Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 8 (2012), 068, 28 pages      arXiv:1209.1134      http://dx.doi.org/10.3842/SIGMA.2012.068
Contribution to the Special Issue “Mirror Symmetry and Related Topics”

Recent Developments in (0,2) Mirror Symmetry

Ilarion Melnikov a, Savdeep Sethi b and Eric Sharpe c
a) Max Planck Institute for Gravitational Physics, Am Mühlenberg 1, D-14476 Golm, Germany
b) Department of Physics, Enrico Fermi Institute, University of Chicago, 5640 S. Ellis Ave., Chicago, IL 60637, USA
c) Department of Physics, MC 0435, 910 Drillfield Dr., Virginia Tech, Blacksburg, VA 24061, USA

Received June 04, 2012, in final form October 02, 2012; Published online October 07, 2012

Abstract
Mirror symmetry of the type II string has a beautiful generalization to the heterotic string. This generalization, known as (0,2) mirror symmetry, is a field still largely in its infancy. We describe recent developments including the ideas behind quantum sheaf cohomology, the mirror map for deformations of (2,2) mirrors, the construction of mirror pairs from worldsheet duality, as well as an overview of some of the many open questions. The (0,2) mirrors of Hirzebruch surfaces are presented as a new example.

Key words: mirror symmetry; (0,2) mirror symmetry; quantum sheaf cohomology.

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References

  1. Adams A., Basu A., Sethi S., (0,2) duality, Adv. Theor. Math. Phys. 7 (2003), 865-950, hep-th/0309226.
  2. Adams A., Distler J., Ernebjerg M., Topological heterotic rings, Adv. Theor. Math. Phys. 10 (2006), 657-682, hep-th/0506263.
  3. Aspinwall P.S., Greene B.R., Morrison D.R., The monomial-divisor mirror map, Int. Math. Res. Not. 1993 (1993), no. 12, 319-337, alg-geom/9309007.
  4. Aspinwall P.S., Morrison D.R., Topological field theory and rational curves, Comm. Math. Phys. 151 (1993), 245-262, hep-th/9110048.
  5. Aspinwall P.S., Plesser M.R., Elusive worldsheet instantons in heterotic string compactifications, arXiv:1106.2998.
  6. Basu A., Sethi S., World-sheet stability of (0,2) linear sigma models, Phys. Rev. D 68 (2003), 025003, 8 pages, hep-th/0303066.
  7. Batyrev V.V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), 493-535, alg-geom/9310003.
  8. Batyrev V.V., Quantum cohomology rings of toric manifolds, Astérisque (1993), no. 218, 9-34, alg-geom/9310004.
  9. Batyrev V.V., Materov E.N., Toric residues and mirror symmetry, Mosc. Math. J. 2 (2002), 435-475, math.AG/0203216.
  10. Beasley C., Witten E., Residues and world-sheet instantons, J. High Energy Phys. 2003 (2003), no. 10, 065, 39 pages, hep-th/0304115.
  11. Blumenhagen R., Flohr M., Aspects of (0,2) orbifolds and mirror symmetry, Phys. Lett. B 404 (1997), 41-48, hep-th/9702199.
  12. Blumenhagen R., Schimmrigk R., Wißkirchen A., (0,2) mirror symmetry, Nuclear Phys. B 486 (1997), 598-628, hep-th/9609167.
  13. Blumenhagen R., Sethi S., On orbifolds of (0,2) models, Nuclear Phys. B 491 (1997), 263-278, hep-th/9611172.
  14. Borisov L.A., Higher-Stanley-Reisner rings and toric residues, Compos. Math. 141 (2005), 161-174, math.AG/0306307.
  15. Borisov L.A., Kaufmann R.M., On CY-LG correspondence for (0,2) toric models, arXiv:1102.5444.
  16. Candelas P., de la Ossa X.C., Green P.S., Parkes L., An exactly soluble superconformal theory from a mirror pair of Calabi-Yau manifolds, Phys. Lett. B 258 (1991), 118-126.
  17. Cox D.A., Katz S., Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, Vol. 68, American Mathematical Society, Providence, RI, 1999.
  18. Dijkgraaf R., Verlinde H., Verlinde E., Notes on topological string theory and 2D quantum gravity, in String Theory and Quantum Gravity (Trieste, 1990), World Sci. Publ., River Edge, NJ, 1991, 91-156.
  19. Donagi R., Guffin J., Katz S., Sharpe E., A mathematical theory of quantum sheaf cohomology, arXiv:1110.3751.
  20. Donagi R., Guffin J., Katz S., Sharpe E., Physical aspects of quantum sheaf cohomology for deformations of tangent bundles of toric varieties, arXiv:1110.3752.
  21. Giveon A., Porrati M., Rabinovici E., Target space duality in string theory, Phys. Rep. 244 (1994), 77-202, hep-th/9401139.
  22. Greene B.R., Plesser M.R., Duality in Calabi-Yau moduli space, Nuclear Phys. B 338 (1990), 15-37.
  23. Guffin J., Quantum sheaf cohomology, a précis, arXiv:1101.1305.
  24. Guffin J., Katz S., Deformed quantum cohomology and (0,2) mirror symmetry, J. High Energy Phys. 2010 (2010), no. 8, 109, 27 pages, arXiv:0710.2354.
  25. Hori K., Katz S., Klemm A., Pandharipande R., Thomas R., Vafa C., Vakil R., Zaslow E., Mirror symmetry, Clay Mathematics Monographs, Vol. 1, Amer. Math. Soc., Providence, RI, 2003.
  26. Hori K., Vafa C., Mirror symmetry, hep-th/0002222.
  27. Kapranov M.M., A characterization of A-discriminantal hypersurfaces in terms of the logarithmic Gauss map, Math. Ann. 290 (1991), 277-285.
  28. Karu K., Toric residue mirror conjecture for Calabi-Yau complete intersections, J. Algebraic Geom. 14 (2005), 741-760, math.AG/0311338.
  29. Katz S., Sharpe E., Notes on certain (0,2) correlation functions, Comm. Math. Phys. 262 (2006), 611-644, hep-th/0406226.
  30. Kontsevich M., Homological algebra of mirror symmetry, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 120-139, alg-geom/9411018.
  31. Kreuzer M., McOrist J., Melnikov I.V., Plesser M.R., (0,2) deformations of linear sigma models, J. High Energy Phys. 2011 (2011), no. 7, 044, 30 pages, arXiv:1001.2104.
  32. Kreuzer M., Skarke H., Complete classification of reflexive polyhedra in four dimensions, Adv. Theor. Math. Phys. 4 (2000), 1209-1230, hep-th/0002240.
  33. McOrist J., The revival of (0,2) sigma models, Internat. J. Modern Phys. A 26 (2011), 1-41, arXiv:1010.4667.
  34. McOrist J., Melnikov I.V., Half-twisted correlators from the Coulomb branch, J. High Energy Phys. 2008 (2008), no. 4, 071, 19 pages, arXiv:0712.3272.
  35. McOrist J., Melnikov I.V., Old issues and linear sigma models, arXiv:1103.1322.
  36. McOrist J., Melnikov I.V., Summing the instantons in half-twisted linear sigma models, J. High Energy Phys. (2009), no. 2, 026, 61 pages, arXiv:0810.0012.
  37. Melnikov I.V., (0,2) Landau-Ginzburg models and residues, J. High Energy Phys. 2009 (2009), no. 9, 118, 25 pages, arXiv:0902.3908.
  38. Melnikov I.V., Plesser M.R., A (0,2) mirror map, J. High Energy Phys. 2011 (2011), no. 2, 001, 15 pages, arXiv:1003.1303.
  39. Melnikov I.V., Sethi S., Half-twisted (0,2) Landau-Ginzburg models, J. High Energy Phys. 2008 (2008), no. 3, 040, 21 pages, arXiv:0712.1058.
  40. Morrison D.R., Plesser M.R., Summing the instantons: quantum cohomology and mirror symmetry in toric varieties, Nuclear Phys. B 440 (1995), 279-354, hep-th/9412236.
  41. Morrison D.R., Plesser M.R., Towards mirror symmetry as duality for two-dimensional abelian gauge theories, in Strings '95 (Los Angeles, CA, 1995), World Sci. Publ., River Edge, NJ, 1996, 374-387, hep-th/9508107.
  42. Periwal V., Strominger A., Kähler geometry of the space of N=2 superconformal field theories, Phys. Lett. B 235 (1990), 261-267.
  43. Sharpe E., Notes on certain other (0,2) correlation functions, Adv. Theor. Math. Phys. 13 (2009), 33-70, hep-th/0605005.
  44. Sharpe E., Notes on correlation functions in (0,2) theories, in Snowbird Lectures on String Geometry, Contemp. Math., Vol. 401, Amer. Math. Soc., Providence, RI, 2006, 93-104, hep-th/0502064.
  45. Silverstein E., Witten E., Criteria for conformal invariance of (0,2) models, Nuclear Phys. B 444 (1995), 161-190, hep-th/9503212.
  46. Strominger A., Yau S.T., Zaslow E., Mirror symmetry is T-duality, Nuclear Phys. B 479 (1996), 243-259, hep-th/9505162.
  47. Szenes A., Vergne M., Toric reduction and a conjecture of Batyrev and Materov, Invent. Math. 158 (2004), 453-495, math.AT/0306311.
  48. Tan M.C., Two-dimensional twisted sigma models and the theory of chiral differential operators, Adv. Theor. Math. Phys. 10 (2006), 759-851, hep-th/0604179.
  49. Tan M.C., Two-dimensional twisted sigma models, the mirror chiral de Rham complex, and twisted generalised mirror symmetry, J. High Energy Phys. 2007 (2007), no. 7, 013, 80 pages, arXiv:0705.0790.
  50. Tan M.C., Yagi J., Chiral algebras of (0,2) sigma models: beyond perturbation theory, Lett. Math. Phys. 84 (2008), 257-273, arXiv:0801.4782.
  51. Witten E., Mirror manifolds and topological field theory, in Essays on Mirror Manifolds, Int. Press, Hong Kong, 1992, 120-158, hep-th/9112056.
  52. Witten E., Phases of N=2 theories in two dimensions, Nuclear Phys. B 403 (1993), 159-222, hep-th/9301042.
  53. Yagi J., Chiral algebras of (0,2) models, arXiv:1001.0118.

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