Communications in Mathematical Physics

, Volume 178, Issue 1, pp 115–134 | Cite as

Stable singularities in string theory

  • Paul S. Aspinwall
  • David R. Morrison
  • Mark Gross


We study a topological obstruction of a very stringy nature concerned with deforming the target space of anN=2 non-linear σ-model. This target space has a singularity which may be smoothed away according to the conventional rules of geometry, but when one studies the associated conformal field theory one sees that such a deformation is not possible without a discontinuous change in some of the correlation functions. This obstruction appears to come from torsion in the homology of the target space (which is seen by deforming the theory by an irrelevant operator). We discuss the link between this phenomenon and orbifolds with discrete torsion as studied by Vafa and Witten.


Neural Network Statistical Physic Field Theory Correlation Function Complex System 
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  1. 1.
    Artin, M., Mumford, D.: Some Elementary Results of Unirational Varieties which are not Rational Proc. London Math. Soc.25, 75–95 (1972)Google Scholar
  2. 2.
    Aspinwall, P.S.: The Moduli Space ofN=2 Superconformal Field Theories. Cornell 1994 preprint CLNS-94/1307, hep-th/9412115, to appear in the proceedings of the Trieste Summer School 1994Google Scholar
  3. 3.
    Aspinwall, P.S.: Resolution of Orbifold Singularities in String Theory. IAS 1994 preprint IASSNS-HEP-94/9, hep-th/9403123, to appear in “Essays on Mirror Manifolds 2”Google Scholar
  4. 4.
    Aspinwall, P.S., Greene, B.R.: On the Geometric Interpretation ofN=2 Superconformal Theories. Nucl. Phys.B437 (1995) 205–230Google Scholar
  5. 5.
    Aspinwall, P.S., Greene, B.R., Morrison, D.R.: The Monomial-Divisor Mirror Map. Internat. Math. Res. Notices1993, 319–338Google Scholar
  6. 6.
    Aspinwall, P.S., Greene, B.R., Morrison, D.R.: Calabi-Yau Moduli Space, Mirror Manifolds and Spacetime Topology Change in String Theory. Nucl. Phys.B416, 414–480 (1994)Google Scholar
  7. 7.
    Aspinwall, P.S., Morrison, D.R.: Chiral Rings Do Not Suffice:N=(2,2) Theories with Nonzero Fundamental Group. Phys. Lett.334B, 79–86 (1994)Google Scholar
  8. 8.
    Batyrev, V.V.: Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties. J. Alg. Geom.3, 493–535 (1994)Google Scholar
  9. 9.
    Bayer, D., Stillman, M.: Macaulay: A System for Computation in Algebraic Geometry and Commutative Algebra. Download from Zariski.harvard.eduGoogle Scholar
  10. 10.
    Borisov, L.: Towards the Mirror Symmetry for Calabi-Yau Complete Intersections in Gorenstein Toric Fano Varieties. Michigan 1993 preprint, alg-geom/9310001Google Scholar
  11. 11.
    Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Berlin, Heidelberg, New York: Springer 1982Google Scholar
  12. 12.
    Candelas, P., de la Ossa, X., Font, A., Katz, S., Morrison, D.R.: Mirror Symmetry for Two Parameter Models-I. Nucl. Phys.B416, 481–562 (1994)Google Scholar
  13. 13.
    Candelas, P., de la Ossa, X.C., Green, P.S., Parkes, L.: A Pair of Calabi-Yau Manifolds as an Exactly Soluble Superconformal Theory. Nucl. Phys.B359, 21–74 (1991)Google Scholar
  14. 14.
    Candelas, P., Green, P., Hübsch, T.: Rolling Among Calabi-Yau Vacua. Nucl. Phys.B330, 49–102 (1990)Google Scholar
  15. 15.
    Cecotti, S., Vafa, C.: Exact Results for Supersymmetric σ Models. Phys. Rev. Lett.68, 903–906 (1992)Google Scholar
  16. 16.
    Clemens, H.: Double Solids. Advances in Math.47, 107–230 (1983)Google Scholar
  17. 17.
    D'Adda, A., Di Vecchia, P., Lüscher, M.: Confinement and Chiral Symmetry Breaking inCP n Models with Quarks. Nucl. Phys.B152, 125–144 (1979)Google Scholar
  18. 18.
    Dixon, L., Harvey, J.A., Vafa, C., Witten, E.: Strings on Orbifolds. Nucl. Phys.B 261, 678–686 (1985) andB274, 285–314 (1986)Google Scholar
  19. 19.
    Dolgachev, I., Gross, M.: Elliptic Threefolds I: Ogg-Shafarevich Theory. J. Alg. Geom.3 39–80 (1994)Google Scholar
  20. 20.
    Green, M., Schwarz, J., Witten, E.: Superstring Theory. Cambridge: Cambridge University Press, 1987, 2 volumesGoogle Scholar
  21. 21.
    Grothendieck, A.: Le Groupe de Brauer, I, II, III, In: Dix Exposés sur la Cohomologie des Schémas. Amsterdam: North Holland, 1968, pp. 46–188Google Scholar
  22. 22.
    Harris, J., Tu, L.: On Symmetric and Skew-symmetric Determinantal Varieties. Topology23, 71–84 (1984)Google Scholar
  23. 23.
    Milne, J.: Étale Cohomology. Princeton University Press, 1980Google Scholar
  24. 24.
    Morrison, D.R., Plesser, M.R.: Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties. Duke and IAS 1994 preprint DUKE-TH-94-78, IASSNS-HEP-94/82, hep-th/9412236, to appear in Nucl. Phys. BGoogle Scholar
  25. 25.
    Strominger, A., Witten, E.: New Manifolds for Superstring Compactification, Commun. Math. Phys.101, 341–361 (1985)Google Scholar
  26. 26.
    Vafa C.: Modular Invariance and Discrete Torsion on Orbifolds. Nucl. Phys.B273, 592–606 (1986)Google Scholar
  27. 27.
    Vafa, C., Warner, N.: Catastrophes and the Classification of Conformal Theories. Phys. Lett218B, 51–58 (1989)Google Scholar
  28. 28.
    Vafa, C., Witten, E.: On Orbifolds with Discrete Torsion. J. Geom. Phys.15, 189–214 (1995)Google Scholar
  29. 29.
    Witten, E.: On the Structure of the Topological Phase of Two Dimensional Gravity, Nucl. Phys.B340, 281–332 (1990)Google Scholar
  30. 30.
    Witten, E.: Mirror Manifolds and Topological Field Theory. In: S.-T. Yau (ed.) Essays on Mirror Manifolds. Hong Kong International Press, 1992Google Scholar
  31. 31.
    Witten, E.: Phases ofN=2 Theories in Two Dimensions. Nucl. Phys.B403, 159–222 (1993)Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Paul S. Aspinwall
    • 1
  • David R. Morrison
    • 2
  • Mark Gross
    • 3
  1. 1.F.R. Newman Lab. of Nuclear StudiesCornell UniversityIthacaUSA
  2. 2.Department of MathematicsDuke UniversityDurhamUSA
  3. 3.Department of MathematicsCornell UniversityIthacaUSA

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