.155791 |
| [2] | M. Kreuzer, “Toric geometry and calabi-yau compactifications,” Ukrainian Journal of Physics, vol. 55, no. 5, pp. 613-625, 2010. |
| [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. · Zbl 1223.14001 |
| [5] | E. Witten, “Phases of N=2 theories in two dimensions,” Nuclear Physics B, vol. 403, no. 1-2, pp. 159-222, 1993. · Zbl 0910.14020 · doi:10.1016/0550-3213(93)90033-L |
| [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. · Zbl 1049.81585 · doi:10.1016/S0550-3213(96)90135-2 |
| [7] | R. Blumenhagen, “Target space duality for (0,2) compactifications,” Nuclear Physics. B, vol. 513, no. 3, pp. 573-590, 1998. · Zbl 0939.32017 · doi:10.1016/S0550-3213(97)00721-9 |
| [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. · Zbl 1314.55012 · doi:10.1063/1.3501132 |
| [9] | S. Jow, “Cohomology of toric line bundles via simplicial Alexander duality,” Journal of Mathematical Physics, vol. 52, no. 3, Article ID 033506, 2011. · Zbl 1315.55010 · doi:10.1063/1.3562523 |
| [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. · Zbl 1314.55013 · doi:10.1063/1.3501135 |
| [11] | cohomCalg package, “High-performance line bundle cohomology computation based on [8],” 2010, http://wwwth.mpp.mpg.de/members/bjurke/cohomcalg/. |
| [12] | M. Cveti\vc, 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. · Zbl 1209.81162 · doi:10.1002/prop.201000093 |
| [13] | R. Blumenhagen, B. Jurke, T. Rahn, and H. Roschy, “Cohomology of line bundles: applications,” http://arxiv.org/abs/1010.3717. · Zbl 1273.81180 |
| [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. · Zbl 1290.81103 · doi:10.1007/JHEP08(2010)079 |
| [15] | P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons, New York, NY, USA, 1994. · Zbl 0836.14001 |
| [16] | E. Witten, “New issues in manifolds of SU(3) holonomy,” Nuclear Physics B, vol. 268, no. 1, pp. 79-112, 1986. · doi:10.1016/0550-3213(86)90202-6 |
| [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. · Zbl 1247.14044 · doi:10.1088/1126-6708/2004/12/054 |
| [18] | J. Distler and S. Kachru, “Duality of (0,2) string vacua,” Nuclear Physics B, vol. 442, no. 1-2, pp. 64-74, 1995. · Zbl 0990.81659 · doi:10.1016/S0550-3213(95)00130-1 |
| [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. · Zbl 0951.81061 · doi:10.1016/S0550-3213(97)00237-X |
| [20] | R. Blumenhagen and T. Rahn, “Landscape study of target space duality of (0,2) heterotic string models,” http://arxiv.org/abs/1106.4998. · Zbl 1301.81188 |
| [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. · Zbl 1196.14007 · doi:10.1016/S0010-4655(03)00491-0 |
| [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. · Zbl 0927.14019 |
| [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. · Zbl 0908.14015 |
| [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. · Zbl 1057.68150 |
| [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. · Zbl 1311.14049 |
| [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. · Zbl 1210.14006 · doi:10.1016/j.aim.2008.11.017 |
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