Mirror symmetry II.

*(English)*Zbl 0905.00079
AMS/IP Studies in Advanced Mathematics. 1. Cambridge, MA: International Press, Providence, RI: American Mathematical Society (AMS). xvi, 844 p. (1997).

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Foreword: Mirror symmetry has undergone dramatic progress since the Mathematical Sciences Research Institute workshop in 1991 whose proceedings constitute volume I of this continuing collection. Tremendous insight has been gained on a number of key issues, and it is the purpose of the present volume to survey some of these results. Some of the contributions are reprints of papers which have appeared elsewhere while others were written specifically for this collection.

The areas covered are organized into four sections, and each presents papers by both physicists and mathematicians. Section I focuses on the present understanding of explicit constructions of mirror manifolds. Paper 1, B. R. Greene and H. Ooguri, Geometry and quantum field theory: a brief introduction (3-27); briefly reviews the notion of path integration to assist those less familiar with this physical tool. Paper 2, B. R. Greene, Constructing mirror manifolds (29-69); reviews the first, and at present, only known construction of mirror manifolds, at the level of conformal field theory. Paper 3, Victor V. Batyrev and Lev. A. Borisov, Dual cones and mirror symmetry for generalized Calabi-Yau manifolds (71-86); discusses a more general construction of mirror pairs, which as yet has not been established in conformal field theory, and paper 4, Per Berglund and Sheldon Katz, Mirror symmetry constructions: a review (87-113); reviews this and other conjectured constructions. Paper 5, Per Berglund and Mans Henningson, On the elliptic genus and mirror symmetry (115-127); discusses mirror symmetry in the context of Landau-Ginsburg theories and paper 6, Shi-shyr Roan, Orbifold Euler characteristic (129-140); reviews properties of the orbifolding operation, from a mathematical perspective.

Section II focuses on work that has honed our understanding of both Calabi-Yau and conformal field theory moduli spaces. Papers 7 and 8, Edward Witten, Phases of \(N=2\) theories in two dimensions (Reprint) (143-211); Paul S. Aspinwall, Brian R. Greene and David R. Morrison, Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory (Reprint) (213-279); discuss properties of the enlarged Kähler moduli space required by conformal field theory, and in particular, establish the first concrete arena for physically smooth spacetime topology change. Paper 9, A. Ceresole, R. D’Auria, S. Ferrara, W. Lerche, J. Louis and T. Regge, Picard-Fuchs equations, special geometry and target space duality (281-353); discusses aspects of geometrical structure of such moduli spaces and, in paper 10, Paul S. Aspinwall, Resolution of orbifold singularities in string theory (355-379); some of the newfound understanding of moduli space is applied to the case of orbifold theories. In paper 11, P. M. H. Wilson, The role of \(c_2\) in Calabi-Yau classification – a preliminary survey (381-392); the classification problem for Calabi-Yau’s is discussed; in paper 12, Z. Ran, Thickening Calabi-Yau moduli spaces (393-400); Witten’s notion of thickening the moduli space is described from a mathematical perspective and in paper 13, Mark Gross, The deformation space of Calabi-Yau \(n\)-folds with canonical singularities can be obstructed (401-411); an example of an obstructed moduli space is discussed. Paper 14, Amit Giveon and Martin Rocek, Introduction to duality (413-426); presents an introductory discussion of duality properties of moduli space, paper 15, E. Kiritsis, C. Kounnas and D. Lust, Non-compact Calabi-Yau spaces and other nontrivial backgrounds for four-dimensional superstrings (427-441); embarks on the issue of noncompact Calabi-Yau spaces while paper 16, R. Schimmrigk, Scaling behavior on the space of Calabi-Yau manifolds (443-453); discusses some interesting Calabi-Yau numerology.

Section III focuses on developments in using mirror symmetry to solve difficult counting problems, i.e. problems in enumerative geometry. Papers 18 and 19, Philip Candelas, Xenia de la Ossa, Anamaria Font, Sheldon Katz and David R. Morrison, Mirror symmetry for two parameter models. I (Reprint) (483-543); S. Hosono, A. Klemm, S. Theisen and S.-T. Yau, Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces (Reprint) (545-606); discuss the methods for counting rational curves for examples whose parameter space is larger than one, while paper 17, David R. Morrison, Making enumerative predictions by means of mirror symmetry (457-482); presents a review of the multiparameter case in general. Paper 20, M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry (Reprint) (607-653); places the physical approach to these enumerative problems on more firm mathematical foundation, as well as applying such methods to a variety of counting problems. Paper 21, M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories (Reprint) (655-682); resolves a number of key issues in mirror symmetry such as the form of the mirror map, in addition to providing a means of extending the domain of accessible counting problems to higher genus curves. In paper 22, P. Deligne, Local behavior of Hodge structures at infinity (683-699); some aspects of the methods used in applying mirror symmetry to enumerative problems are placed in an appropriate mathematical framework.

Section IV focuses on the extension of mirror symmetry away from the familiar case of complex dimension three to both lower and higher dimension. Papers 23 and 24, Paul S. Aspinwall and David R. Morrison, String theory on \(K3\) surfaces (703-716); Ciprian Borcea, \(K3\) surfaces with involution and mirror pairs of Calabi-Yau manifolds (717-743); discuss mirror symmetry for complex dimension 2 and papers 25 and 26, Brian R. Greene, David R. Morrison and M. Ronen Plesser, Mirror manifolds in higher dimension (Reprint) (745-791); S. Sethi, Supermanifolds, rigid manifolds and mirror symmetry (Reprint) (793-815); discuss various aspects of mirror symmetry in dimension greater than 3. Due to space limitations, there are a number of equally interesting and important developments that have not been included. The papers of this volume, though, will undoubtedly allow the reader to gain much insight into both the physics and the mathematics of the remarkable structure of mirror symmetry.

Indexed articles:

Greene, B. R.; Ooguri, H., Geometry and quantum field theory: a brief introduction, 3-27 [Zbl 1072.81502]

Greene, B. R., Constructing mirror manifolds, 29-69 [Zbl 0961.32017]

Batyrev, Victor V.; Borisov, Lev A., Dual cones and mirror symmetry for generalized Calabi-Yau manifolds, 71-86 [Zbl 0927.14019]

Berglund, Per; Katz, Sheldon, Mirror symmetry constructions: A review, 87-113 [Zbl 0919.14023]

Berglund, Per; Henningson, Måns, On the elliptic genus and mirror symmetry, 115-127 [Zbl 0931.81022]

Roan, Shi-shyr, Orbifold Euler characteristic, 129-140 [Zbl 0939.14020]

Witten, Edward, Phases of \(N=2\) theories in two dimensions, 143-211 [Zbl 0910.14019]

Aspinwall, Paul S.; Greene, Brian R.; Morrison, David R., Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory, 213-279 [Zbl 0923.32021]

Ceresole, A.; D’Auria, R.; Ferrara, S.; Lerche, W.; Louis, J.; Regge, T., Picard-Fuchs equations, special geometry and target space duality, 281-353 [Zbl 0927.14020]

Aspinwall, Paul S., Resolution of orbifold singularities in string theory, 355-379 [Zbl 0920.14004]

Wilson, P. M. H., The role of \(c_2\) in Calabi-Yau classification. A preliminary survey, 381-392 [Zbl 0936.14029]

Ran, Z., Thickening Calabi-Yau moduli spaces, 393-400 [Zbl 0927.32011]

Gross, Mark, The deformation space of Calabi-Yau \(n\)-folds with canonical singularities can be obstructed, 401-411 [Zbl 0921.14027]

Giveon, Amit; Roček, Martin, Introduction to duality, 413-426 [Zbl 1129.81341]

Kiritsis, E.; Kounnas, C.; Lüst, D., Non-compact Calabi-Yau spaces and other non-trivial backgrounds for four-dimensional superstrings, 427-441 [Zbl 0920.14015]

Schimmrigk, R., Scaling behavior on the space of Calabi-Yau manifolds, 443-453 [Zbl 0936.14030]

Morrison, David R., Making enumerative predictions by means of mirror symmetry, 457-482 [Zbl 0932.14021]

Candelas, Philip; de la Ossa, Xenia; Font, Anamaría; Katz, Sheldon; Morrison, David R., Mirror symmetry for two parameter models. I, 483-543 [Zbl 0917.14023]

Hosono, S.; Klemm, A.; Theisen, S.; Yau, S.-T., Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, 545-606 [Zbl 0920.32020]

Kontsevich, M.; Manin, Yu., Gromov-Witten classes, quantum cohomology, and enumerative geometry, 607-653 [Zbl 0931.14030]

Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C., Holomorphic anomalies in topological field theories, 655-682 [Zbl 0919.58067]

Deligne, P., Local behavior of Hodge structures at infinity, 683-699 [Zbl 0939.14005]

Aspinwall, Paul S.; Morrison, David R., String theory on K3 surfaces, 703-716 [Zbl 0931.14020]

Borcea, Ciprian, K3 surfaces with involution and mirror pairs of Calabi-Yau manifolds, 717-743 [Zbl 0939.14021]

Greene, Brian R.; Morrison, David R.; Plesser, M. Ronen, Mirror manifolds in higher dimension, 745-791 [Zbl 0923.32022]

Sethi, S., Supermanifolds, rigid manifolds and mirror symmetry, 793-815 [Zbl 0929.32014]

##### MSC:

00B25 | Proceedings of conferences of miscellaneous specific interest |

81-06 | Proceedings, conferences, collections, etc. pertaining to quantum theory |

32-06 | Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces |

14-06 | Proceedings, conferences, collections, etc. pertaining to algebraic geometry |