×

zbMATH — the first resource for mathematics

Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory. (English) Zbl 0899.32006
Summary: We analyze the moduli spaces of Calabi-Yau three-folds and their associated conformally invariant nonlinear \(\sigma{}\)-models and show that they are described by an unexpectedly rich geometrical structure. Specifically, the Kähler sector of the moduli space of such Calabi-Yau conformal theories admits a decomposition into adjacent domains some of which correspond to the (complexified) Kähler cones of topologically distinct manifolds. These domains are separated by walls corresponding to singular Calabi-Yau spaces in which the spacetime metric has degenerated in certain regions. We show that the union of these domains is isomorphic to the complex structure moduli space of a single topological Calabi-Yau space – the mirror. In this way we resolve a puzzle for mirror symmetry raised by the apparent asymmetry between the Kähler and complex structure moduli spaces of a Calabi-Yau manifold. Furthermore, using mirror symmetry, we show that we can interpolate in a physically smooth manner between any two theories represented by distinct points in the Kähler moduli space, even if such points correspond to topologically distinct spaces. Spacetime topology change in string theory, therefore, is realized for the most basic operation of deformation by a truly marginal operator. Finally, this work also yields some important insights on the nature of orbifolds in string theory.

MSC:
32G20 Period matrices, variation of Hodge structure; degenerations
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
32G81 Applications of deformations of analytic structures to the sciences
14J30 \(3\)-folds
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
32J81 Applications of compact analytic spaces to the sciences
14J10 Families, moduli, classification: algebraic theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Dixon, L.; Harvey, J.; Vafa, C.; Witten, E.; Dixon, L.; Harvey, J.; Vafa, C.; Witten, E., Nucl. phys., Nucl. phys., B274, 285, (1986)
[2] Dixon, L., (), 67
[3] Lerche, W.; Vafa, C.; Warner, N., Nucl. phys., B324, 427, (1989)
[4] Candelas, P.; Lynker, M.; Schimmrigk, R., Nucl. phys., B341, 383, (1990)
[5] Greene, B.R.; Plesser, M.R., Nucl. phys., B338, 15, (1990)
[6] Candelas, P.; de la Ossa, X.C.; Strominger, A.; Cecotti, S.; Ferrara, S.; Girardello, L., Nucl. phys., Commun. math. phys., Phys. lett., B213, 443, (1989)
[7] Viehweg, E., Invent. math., 101, 521, (1990)
[8] Aspinwall, P.S.; Greene, B.R.; Morrison, D.R., Phys. lett., B303, 249, (1993)
[9] Tian, G.; Yau, S.-T., (), 543
[10] Kollár, J., Nagoya math. J., 113, 15, (1989)
[11] Candelas, P.; Green, P.; Hübsch, T., Nucl. phys., B330, 49, (1990)
[12] Witten, E., Nucl. phys., B403, 159, (1993)
[13] Roan, S.-S.; Roan, S.-S., Topological properties of Calabi-Yau mirror manifolds, Int. J. math., MAX-Planck-institut preprint, 2, 439, (1992)
[14] Batyrev, V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, Essen preprint, (November, 1992)
[15] Gepner, D., Phys. lett., B199, 380, (1987)
[16] Greene, B.; Vafa, C.; Warner, N., Nucl. phys., B324, 371, (1989)
[17] Martinec, E.; Martinec, E., (), Phys. lett., B217, 431, (1989)
[18] Aspinwall, P.S., Commun. math. phys., 128, 593, (1990)
[19] Aspinwall, P.S.; Lütken, C.A., Nucl. phys., B355, 482, (1991)
[20] Yau, S.-T., (), 1798
[21] Bogomolov, F.A.; Tian, G.; Todorov, A., (), Commun. math. phys., 126, no. 5, 325, (1989)
[22] Green, P.; Hübsch, T., Commun. math. phys., 113, 505, (1987)
[23] Candelas, P.; de la Ossa, X.C., Nucl. phys., B342, 246, (1990)
[24] Guillemin, V.; Sternberg, S., Invent. math., 97, 485, (1989)
[25] Friedman, R., (), 103
[26] Strominger, A.; Witten, E., Commun. math. phys., 101, 341, (1985)
[27] Strominger, A., Phys. rev. lett., 55, 2547, (1985)
[28] Dine, M.; Seiberg, N.; Wen, X.-G.; Witten, E.; Dine, M.; Seiberg, N.; Wen, X.-G.; Witten, E., Nucl. phys., Nucl. phys., B289, 319, (1987)
[29] Candelas, P.; de la Ossa, X.C.; Green, P.S.; Parkes, L.; Candelas, P.; de la Ossa, X.C.; Green, P.S.; Parkes, L., Phys. lett., Nucl. phys., B359, 21, (1991)
[30] Aspinwall, P.S.; Morrison, D.R., Commun. math. phys., 151, 245, (1993)
[31] Morrison, D.R., J. amer. math. soc., 6, 223, (1993)
[32] Morrison, D.R.; Font, A.; Klemm, A.; Theisen, S.; Libgober, A.; Teitelbaum, J.; Batyrev, V.; van Straten, D.; Candelas, P.; de la Ossa, X.; Font, A.; Katz, S.; Morrison, D.R., Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties, (), Nucl. phys., Nucl. phys., Int. math. res. notices, Essen preprint, Nucl. phys., B416, 481, (1994)
[33] S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Harvard preprint HUTMP-93/0801
[34] Oda, T., Convex bodies and algebraic geometry, (1988), Springer Berlin
[35] Fulton, W., Introduction to toric varieties, () · Zbl 1083.14065
[36] Reid, M., (), 273
[37] Aspinwall, P.S.; Greene, B.R.; Morrison, D.R., Int. mat. res. notices, 319, (1993)
[38] Roan, S.-S., Int. J. math., 1, 211, (1990)
[39] Cox, D.A., The homogeneous coordinate ring of a toric variety, Amherst preprint, (1992)
[40] Audin, M., The topology of torus actions on symplectic manifolds, () · Zbl 0726.57029
[41] Oda, T.; Park, H.S., Tôhoku math. J., 43, 375, (1991)
[42] Billera, L.J.; Filliman, P.; Sturmfels, B., Adv. math., 83, 155, (1990)
[43] Witten, E.; Witten, E., (), B268, 120, (1985)
[44] Distler, J.; Greene, B., Nucl. phys., B309, 295, (1988)
[45] P.S. Aspinwall, B.R. Greene and D.R. Morrison, Measuring small distances in N = 2 sigma models, IASSNS-HEP-93/49 · Zbl 0990.81689
[46] Markushevich, D.; Olshanetsky, M.; Perelomov, A., Commun. math. phys., 111, 247, (1987)
[47] Candelas, P., Nucl. phys., B298, 458, (1988)
[48] Vafa, C.; Warner, N., Phys. lett., B218, 51, (1989)
[49] Batyrev, V., Quantum cohomology rings of toric manifolds, MSRI preprint, (1993) · Zbl 0806.14041
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.