Topological field theory and rational curves. (English) Zbl 0776.53043

Summary: We analyze the quantum field theory corresponding to a string propagating on a Calabi-Yau threefold. This theory naturally leads to the consideration of Witten’s topological nonlinear \(\sigma\)-model and the structure of rational curves on the Calabi-Yau manifold. We study in detail the case of the world-sheet of the string being mapped to a multiple cover of an isolated rational curve and we show that a natural compactification of the moduli space of such a multiple cover leads to a formula in agreement with a conjecture by Candelas, de la Ossa, Green and Parkes.


53Z05 Applications of differential geometry to physics
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
Full Text: DOI arXiv


[1] Candelas, P., Horowitz, G., Strominger, A., Witten, E.: Vacuum Configuration for Superstrings. Nucl. Phys.B258, 46–74 (1985)
[2] Callan, C.G., Friedan, D., Martinec, E.J., Parry, M.J.: Strings in Background Fields. Nucl. Phys.B262, 593–609 (1985)
[3] Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory. Nucl. Phys.B241, 333–380 (1984) · Zbl 0661.17013
[4] Friedan, D., Martinec, E., Shenker, S.: Conformal Invariance, Supersymmetry and String Theory. Nucl. Phys.B271, 93–165 (1986)
[5] Gepner, D.: Exactly Solvable String Compactifications on Manifolds ofSU(N) Holonomy. Phys. Lett.199B, 380–388 (1987)
[6] Aspinwall, P.S., Lütken, C.A.: Quantum Algebraic Geometry of Superstring Compactifications. Nucl. Phys.B355, 482–510 (1991)
[7] Greene, B.R., Plesser, M.R.: Duality in Calabi-Yau Moduli Space. Nucl. Phys.B338, 15–37 (1990)
[8] Aspinwall, P.S., Lütken, C.A., Ross, G.G.: Construction and Couplings of Mirror Manifolds. Phys. Lett.241B, 373–380 (1990)
[9] 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) · Zbl 1098.32506
[10] Witten, E.: Topological Quantum Field Theory. Commun. Math. Phys.117, 353–386 (1988) · Zbl 0656.53078
[11] Donaldson, S.K.: An Application of Gauge Theory to the Topology of Four-Manifolds. J. Diff. Geom.18, 269–277 (1983); The Orientation of Yang-Mills Moduli Spaces and 4-Manifold Topology. J. Diff. Geom.26, 397–428 (1987) · Zbl 0504.49027
[12] Donaldson, S.K.: Polynomial Invariants for Smooth Four-Manifolds. Topology29, 257–315 (1990) · Zbl 0715.57007
[13] Witten, E.: Topological Sigma Models. Commun. Math. Phys.118, 411–449 (1988) · Zbl 0674.58047
[14] Zumino, B.: Supersymmetry and Kähler Manifolds. Phys. Lett.87B, 203–211 (1979)
[15] Witten, E.: On the Structure of the Topological Phase of Two Dimensional Gravity. Nucl. Phys.B340, 281–332 (1990)
[16] Strominger, A., Witten, E.: New Manifolds for Superstring Compactification. Commun. Math. Phys.101, 341–361 (1985)
[17] Dine, M., Seiberg, N., Wen, X.G., Witten, E.: Nonperturbative Effects on the String World-Sheet. Nucl. Phys.B278, 769–789 (1986) and Nucl. Phys.B289, 319–363 (1987)
[18] Clemens, H., Kollár, J., Mori, S.: Higher Dimensional Complex Geometry. Astérisque166 (1988)
[19] Distler, J., Greene, B.: Some Exact Results on the Superpotential from Calabi-Yau compactifications. Nucl. Phys.B309, 295–316 (1988)
[20] Witten, E.: TheN-Matrix Model and GaugedWZW Models. Nucl. Phys.B371, 191–245 (1992)
[21] Atiyah, M.F., Jeffrey, L.: Topological Lagrangians and Cohomology. J. Geom. Phys.7, 119–136 (1990) · Zbl 0721.58056
[22] Bott, R., Tu, L.W.: Differential Forms in Algebraic Topology. Graduate Texts in Mathematics vol.82, Berlin, Heidelberg, New York: Springer 1982 · Zbl 0496.55001
[23] Mathai, V., Quillen, D.: Superconnections, Thom Classes and Equivariant Differential Forms. Topology25, 85–110 (1986) · Zbl 0592.55015
[24] McDuff, D.: Examples of Symplectic Structures. Invent. Math.89, 13–36 (1987); Elliptic Methods in Symplectic Geometry. Bull. (N.S.) Am. Math. Soc.23, 311–358 (1990) · Zbl 0625.53040
[25] Klingenberg, W.: Lectures on Closed Geodesics. Grundlehren der Math. Wiss. vol.230, Berlin, Heidelberg, New York: Springer 1978 · Zbl 0397.58018
[26] Gromov, M.: Pseudo-holomorphic Curves on Almost Complex Manifolds. Invent. Math.82, 307–347 (1985); Soft and Hard Symplectic Geometry. Proc. Intern. Congress Math., Berkeley 1986, Providence, RJ: American Mathematical Society (1987), pp. 81–98 · Zbl 0592.53025
[27] Wolfson, J.G.: Gromov’s Compactness of Pseudo-holomorphic Curves and Symplectic Geometry. J. Diff. Geom.28, 383–405 (1988) · Zbl 0661.53024
[28] Morrison, D.R.: Mirror Symmetry and Rational Curves on Quintic Threefolds: A Guide for Mathematicians. Duke preprint DUK-M-91-01
[29] Clemens, H.: Some Results on Abel-Jacobi Mappings. In: Topics in Transcendental Algebraic Geometry. Princeton, NJ: Princeton University Press, 1984 · Zbl 0575.14007
[30] Albano, A., Katz, S.: Lines on the Fermat Quintic Threefold and the Infinitesimal Generalized Hodge Conjecture. Trans. Am. Math. Soc.324, 353–368 (1991) · Zbl 0767.14016
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.