Journal of High Energy Physics

, Volume 2011, Issue 2, pp 1–15 | Cite as

A (0,2) mirror map

  • Ilarion V. Melnikov
  • M. Ronen Plesser


We study the linear sigma model subspace of the moduli space of (0,2) super-conformal world-sheet theories obtained by deforming (2,2) theories based on Calabi-Yau hypersurfaces in reflexively plain toric varieties. We describe a set of algebraic coordinates on this subspace, formulate a (0,2) generalization of the monomial-divisor mirror map, and show that the map exchanges principal components of singular loci of the mirror half-twisted theories. In non-reflexively plain examples the proposed map yields a mirror isomorphism between subfamilies of linear sigma models.


Superstrings and Heterotic Strings Differential and Algebraic Geometry 


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  1. [1]
    P.S. Aspinwall, B.R. Greene and D.R. Morrison, The monomial divisor mirror map, alg-geom/9309007 [SPIRES].
  2. [2]
    V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom. 3 (1994) 493 [alg-geom/9310003] [SPIRES].MathSciNetzbMATHGoogle Scholar
  3. [3]
    E. Witten, Phases of N = 2 theories in two dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    D.R. Morrison and M. Ronen Plesser, Summing the instantons: quantum cohomology and mirror symmetry in toric varieties, Nucl. Phys. B 440 (1995) 279 [hep-th/9412236] [SPIRES].ADSCrossRefGoogle Scholar
  5. [5]
    D.R. Morrison and M.R. Plesser, Towards mirror symmetry as duality for two dimensional abelian gauge theories, Nucl. Phys. Proc. Suppl. 46 (1996) 177 [hep-th/9508107] [SPIRES].MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. [6]
    M. Kreuzer, J. McOrist, I.V. Melnikov and M.R. Plesser, (0,2) deformations of linear σ-models, arXiv:1001.2104 [SPIRES].
  7. [7]
    V.V. Batyrev and E.N. Materov, Toric residues and mirror symmetry, Mosc. Math. J. 2 (2002) 435 [math.AG/0203216] [SPIRES].MathSciNetzbMATHGoogle Scholar
  8. [8]
    A. Szenes and M. Vergne, Toric reduction and a conjecture of Batyrev and Materov, Invent. Math. 158 (2004) 453.MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. [9]
    L.A. Borisov, Higher Stanley-Reisner rings and toric residues, Compos. Math. 141 (2005) 161.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    K. Karu, Toric residue mirror conjecture for Calabi-Yau complete intersections, J. Algebraic Geom. 14 (2005) 741 [math.AG/0311338].MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    J. McOrist and I.V. Melnikov, Summing the instantons in half-twisted linear σ-models, JHEP 02 (2009) 026 [arXiv:0810.0012] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  12. [12]
    E. Silverstein and E. Witten, Criteria for conformal invariance of (0,2) models, Nucl. Phys. B 444 (1995) 161 [hep-th/9503212] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  13. [13]
    A. Basu and S. Sethi, World-sheet stability of (0,2) linear σ-models, Phys. Rev. D 68 (2003) 025003 [hep-th/0303066] [SPIRES].MathSciNetADSGoogle Scholar
  14. [14]
    C. Beasley and E. Witten, Residues and world-sheet instantons, JHEP 10 (2003) 065 [hep-th/0304115] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    J. Distler, Notes on (0,2) superconformal field theories, hep-th/9502012 [SPIRES].
  16. [16]
    D.A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995) 17 [alg-geom/9210008] [SPIRES].
  17. [17]
    D.A. Cox and S. Katz, Mirror symmetry and algebraic geometry, American Mathematical Society, Providence U.S.A. (2000) [SPIRES].Google Scholar
  18. [18]
    J. Harris, Algebraic geometry: a first course, Springer, New York U.S.A. (1992).zbMATHGoogle Scholar
  19. [19]
    J. McOrist and I.V. Melnikov, Half-twisted correlators from the Coulomb branch, JHEP 04 (2008) 071 [arXiv:0712.3272] [SPIRES].MathSciNetCrossRefGoogle Scholar
  20. [20]
    M.M. Kapranov, A characterization of A-discriminantal hypersurfaces in terms of the logarithmic Gauss map, Math. Ann. 290 (1991) 277.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    A. Adams, A. Basu and S. Sethi, (0,2) duality, Adv. Theor. Math. Phys. 7 (2004) 865 [hep-th/0309226] [SPIRES].MathSciNetGoogle Scholar
  22. [22]
    A. Adams, J. Distler and M. Ernebjerg, Topological heterotic rings, Adv. Theor. Math. Phys. 10 (2006) 657 [hep-th/0506263] [SPIRES].MathSciNetzbMATHGoogle Scholar
  23. [23]
    J. Distler and S. Kachru, (0,2) Landau-Ginzburg theory, Nucl. Phys. B 413 (1994) 213 [hep-th/9309110] [SPIRES].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    J. Distler and S. Kachru, Duality of (0,2) string vacua, Nucl. Phys. B 442 (1995) 64 [hep-th/9501111] [SPIRES].MathSciNetADSGoogle Scholar
  25. [25]
    R. Blumenhagen, R. Schimmrigk and A. Wisskirchen, (0,2) mirror symmetry, Nucl. Phys. B 486 (1997) 598 [hep-th/9609167] [SPIRES].MathSciNetADSCrossRefGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2011

Authors and Affiliations

  1. 1.Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut)GolmGermany
  2. 2.Center for Geometry and Theoretical PhysicsDuke UniversityDurhamU.S.A.

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