Abstract
A recently introduced framework for the compactification of supersymmetric string theory involving noncritical manifolds of complex dimension 2k + D Crit, k ≥ 1, is reviewed. These higher dimensional manifolds are spaces with quantized positive Ricci curvature and therefore do not, a priori, describe consistent string vacua. It is nevertheless possible to derive from these manifolds the massless spectra of critical string ground states. For a subclass of these noncritical theories it is also possible to explicitly construct Calabi-Yau manifolds from the higher dimensional spaces. Thus the new class of theories makes contact with the standard framework of string compactification. This class of manifolds is more general than that of Calabi-Yau manifolds because it contains spaces which correspond to critical string vacua with no Kähler deformations, i.e. no antigenerations, thus providing mirrors of rigid Calabi-Yau manifolds. The constructions reviewed here lead to new insights into the relation between exactly solvable models and their mean field theories on the one hand and Calabi-Yau manifolds on the other, leading, for instance, to a modification of Gepner’s conjecture. They also raise fundamental questions about the Kaluza-Klein concept of string compactification, in particular regarding the rôle played by the dimension of the internal theories.
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References
P. Candelas, G. Horowitz, A. Strominger and E. Witten, Vacuum configurations for superstrings, Nucl. Phys. B258 (1985) 49.
P. Candelas, M. Lynker and R. Schimmrigk, Calabi-Yau manifolds in weighted P(4), Nucl. Phys. B341 (1990) 383.
B.R. Greene and R. Plesser, Duality in Calabi-Yau moduli space, Nucl. Phys. B338 (1990) 15.
M. Lynker and R. Schimmrigk, Landau-Ginzburg theories as orbifolds, Phys. Lett. 249B (1990) 237.
A. Klemm and R. Schimmrigk, Landau-Ginzburg String Vacua, CERN preprint CERN-TH-6459/92 and Heidelberg preprint HD-THEP-92-13.
M. Kreuzer and H. Skarke, No Mirror Symmetry among Landau-Ginzburg Vacua, CERN preprint CERN-TH-6461/92.
P. Candelas, Talk at the Workshop on Geometry and Quantum Field Theory, Baltimore, March 1992.
P. Candelas, E. Derrick and L. Parkes, in preparation.
C. Vafa, Topological Mirrors and Quantum Rings, Harvard preprint HUTP-91/A059.
P. Candelas, X. de la Ossa, P. Green and L. Parkes, An exactly soluble superconformal theory from a mirror pair of Calabi-Yau manifolds, Phys. Lett. 258B (1991) 118.
P. Candelas, X. de la Ossa, P. Green and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B359 (1991) 21.
G. Ellingsrud and S.A. Stromme, The Number of twisted cubic curves on the general quintic 3-fold.
S. Katz, private communication.
B. Zumino, Phys. Lett. 87B (1979) 203.
R. Schimmrigk, Critical String Vacua from Noncritical Manifolds: A Novel Framework for String Compactification, Heidelberg preprint HD-THEP-92-29.
E. Martinec, Algebraic geometry and effective lagrangians, Phys. Lett. 217 (1989) 431.
C. Vafa and N. Warner, Catastrophes and the classification of conformal theories, Phys. Lett. 218 (1989) 51.
C. Vafa, String vacua and orbifoldized L-G models, Mod. Phys. Lett. A4 (1989) 1169.
P. Candelas, Yukawa couplings between (2,1) forms, Nucl. Phys. B298 (1988) 488.
P. Berglund, B.R. Greene and T. Hübsch, Classical vs. Landau-Ginzburg Geometry of Compactification, Mod. Phys. Lett. A7 (1992) 1855.
R. Schimmrigk, work in progress.
B.R. Greene, C. Vafa and N. Warner, Calabi-Yau manifolds and renormalization group flows, Nucl. Phys. B324 (1989) 371.
D. Gepner, Space-time supersymmetry in compactified string theory and superconformal models, Nucl. Phys. B296 (1988) 757.
R. Schimmrigk, A new construction of a three generation Calabi-Yau manifold, Phys. Lett. 193B (1987) 175.
D. Gepner, String Theory on Calabi-Yau Manifolds: The Three Generation Case, Princeton University preprint, December 1987.
R. Schimmrigk, Heterotic RG flow fixed points with nondiagonal affine invariants, Phys. Lett. 229B (1989) 227.
P. Candelas, A. Dale, C.A. Lütken and R. Schimmrigk, Complete intersection Calabi-Yau manifolds, Nucl. Phys. B298 (1988) 493.
V.V. Batyrev, Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties, University of Essen preprint.
V.V. Batyrev, Variations of the Mixed Hodge Structure of Affine Hypersurfaces in Algebraic Tori, University of Essen preprint.
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© 1993 Springer Science+Business Media New York
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Schimmrigk, R. (1993). Noncritical Dimensions for Critical String Theory: Life Beyond the Calabi-Yau Frontier. In: Osborn, H. (eds) Low-Dimensional Topology and Quantum Field Theory. NATO ASI Series, vol 315. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1612-9_12
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DOI: https://doi.org/10.1007/978-1-4899-1612-9_12
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