zbMATH — the first resource for mathematics

A (0,2) mirror map. (English) Zbl 1294.81223
Summary: 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.

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
81T10 Model quantum field theories
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14J33 Mirror symmetry (algebro-geometric aspects)
Full Text: DOI arXiv
[1] P.S. Aspinwall, B.R. Greene and D.R. Morrison, The monomial divisor mirror map, alg-geom/9309007 [SPIRES].
[2] Batyrev, VV, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Alg. Geom., 3, 493, (1994)
[3] Witten, E., Phases of N = 2 theories in two dimensions, Nucl. Phys., B 403, 159, (1993)
[4] Morrison, DR; Ronen Plesser, M., Summing the instantons: quantum cohomology and mirror symmetry in toric varieties, Nucl. Phys., B 440, 279, (1995)
[5] Morrison, DR; Plesser, MR, Towards mirror symmetry as duality for two dimensional abelian gauge theories, Nucl. Phys. Proc. Suppl., 46, 177, (1996)
[6] M. Kreuzer, J. McOrist, I.V. Melnikov and M.R. Plesser, (0,2) deformations of linear σ-models, arXiv:1001.2104 [SPIRES].
[7] Batyrev, VV; Materov, EN, Toric residues and mirror symmetry, Mosc. Math. J., 2, 435, (2002)
[8] Szenes, A.; Vergne, M., Toric reduction and a conjecture of Batyrev and Materov, Invent. Math., 158, 453, (2004)
[9] Borisov, LA, Higher Stanley-Reisner rings and toric residues, Compos. Math., 141, 161, (2005)
[10] Karu, K., Toric residue mirror conjecture for Calabi-Yau complete intersections, J. Algebraic Geom., 14, 741, (2005)
[11] McOrist, J.; Melnikov, IV, Summing the instantons in half-twisted linear σ-models, JHEP, 02, 026, (2009)
[12] Silverstein, E.; Witten, E., Criteria for conformal invariance of (0, 2) models, Nucl. Phys., B 444, 161, (1995)
[13] Basu, A.; Sethi, S., World-sheet stability of (0, 2) linear σ-models, Phys. Rev., D 68, 025003, (2003)
[14] Beasley, C.; Witten, E., Residues and world-sheet instantons, JHEP, 10, 065, (2003)
[15] J. Distler, Notes on (0,2) superconformal field theories, hep-th/9502012 [SPIRES].
[16] D.A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom.4 (1995) 17 [alg-geom/9210008] [SPIRES].
[17] D.A. Cox and S. Katz, Mirror symmetry and algebraic geometry, American Mathematical Society, Providence U.S.A. (2000) [SPIRES].
[18] J. Harris, Algebraic geometry: a first course, Springer, New York U.S.A. (1992).
[19] McOrist, J.; Melnikov, IV, Half-twisted correlators from the Coulomb branch, JHEP, 04, 071, (2008)
[20] Kapranov, MM, A characterization of A-discriminantal hypersurfaces in terms of the logarithmic Gauss map, Math. Ann., 290, 277, (1991)
[21] Adams, A.; Basu, A.; Sethi, S., (0, 2) duality, Adv. Theor. Math. Phys., 7, 865, (2004)
[22] Adams, A.; Distler, J.; Ernebjerg, M., Topological heterotic rings, Adv. Theor. Math. Phys., 10, 657, (2006)
[23] Distler, J.; Kachru, S., (0, 2) Landau-Ginzburg theory, Nucl. Phys., B 413, 213, (1994)
[24] Distler, J.; Kachru, S., Duality of (0, 2) string vacua, Nucl. Phys., B 442, 64, (1995)
[25] Blumenhagen, R.; Schimmrigk, R.; Wisskirchen, A., (0, 2) mirror symmetry, Nucl. Phys., B 486, 598, (1997)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.