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Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties. (English) Zbl 0908.14014
Summary: We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety \(V\) or a Calabi-Yau hypersurface \(M\subset|V\). In the linear model the instanton moduli spaces are relatively simple objects and the correlators are explicitly computable; moreover, the instantons can be summed, leading to explicit solutions for both kinds of models. In the case of smooth \(V\), our results reproduce and clarify an algebraic solution of the \(V\) model due to Batyrev. In addition, we find an algebraic relation determining the solution for \(M\) in terms of that for \(V\). Finally, we propose a modification of the linear model which computes instanton expansions about any limiting point in the moduli space. In the smooth case this leads to a (second) algebraic solution of the \(M\) model. We use this description to prove some conjectures about mirror symmetry, including the previously conjectured “monomial-divisor mirror map” of P. S. Aspinwall, B. R. Greene and D. R. Morrison [Int.Math. Res. Not. 1993, No. 12, 319-337 (1993; Zbl 0798.14030)].

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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[1] Grisaru, M.T.; Siegel, W.; Roček, M.; Seiberg, N., Nucl. phys. B, Phys. lett. B, 318, 469, (1993), hep-ph/9309335
[2] Essays on mirror manifolds II, ed. B.R. Greene and S.-T. Yau (International Press, Hong Kong) to appear.
[3] Witten, E., Commun. math. phys., 118, 411, (1988)
[4] Witten, E., Mirror manifolds and topological field theory, (), 120, hep-th/9112056 · Zbl 0834.58013
[5] Distler, J.; Greene, B., Nucl. phys. B, 309, 295, (1988)
[6] Witten, E., Nucl. phys. B, 403, 159, (1993), hep-th/9301042
[7] D.R. Morrison and M.R. Plesser, work in progress.
[8] Batyrev, V.V.; Batyrev, V.V., Quantum cohomology rings of toric manifolds, (), 9, (Juillet 1992), alg-geom/9310004
[9] Aspinwall, P.S.; Morrison, D.R., Phys. lett. B, 334, 79, (1994), hep-th/9406032
[10] Witten, E., Nucl. phys. B, 371, 191, (1992)
[11] Curtis, C.W.; Reiner, I., Representation theory of finite groups and associative algebras, (1962), Interscience Publishers New York · Zbl 0131.25601
[12] Karpilovsky, G., Symmetric and G-algebras: with applications to group representations, (1990), Kluwer Academic Publishers Dordrecht · Zbl 0705.16001
[13] Dubrovin, B., Nucl. phys. B, 379, 627, (1992)
[14] Nakayama, T., Ann. of math., 40, 611, (1939)
[15] S. Katz, Rational curves on Calabi-Yau threefolds, (revised version), alg-geom/9312009.
[16] Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C., Nucl. phys. B, 405, 298, (1993), hep-th/9302103
[17] Katz, S., Rational curves on Calabi-Yau manifolds: verifying predictions of mirror symmetry, (), 231, alg-geom/9301006 · Zbl 0839.14043
[18] Ruan, Y., Topological sigma model and Donaldson type invariants in Gromov theory, (1993), preprint
[19] Ruan, Y.; Tian, G., Math. res. lett., 1, 269-278, (1994)
[20] McDuff, D.; Salamon, D., J-holomorphic curves and quantum cohomology, (1994), American Mathematical Society · Zbl 0809.53002
[21] Kontsevich, M.; Manin, Yu., Commun. math. phys., 164, 525, (1994), hep-th/9402147
[22] M. Kontsevich, Enumeration of rational curves via torus actions, hep-th/9405035. · Zbl 0885.14028
[23] Yu.I. Manin, Generating functions in algebraic geometry and sums over trees, alg-geom/9407005. · Zbl 0871.14022
[24] Greene, B.R.; Plesser, M.R., Nucl. phys. B, 338, 15, (1990)
[25] Berglund, P.; Hübsch, T., A generalized construction of mirror manifolds, (), 388, hep-th/9201014 · Zbl 0842.32023
[26] Batyrev, V.V., J. alg. geom., 3, 493-535, (1994), alg-geom/9310003
[27] L.A. Borisov, Towards the mirror symmetry for Calabi-Yau complete intersections in Gorenstein toric Fano varieties, alg-geom/9310001.
[28] V.V. Batyrev and L.A. Borisov, Dual cones and mirror symmetry for generalized Calabi-Yau manifolds, alg-geom/9402002.
[29] P. Berglund and S. Katz, Mirror symmetry constructions: a review, preprint IASSNS-HEP-94/38, OSUM-94/2, to appear in Essays in mirror manifolds II, ed. B.R. Greene and S.-T. Yau (International Press, Hong Kong) hep-th/9406008.
[30] Aspinwall, P.S.; Greene, B.R.; Morrison, D.R., Nucl. phys. B, 416, 414, (1994), hep-th/9309097
[31] de Wit, B.; Van Proeyen, A., Nucl. phys. B, 245, 89, (1985)
[32] de Wit, B.; Lauwers, P.; Van Proeyen, A., Nucl. phys. B, 255, 569-608, (1985)
[33] Cremmer, E.; Kounnas, C.; Van Proeyen, A.; Derendinger, J.P.; de Wit, B.; Girardello, L., Nucl. phys. B, 250, 385-426, (1985)
[34] Strominger, A., Commun. math. phys., 133, 163, (1990)
[35] Cecotti, S.; Vafa, C., Nucl. phys. B, 367, 359-461, (1991)
[36] Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C., Commun. math. phys., 165, 311, (1994), hep-th/9309140
[37] B.R. Greene, D.R. Morrison and M.R. Plesser, Mirror symmetry in higher dimension, preprint CLNS-93/1253, IASSNS-HEP-94/2, YCTP-P31-92, hep-th/9402119.
[38] Candelas, P.; de la Ossa, X.C.; Green, P.S.; Parkes, L., Nucl. phys. B, 359, 21, (1991)
[39] Aspinwall, P.S.; Greene, B.R.; Morrison, D.R., Int. math. res. notices, 319-337, (1993), alg-geom/9309007
[40] Mumford, D., Geometric invariant theory, () · Zbl 0147.39304
[41] Kirwan, F.C., Cohomology of quotients in symplectic and algebraic geometry, (), no. 31 · Zbl 0553.14020
[42] Ness, L., Amer. J. math., 106, 1281-1329, (1984)
[43] Fulton, W., Introduction to toric varieties, () · Zbl 1083.14065
[44] Cox, D.A., The homogeneous coordinate ring of a toric variety, (1992), alg-geom/9210008
[45] Candelas, P.; de la Ossa, X.; Font, A.; Katz, S.; Morrison, D.R., Nucl. phys. B, 416, 481-538, (1994), hep-th/9308083
[46] 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, hep-th/9308122. · Zbl 0814.53056
[47] Batyrev, V.V., Tôhoku math. J., 43, 569, (1991)
[48] Guillemin, V.; Sternberg, S., Invent. math., 97, 485, (1989)
[49] Duistermaat, J.J.; Heckman, G.J.; Duistermaat, J.J.; Heckman, G.J., Invent. math., Invent. math., 72, 153, (1983)
[50] Oda, T.; Park, H.S., Tôhoku math. J., 43, 375, (1991)
[51] N. Seiberg, The power of holomorphy: exact results in 4D SUSY field theories, preprint RU-94-64, IASSNS-HEP-94/67, hep-th/9408013, to appear in proceedings of PASCOS 94.
[52] Coleman, S., Ann. phys., 101, 239, (1976)
[53] Cecotti, S.; Vafa, C., Phys. rev. lett., 68, 903, (1992), hep-th/9111016
[54] Intriligator, K.; Leigh, R.; Seiberg, N., Phys. rev., D50, 1092, (1994), hep-th/9403198
[55] S. Cordes, G. Moore and S. Ramgoolam, Lectures on 2D Yang-Mills theory, equivariant cohomology and topological field theories, preprint YCTP-P11-94, hep-th/9411210.
[56] Bradlow, S.B.; Daskalopoulos, G.D., Int. J. math., 2, 477, (1991)
[57] García-Prado, O., Int. J. math., 5, 1-52, (1994)
[58] Cox, D.A., The functor of a smooth toric variety, (1993), alg-geom/9312001
[59] E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, preprint IASSNS-HEP-93/41, hep-th/9312004.
[60] Aspinwall, P.S.; Greene, B.R.; Morrison, D.R., Nucl. phys. B, 420, 184, (1994), hep-th/9311042
[61] B. Siebert and G. Tian, On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, alg-geom/9403010. · Zbl 0974.14040
[62] Witten, E., Nucl. phys. B, 340, 281, (1990)
[63] P.S. Aspinwall, Resolution of orbifold singularities in string theory, to appear in Essays on mirror manifolds II, ed. B.R. Greene and S.-T. Yau (International Press, Hong Kong) hep-th/9403123.
[64] Silverstein, E.; Witten, E., Phys. lett. B, 328, 307, (1994), hep-th/9403054
[65] Berglund, P.; Katz, S., Nucl. phys. B, 420, 289, (1994), hep-th/9311014
[66] Batyrev, V.V., Duke math. J., 69, 349, (1993)
[67] Batyrev, V.V.; Cox, D.A., Duke math. J., 75, 293, (1994), alg-geom/9306011
[68] Cox, D.A., Toric residues, (1994), alg-geom/9410017 · Zbl 0904.14029
[69] Gel’fand, I.M.; Zelevinskiľ, A.V.; Kapranov, M.M., Leningrad math. J., 2, 449, (1991), (from Russian in Algebra i analiz 2)
[70] Horn, J., Math. ann., 34, 544, (1889)
[71] Kapranov, M.M., Math. ann., 290, 277, (1991)
[72] Hübsch, T.; Yau, S.-T., An \(SL(2, C)\) action on certain Jacobian rings and the mirror map, (), 372 · Zbl 0843.32017
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