Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties. (English) Zbl 0843.14016

Extending known results of P. Candelas, X. C. de la Ossa, P. S. Green and L. Parkes [“A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory” in: Essays on mirror manifolds, 31-95 (1992; Zbl 0826.32016), see also Nuclear Phys., Particle Physics, B 359, No. 10, 21-74 (1991)], D. R. Morrison [“Mirror symmetry and rational curves on quintic threefolds: A guide for mathematicians”, J. Am. Math. Soc. 6, No. 1, 223-247 (1993; Zbl 0843.14005) and “Picard Fuchs equations and mirror maps for hypersurfaces”, in: Essays on mirror manifolds, 241-264 (1992; Zbl 0841.32013)] and many others, the authors formulate interesting conjectures on the mirror symmetry and generalized hypergeometric series for Calabi-Yau complete intersections in toric varieties. To state these conjectures, we need to introduce necessary ingredients as follows: Let \(P_\Sigma\) be a \((d + r)\)-dimensional projective toric variety corresponding to a complete simplicial fan \(\Sigma\) for a free \(\mathbb{Z}\)-module \(N\) of \(\text{rank} d + r\). Denote by \(E = \{v_1, \dots, v_k\}\) the set of primitive generators of one-dimensional cones in the fan \(\Sigma\), and let \(D_j\) be the irreducible torus-invariant Weil divisor on \(P_\Sigma\) corresponding to the one-dimensional cone spanned by \(v_j \in E\). Split \(E\) into a disjoint union \(E = E_1 \cup E_2 \cup \cdots \cup E_r\) and denote also by \(E_i\) the set of indices \(\{j \mid v_j \in E_i\}\). Assume that \(\sum_{j \in E_i} D_j\) for each \(1 \leq i \leq r\) is numerically effective (or equivalently, base-point-free in the present context) and is linearly equivalent to a hypersurface \(V_i \subset P_\Sigma\). Then the complete intersection \(V : = V_1 \cap V_2 \cap \cdots \cap V_r\) is a \(d\)-dimensional Calabi-Yau variety possibly with Gorenstein toroidal singularities, since \(\sum^k_{j = 1} D_j = \sum_i (\sum_{j \in E_i} D_j)\) is an anticanonical divisor of \(P_\Sigma\). When \(P_\Sigma\) is smooth, the kernel of the surjective homomorphism \(Z^k \ni \lambda = (\lambda_1, \dots, \lambda_k) \mapsto \sum^k_{j = 1} \lambda_j v_j \in N\) is known to coincide with the \(\mathbb{Z}\)-module \(R(E)\) of algebraic 1-cycles on \(P_\Sigma\). \(\lambda_j = \langle D_j, \lambda \rangle\) is the intersection number of the algebraic 1-cycle \(\lambda \in R(E)\) with the divisor \(D_j\). Thus \(R^+ (E) : = R(E) \cap (\mathbb{Z}_{\geq 0})^k\) is the submonoid of nef 1-cycles, where \(\mathbb{Z}_{\geq 0}\) is the set of nonnegative integers. We can choose a \(\mathbb{Z}\)-basis \(\{\lambda^{(1)}, \dots, \lambda^{(t)}\}\) of \(R(E)\) so that effective algebraic 1-cycles on \(P_\Sigma\), hence elements in \(R^+ (E)\) in particular, are nonnegative linear combinations of \(\lambda^{(1)}, \dots, \lambda^{(t)}\). Let us introduce a generalized hypergeometric series in complex variables \(u_1, \dots, u_k\) by \[ \Phi_0 (u) : = \sum_{\lambda \in R^+ (E)} \prod^r_{i = 1} \left( \sum_{j \in E_i} \lambda_j \right)! \left( \left.\prod_{j \in E_i} u_j^{ \lambda_j}\right/\lambda_j! \right). \] Let \(T : = \operatorname{Hom}_\mathbb{Z} (N,C^\times)\) be the \((d + r)\)-dimensional algebraic torus with the character group \(N\). Denote by \(X^v\) the Laurent monomial corresponding to \(v \in N\). Then in terms of the Laurent polynomials \(P_{E_i} (X) : = 1 - \sum_{j \in E_i} u_j X^{v_j}\), \(i = 1, 2, \dots, r\), we have an integral representation \[ \Phi_0 (u) = {1 \over (2i \sqrt {-1})^{d + r}} \int_{|X_1 |= 1, \dots, |X_{d + r} |= 1} {1 \over P_{ E_1} (X) \cdots P_{E_r}(X)} {dX_1 \over X_1} \wedge \cdots \wedge {dX_r \over X_r}, \] where \(X_1, \dots, X_{d + r}\) are suitable coordinates for \(T\).
In terms of a new set of complex variables \(z_1, \dots, z_t\) defined by \(z_s : = \prod^r_{i = 1} \prod_{j \in E_i} u^{\lambda_j^{(s)}}_j\), \(s = 1, \dots, t\), \(\Phi_0 (u)\) can be expressed as a power series \[ \Phi_0 (z) = \sum_{\lambda \in R^+ (E)} (\langle V_1, \lambda \rangle! \cdots \langle V_r, \lambda \rangle!/ \langle D_1, \lambda \rangle! \cdots \langle V_k, \lambda \rangle!) z^\lambda, \] where \(z^\lambda = z_1^{c_1} \cdots z_t^{c_t}\) with \(\lambda = c_1 \lambda^{(1)} + \cdots + c_t \lambda^{(t)}\), and \(\langle V_i, \lambda \rangle\) is the intersection number of \(V_i\) with the nef 1-cycle \(\lambda\).
Assume further that \(V\) is smooth and that the restriction map \(\text{Pic} (\mathbb{P}_\Sigma) \leftarrow \text{Pic} (V)\) is injective. There exists a flat “\(A\)-model connection” \(\nabla_{AP}\) on \(H^* (\mathbb{P}_\Sigma, \mathbb{C})\) which defines a quantum variation of Hodge structures on \(H^* (\mathbb{P}_\Sigma, \mathbb{C})\). Likewise, there exists a flat “\(A\)-model connection” \(\nabla_{AV}\) on \(H^* (V,\mathbb{C})\) which defines a quantum variation of Hodge structures on \(H^* (V,\mathbb{C})\). The complex variables \(z_1, \dots, z_t\) can be identified with \(\nabla_{AP}\)-flat coordinates on the image \(\widetilde H^2\) of \(H^2 (\mathbb{P}_\Sigma, \mathbb{C})\) and \(H^2 (V,\mathbb{C})\). – Here are some of the authors’ conjectures in terms of these ingredients:
(1) The generalized hypergeometric series \(\Phi_0 (z)\) in terms of \(z_1, \dots, z_r\) is a solution of the differential system \({\mathcal D}\) defined by the restriction of \(\nabla_{ AV}\) to \(\widetilde H^2\).
(2) The differential system \({\mathcal D}\) has logarithmic solutions of the form \(\Phi_s (z) = (\log z_s) \Phi_0 (z) + \Psi_s (z)\), \(s = 1, \dots, t\), with \(\Psi_s (z)\) holomorphic at \(z = 0\) and \(\Psi_s (0) = 0\). We can then define \(\nabla_{AV}\)-flat coordinates on \(\widetilde H^2\) by \(q_s : = \exp (\Phi_s (z)/ \Phi_0 (z))\), \(s = 1, \dots, t\). The coefficients of \(q_1, \dots, q_s\) with respect to \(z_1, \dots, z_t\) are integers.
(3) The Calabi-Yau variety mirror symmetric to \(V\) is obtained as a Calabi-Yau compactification of the complete intersection \(\{P_{E_1} (X) = 0\} \cap \cdots \cap \{P_{E_r} (X) = 0\}\) of affine hypersurfaces in the \((d + r)\)-dimensional algebraic torus \(T\).
The authors go on to check these conjectures by dealing with many examples of Calabi-Yau threefolds obtained as complete intersections in products of projective spaces.
Reviewer: T.Oda (Sendai)


14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
33C20 Generalized hypergeometric series, \({}_pF_q\)
Full Text: DOI arXiv


[1] Batyrev, V.V.: Dual Polyhedra and Mirror Symmetry for Calabi-Yau Hypersurfaces in Toric Varieties. Nov. 18 (1992) Preprint, Uni-Essen · Zbl 0829.14023
[2] Batyrev, V.V.: Variations of the Mixed Hodge Structure of Affine Hypersurfaces in Algebraic Tori. Duke Math. J.69, 349–409 (1993) · Zbl 0812.14035
[3] Batyrev, V.V.: Quantum Cohomology Rings of Toric Manifolds. Preprint MSRI, (1993) · Zbl 0806.14041
[4] Borisov, L.A.: Towards the Mirror Symmetry for Calabi-Yau Complete Intersection in Gorenstein Toric Fano Varieties. Preprint 1993, alg-geom 9310001
[5] Candelas, P., Dale, A.M., Lütken, C.A., Schimmrigk, R.: Complete Intersection Calabi-Yau Manifolds I, II. Nucl. Phys.B 298, 493–525 (1988), Nucl. Phys.B 306, 113–136 (1988)
[6] Candelas, P., Lynker, M., Schimmrigk, R.: Calabi-Yau Manifolds in WeightedP 4. Nucl. Phys.B 341, 383–402 (1990) · Zbl 0962.14029
[7] Candelas, P., He, A.: On the Number of Complete Intersection of Calabi-Yau Manifolds. Commun. Math. Phys.135, 193–199 (1990) · Zbl 0722.53061
[8] Candelas, P., de la Ossa, X.: Nucl. Phys.342, 246 (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.B 359, 21–74 (1991) · Zbl 1098.32506
[10] Ceresole, A., D’Auria, R., Ferrara, S., Lerche, W., Louis, J.: Picard-Fuchs Equations and Special Geometry. CERN-TH.6441/92, UCLA/92/TEP/8, CALT-8-1776, POLFIS-TH.8/92
[11] Deligne, P.: Letter to D. Morrison, 6 November 1991
[12] Dixon, L.: In Superstrings. Unified Theories and Cosmology 1987, G. Furlan et al., eds., Singapore: World Scientific, 1988
[13] Ferrara, S., Louis, J.: Phys. Lett.B 273, 246 (1990)
[14] Font, A.: Periods and duality symmetries in Calabi-Yau compactifications. Nucl. Phys.B 389, 153 (1993) · Zbl 1360.32009
[15] Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Hypergeometric functions and toric manifolds. Funct. Anal. Appl.28:2, 94–106 (1989) · Zbl 0721.33006
[16] Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants of polynomials in several variables and triangulations of Newton polytopes. Algebra i analiz (Leningrad Math. J.)2, 1–62 (1990)
[17] Greene, B.R., Plesser, M.R.: Duality in Calabi-Yau moduli space. Nucl. Phys.B 338, 15–37 (1990)
[18] Greene, B.R., Morrison, D.R., Plesser, M.R.: Mirror manifolds in higher dimension. In preparation · Zbl 0923.32022
[19] Katz, S.: Rational curves on Calabi-Yau 3-folds. In: Essays on Mirror Manifolds (Ed. S.-T. Yau), Hong Kong: Int. Press Co., 1992 pp. 168–180 · Zbl 0835.14015
[20] Katz, S.: Rational curves on Calabi-Yau manifolds: Verifying of predictions of Mirror Symmetry. Preprint, alg-geom/9301006 (1993)
[21] Klemm, A., Theisen, S.: Consideration of One Modulus Calabi-Yau Compactification: Picard-Fuchs Equation, Kähler Potentials and Mirror Maps. Nucl. Phys.B 389, 753 (1993)
[22] Klemm, A., Theisen, S.: Mirror Maps and Instanton Sums for Intersections in Weighed Projective Space. LMU-TPW 93-08, Preprint April 1993 · Zbl 1020.32507
[23] Klemm, A., Schimmrigk, R.: Landau-Ginzburg String Vacua. CERN-TH-6459
[24] Kim, J.K., Park, C.J., Yoon, Y.: Calabi-Yau manifolds from complete intersections in products of weighted projective spaces. Phys. Lett.B 224, 108–114 (1989)
[25] Kreuzer, M., Schimmrigk, R., Skarke, H.: Abelian Landau-Ginzburg Orbifolds and Mirror Symmetry. Nucl. Hys.B 372, 61–86 (1992)
[26] Kreuzer, M., Skarke, H.: No Mirror Symmetry in Landau-Ginzburg Spectra!. CERN-TH 6461/92, to appear in Nucl. Phys. B
[27] Lerche, W., Vafa, C., Warner, N.: Nucl. Phys.B 324, 427 (1989)
[28] Libgober, A., Teitelbaum, J.: Lines on Calabi-Yau complete intersections, mirror symmetry, and Picard-Fuchs equations. Duke Math. J., Int. Math. Res Notices129, (1993) · Zbl 0789.14005
[29] Lynker, M., Schimmrigk, R.: Phys. Lett.249, 237 (1990)
[30] Manin, Yu.I.: Private communication
[31] Mathai, A.M., Saxens, R.K.: Generalized Hypergeometric Functions with Applications in Statistic and Physical Sciences. Lect. Notes Math348, (1973)
[32] Morrison, D.: Mirror symmetry and rational curves on quintic threefolds: A guide for mathematicians. J. Arn. Math. Soc.6, 223–247 (1993) · Zbl 0843.14005
[33] Morrison, D.: Picard-Fuchs equations and mirror maps for hypersurfaces. In: Essay on Mirror Manifolds. Ed. Yau, S.-T., Hong Kong: Int. Press. Co., 1992, pp. 241–264 · Zbl 0841.32013
[34] Morrison, D.: Compactifications of moduli spaces inspired by mirror symmetry. Preprint (1993) · Zbl 0824.14007
[35] Morrison, D.: Hodge-theoretic aspects of mirror symmetry. In preparation
[36] Roan, S.-S.: The mirror of Calabi-Yau Orbifold. Internat. J. Math.2, 439–455 (1991) · Zbl 0817.14018
[37] Roan, S.-S.: Topological Properties of Calabi-Yau Mirror Manifolds. Preprint (1992) · Zbl 0852.14013
[38] Schimmrigk, R.: A new construction of a three-generation Calabi-Yau manifold. Phys. Lett.B 193, 175–180 (1987)
[39] Slater, L.J.: Generalized Hypergeometric Functions. Cambridge: Cambridge University Press, XIII, 1966 · Zbl 0135.28101
[40] Stienstra, J., Beukers, F.: On the Picard-Fuchs Equation and the Formal Brauer Group of Cer tain Elliptic K3-surfaces. Math. Ann.271, 269–304 (1985) · Zbl 0555.14006
[41] Witten, E.: Topological sigma models. Commun. Math. Phys.118, 411–449 (1988) · Zbl 0674.58047
[42] Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surveys in Diff. Geom.1, 243–310 (1991) · Zbl 0757.53049
[43] Witten, E.: Mirror Manifolds and Topological Field Theory. In: Essays on Mirror Manifolds, Ed. S.-T. Yau, Hong Kong: Int. Press Co., 1992, pp. 120–180 · Zbl 0834.58013
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.