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Mirror symmetry and the type II string. (English) Zbl 0957.81596
Summary: If \(X\) and \(Y\) are a mirror pair of Calabi-Yau threefolds, mirror symmetry should extend to an isomorphism between the type IIA string theory compactified on \(X\) and the type IIB string theory compactified on \(Y\), with all nonperturbative effects included. We study the implications which this proposal has for the structure of the semiclassical moduli spaces of the compactified type II theories. For the type IIB theory, the form taken by discrete shifts in the Ramond-Ramond scalars exhibits an unexpected dependence on the \(B\)-field.

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14K30 Picard schemes, higher Jacobians
32G81 Applications of deformations of analytic structures to the sciences
32J81 Applications of compact analytic spaces to the sciences
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[1] Aspinwall, P.S.; Morrison, D.R., U-duality and integral structures, Phys. lett. B, 355, 141-149, (1995)
[2] Cecotti, S.; Ferrara, S.; Girardello, L., Geometry of type II superstrings and the moduli of superconformal field theories, Int. J. mod. phys. A, 4, 2475-2529, (1989) · Zbl 0681.58044
[3] de Wit, B.; Van Proeyen, A., Hidden symmetries, special geometry and quaternionic manifolds, Int. J. mod. phys. D, 3, 31-48, (1994)
[4] Strominger, A., Massless black holes and conifolds in string theory, Nucl. phys. B, 451, 96-108, (1995) · Zbl 0925.83071
[5] Aspinwall, P.S.; Lütken, C.A., Quantum algebraic geometry of superstring compactifications, Nucl. phys. B, 355, 482-510, (1991)
[6] Morrison, D.R., Mirror symmetry and rational curves on quintic threefolds: A guide for mathematicians, J. amer. math. soc., 6, 223-247, (1993) · Zbl 0843.14005
[7] Candelas, P.; de la Ossa, X.; Font, A.; Katz, S.; Morrison, D.R., Mirror symmetry for two parameter models (I), Nucl. phys. B, 416, 481-562, (1994) · Zbl 0899.14017
[8] Morrison, D.R., Making enumerative predictions by means of mirror symmetry, Essays on Mirror Manifolds II, to appear · Zbl 0932.14021
[9] Witten, E., Dyons of charge eθ/2π, Phys. lett. B, 86, 283-287, (1979)
[10] Dixon, L.J., Some world-sheet properties of superstring compactifications, on orbifolds and otherwise, (), 67-126
[11] Lerche, W.; Vafa, C.; Warner, N.P., Chiral rings in N=2 superconformal theories, Nucl. phys. B, 324, 427-474, (1989)
[12] Candelas, P.; Lynker, M.; Schimmrigk, R., Calabi-Yau manifolds in weighted P4, Nucl. phys. B, 341, 383-402, (1990) · Zbl 0962.14029
[13] Greene, B.R.; Plesser, M.R., Duality in Calabi-Yau moduli space, Nucl. phys. B, 338, 15-37, (1990)
[14] Dine, M.; Huet, P.; Seiberg, N., Large and small radius in string theory, Nucl. phys. B, 322, 301-316, (1989)
[15] Dai, J.; Leigh, R.; Polchinski, J., New connections between string theories, Mod. phys. lett. A, 4, 2073-2083, (1989)
[16] Papadopoulos, G.; Townsend, P.K., Compactification of D=11 supergravity on spaces of exceptional holonomy, Phys. lett. B, 357, 300-306, (1995)
[17] D. D. Joyce, Compact Riemannian 7-manifolds with holonomy G_{2}: I, II, J. Differential Geom., to appear. · Zbl 0861.53023
[18] Aspinwall, P.S.; Morrison, D.R., String theory on K3 surfaces, Essays on Mirror Manifolds II, to appear · Zbl 0931.14020
[19] Narain, K.S., New heterotic string theories in uncompactified dimensions < 10, Phys. lett. B, 169, 41-46, (1986)
[20] Narain, K.S.; Samadi, M.H.; Witten, E., A note on the toroidal compactification of heterotic string theory, Nucl. phys. B, 279, 369-379, (1987)
[21] Seiberg, N., Observations on the moduli space of superconformal field theories, Nucl. phys. B, 303, 286-304, (1988)
[22] D. D. Joyce, Compact Riemannian 8-manifolds with holonomy Spin(7), Invent. Math., to appear. · Zbl 1040.53062
[23] Shatashvili, S.L.; Vafa, C., Superstrings and manifold of exceptional holonomy · Zbl 0839.53060
[24] Seiberg, N.; Witten, E., Spin structures in string theory, Nucl. phys. B, 276, 272-290, (1986)
[25] Vafa, C., Modular invariance and discrete torsion on orbifolds, Nucl. phys. B, 273, 592-606, (1986) · Zbl 0992.81515
[26] Dijkgraaf, R.; Witten, E., Topological gauge theories and group cohomology, Commun. math. phys., 129, 393-429, (1990) · Zbl 0703.58011
[27] Aspinwall, P.S.; Morrison, D.R., Chiral rings do not suffice: N=(2,2) theories with nonzero fundamental group, Phys. lett. B, 334, 79-86, (1994)
[28] Distler, J.; Greene, B., Some exact results on the superpotential from Calabi-Yau compactifications, Nucl. phys. B, 309, 295-316, (1988)
[29] Aspinwall, P.S.; Greene, B.R.; Morrison, D.R., Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory, Nucl. phys. B, 416, 414-480, (1994) · Zbl 0899.32006
[30] Morrison, D.R., Where is the large radius limit?, (), 311-315 · Zbl 0844.58015
[31] Griffiths, P.; Griffiths, P., Periods of integrals on algebraic manifolds, I, II, Amer. J. math., Amer. J. math., 90, 805-865, (1968) · Zbl 0183.25501
[32] Donagi, R.; Markman, E., Cubics, integrable systems and Calabi-Yau threefolds, Hirzebruch’s 65th Birthday Workshop in Algebraic Geometry, to appear · Zbl 0878.14031
[33] Ferrara, S.; Sabharwal, S., Quaternionic manifolds for type II superstring vacua of Calabi-Yau spaces, Nucl. phys. B, 332, 317-332, (1990)
[34] Greene, B.R.; Morrison, D.R.; Strominger, A., Black hole condensation and the unification of string vacua, Nucl. phys. B, 451, 109-120, (1995) · Zbl 0908.53041
[35] Becker, K.; Becker, M.; Strominger, A., Fivebranes, membranes and non-perturbative string theory, Nucl. phys. B, 456, 130-152, (1995) · Zbl 0925.81161
[36] Landman, A., On the Picard-Lefschetz transformations, Trans. amer. math. soc., 181, 89-126, (1973)
[37] Schmid, W., Variation of Hodge structure: the singularities of the period mapping, Invent. math., 22, 211-319, (1973) · Zbl 0278.14003
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