# American Institute of Mathematical Sciences

2011, 18: 131-143. doi: 10.3934/era.2011.18.131

## Semisimplicity of the quantum cohomology for smooth Fano toric varieties associated with facet symmetric polytopes

 1 Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, United Kingdom 2 Yale University, MathematicsDepartment, PO Box 208283, New Haven, Connecticut 06520, United States

Received  March 2011 Revised  July 2011 Published  September 2011

The degree zero part of the quantum cohomology algebra of a smooth Fano toric symplectic manifold is determined by the superpotential function, $W$, of its moment polytope. In particular, this algebra is semisimple, i.e. splits as a product of fields, if and only if all the critical points of $W$ are non-degenerate. In this paper, we prove that this non-degeneracy holds for all smooth Fano toric varieties with facet-symmetric duals to moment polytopes.
Citation: Maksim Maydanskiy, Benjamin P. Mirabelli. Semisimplicity of the quantum cohomology for smooth Fano toric varieties associated with facet symmetric polytopes. Electronic Research Announcements, 2011, 18: 131-143. doi: 10.3934/era.2011.18.131
##### References:
 [1] Matthew Strom Borman, Quasi-states, quasi-morphisms, and the moment map, arXiv:1105.1805. [2] T. Delzant, Hamiltoniens periodiques et image convexes de l'application moment, Bulletin de la Societe Mathmatique de France, 116 (1988), 315-339. [3] David Cox, Minicourse on Toric Varieties, given at the University of Buenos Aires. Available from: http://www.cs.amherst.edu/ dac/lectures/toric.ps. [4] Y. Eliashberg and L. Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifolds, arXiv:1006.2501. [5] M. Entov and L. Polterovich, Symplectic quasi-states and semi-simplicity of quantum homology, in "Toric Topology," Contemp. Math., 460, American. Math. Society, Providence, RI, (2008), 47-70. [6] Günter Ewald, On the classification of Toric Fano varieties, Discrete and Computational Goemetry, 3 (1988), 49-54. doi: 10.1007/BF02187895. [7] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Spectral invariants with bulk quasimorphisms and Lagrangian Floer theory, arXiv:1105.5123. [8] William Fulton, "Introduction to Toric Varieties," Annals of Mathematics Studies, 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993. [9] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, in "Algebraic Geometry-Santa Cruz 1995," Proc. of Symp. in Pure Math., 62 Part 2, Amer. Math. Soc., Providence, RI, (1997), 45-96. [10] Alexander Givental, A tutorial on quantum cohomology, in "Symplectic Geometry and Topology" (Park City, UT, 1997), 231-264, IAS/Park City Math. Ser., 7, Amer. Math. Soc., Providence, RI, 1999. [11] Robin Hartshorne, "Algebraic Geometry," Graduate Texts in Math., 52, Springer-Verlag, New York-Heidelberg, 1977. [12] J. Lagarias and G. Ziegler, Bounds for lattice polytopes containing a fixed number of interior points in a sublattice, Can. J. Math., 43 (1991), 1022-1035. doi: 10.4153/CJM-1991-058-4. [13] Wilhelm Ljunggren, On the irreducibility of certain trinomials and quadrinomials, Math. Scand., 8 (1960), 65-70. [14] D. McDuff and D. Salamon, "J-holomorphic Curves and Symplectic Topology," American Math. Society Collaquium Publications, 52, American Mathematical Society, Providence, RI, 2004. [15] Benjamin Nill, "Reflexive Polytopes - Combinatorics and Convex Goemetry." Available from: http://personales.unican.es/santosf/anogia05/slides/Nill-anogia05.pdf. [16] Mikkel Obro, "Classification of smooth Fano polytopes,'' Dissertation, Univ. of Aarhus, 2007. Available from: http://imf.au.dk/publication/publid/668/. [17] Yaron Ostrover, Calabi quasi-morphisms for some non-monotone symplectic manifolds, Algebr. Geom. Topol., 6 (2006), 405-434. doi: 10.2140/agt.2006.6.405. [18] Y. Ostrover and I. Tyomkin, On the quantum homology algebra of toric Fano manifolds, Selecta Math. (N.S.), 15 (2009), 121-149. [19] Michael Usher, Deformed Hamiltonian Floer theory, capacity estimates, and Calabi quasimorphisms, arXiv:1006.5390.

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##### References:
 [1] Matthew Strom Borman, Quasi-states, quasi-morphisms, and the moment map, arXiv:1105.1805. [2] T. Delzant, Hamiltoniens periodiques et image convexes de l'application moment, Bulletin de la Societe Mathmatique de France, 116 (1988), 315-339. [3] David Cox, Minicourse on Toric Varieties, given at the University of Buenos Aires. Available from: http://www.cs.amherst.edu/ dac/lectures/toric.ps. [4] Y. Eliashberg and L. Polterovich, Symplectic quasi-states on the quadric surface and Lagrangian submanifolds, arXiv:1006.2501. [5] M. Entov and L. Polterovich, Symplectic quasi-states and semi-simplicity of quantum homology, in "Toric Topology," Contemp. Math., 460, American. Math. Society, Providence, RI, (2008), 47-70. [6] Günter Ewald, On the classification of Toric Fano varieties, Discrete and Computational Goemetry, 3 (1988), 49-54. doi: 10.1007/BF02187895. [7] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono, Spectral invariants with bulk quasimorphisms and Lagrangian Floer theory, arXiv:1105.5123. [8] William Fulton, "Introduction to Toric Varieties," Annals of Mathematics Studies, 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993. [9] W. Fulton and R. Pandharipande, Notes on stable maps and quantum cohomology, in "Algebraic Geometry-Santa Cruz 1995," Proc. of Symp. in Pure Math., 62 Part 2, Amer. Math. Soc., Providence, RI, (1997), 45-96. [10] Alexander Givental, A tutorial on quantum cohomology, in "Symplectic Geometry and Topology" (Park City, UT, 1997), 231-264, IAS/Park City Math. Ser., 7, Amer. Math. Soc., Providence, RI, 1999. [11] Robin Hartshorne, "Algebraic Geometry," Graduate Texts in Math., 52, Springer-Verlag, New York-Heidelberg, 1977. [12] J. Lagarias and G. Ziegler, Bounds for lattice polytopes containing a fixed number of interior points in a sublattice, Can. J. Math., 43 (1991), 1022-1035. doi: 10.4153/CJM-1991-058-4. [13] Wilhelm Ljunggren, On the irreducibility of certain trinomials and quadrinomials, Math. Scand., 8 (1960), 65-70. [14] D. McDuff and D. Salamon, "J-holomorphic Curves and Symplectic Topology," American Math. Society Collaquium Publications, 52, American Mathematical Society, Providence, RI, 2004. [15] Benjamin Nill, "Reflexive Polytopes - Combinatorics and Convex Goemetry." Available from: http://personales.unican.es/santosf/anogia05/slides/Nill-anogia05.pdf. [16] Mikkel Obro, "Classification of smooth Fano polytopes,'' Dissertation, Univ. of Aarhus, 2007. Available from: http://imf.au.dk/publication/publid/668/. [17] Yaron Ostrover, Calabi quasi-morphisms for some non-monotone symplectic manifolds, Algebr. Geom. Topol., 6 (2006), 405-434. doi: 10.2140/agt.2006.6.405. [18] Y. Ostrover and I. Tyomkin, On the quantum homology algebra of toric Fano manifolds, Selecta Math. (N.S.), 15 (2009), 121-149. [19] Michael Usher, Deformed Hamiltonian Floer theory, capacity estimates, and Calabi quasimorphisms, arXiv:1006.5390.
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