Geometric transitions and mixed Hodge structures.

*(English)*Zbl 1139.81379From the text: We formulate and prove a \(\mathbf B\)-model disc level large \(N\) duality result for general conifold transitions between compact Calabi-Yau spaces using degenerations of Hodge structures.

Large \(N\) duality is a relation between open and closed string theories on two different Calabi-Yau manifolds connected by an extremal transition [R. Gopakumar and C. Vafa, Adv. Theor. Math. Phys. 3, No. 5, 1415–1443 (1999; Zbl 0972.81135), see also AMS/IP Stud. Adv. Math. 23, 45–63 (2001; Zbl 1026.81029)]. This relation was originally formulated in the context of topological \(\mathbf A\)-model for a local conifold transition [loc. cit.] and it was extended to more general noncompact toric Calabi-Yau manifolds in [M. Aganagic et al., D.-E. Diaconescu et al.].

This paper is part of a long term project aimed at understanding large \(N\) duality for extremal transitions between compact Calabi-Yau manifolds. A first step in this direction has been made in [D. Diaconescu and B. Florea, Adv. Theor. Math. Phys. 9, No. 1, 31–128 (2005; Zbl 1169.14322)] for topological \(\mathbf A\)-models. Here we will be concerned with large \(N\) duality in the topological \(\mathbf B\)-model. Open-closed duality for topological \(\mathbf B\)-strings was first developed in [R. Dijkgraaf and C. Vafa, Nucl. Phys., B 644, No. 1–2, 3–20 (2002; Zbl 0999.81068), ibid. 644, No. 1–2, 21–39 (2002; Zbl 0999.81069)] for a special class of noncompact toric Calabi-Yau manifolds employing a remarkable relation between holomorphic Chern-Simons theory and random matrix models.

In contrast with the \(\mathbf A\)-model, the topological \(\mathbf B\)-model on compact Calabi-Yau spaces has not been given so far a rigorous mathematical description. However it is well known that the genus zero topological closed string amplitudes can be expressed in Hodge theoretic terms using the formalism of special geometry. On the other hand disc level topological open string amplitudes associated to D-branes wrapping curves in Calabi-Yau threefolds can also be given a geometric interpretation in terms of Abel-Jacobi maps. Higher genus amplitudes do not have a pure geometric interpretation. In principle one would have to quantize Kodaira-Spencer theory coupled to holomorphic Chern-Simons theory on a compact Calabi-Yau space, which is a very hard task at best.

In this paper we will formulate and prove a first order \(\mathbf B\)-model duality statement for general conifold transitions between compact Calabi-Yau spaces. By first order duality we mean a correspondence between topological disc amplitudes on the open string side and first order terms in a suitable expansion of the holomorphic prepotential on the closed string side. The expansion is taken around an appropriate stratum parameterizing nodal Calabi-Yau spaces that admit a projective crepant resolution.

Using special geometry, in section two we show that the first order terms in this expansion admit a intrinsic geometric interpretation in terms of degenerations of Hodge structures. In section three we will show that the first order duality statement follows from a Hodge theoretic result relating two different mixed Hodge structures. The main element in the proof is the Clemens-Schmid exact sequence.

A connection between mixed Hodge structures and \(\mathbf B\)-model topological disc amplitudes on toric Calabi-Yau manifolds has been previously developed in [W. Lerche, P. Mayr and N. Warner, Holomorphic \(N=1\) special geometry of open-closed type II strings, 2002, http://arxiv.org/abs/hep-th/0207259, \(N=1\) special geometry, mixed Hodge variations and toric geometry, 2002, http://arxiv.org/abs/hep-th/0208039, and B. Forbes, Open string mirror maps from Picard-Fuchs equations on relative cohomology, 2003, http://arxiv.org/abs/hep-th/0307167]. This machinery has been applied to first order large \(N\) duality for toric Calabi-Yau manifolds in [B. Forbes, Mod. Phys. Lett. A 20, No. 35, 2685–2697 (2005; Zbl 1075.81051), see also http://arxiv.org/abs/hep-th/0408167]. Our approach is different and can be used to extend the B-model large \(N\) duality beyond disc level. Some progress along these lines for an interesting class of noncompact transitions is reported in the companion paper [D.-E. Diaconescu et al., Nucl. Phys., B 752, No. 3, 329–390 (2006; Zbl 1215.14048)].

Large \(N\) duality is a relation between open and closed string theories on two different Calabi-Yau manifolds connected by an extremal transition [R. Gopakumar and C. Vafa, Adv. Theor. Math. Phys. 3, No. 5, 1415–1443 (1999; Zbl 0972.81135), see also AMS/IP Stud. Adv. Math. 23, 45–63 (2001; Zbl 1026.81029)]. This relation was originally formulated in the context of topological \(\mathbf A\)-model for a local conifold transition [loc. cit.] and it was extended to more general noncompact toric Calabi-Yau manifolds in [M. Aganagic et al., D.-E. Diaconescu et al.].

This paper is part of a long term project aimed at understanding large \(N\) duality for extremal transitions between compact Calabi-Yau manifolds. A first step in this direction has been made in [D. Diaconescu and B. Florea, Adv. Theor. Math. Phys. 9, No. 1, 31–128 (2005; Zbl 1169.14322)] for topological \(\mathbf A\)-models. Here we will be concerned with large \(N\) duality in the topological \(\mathbf B\)-model. Open-closed duality for topological \(\mathbf B\)-strings was first developed in [R. Dijkgraaf and C. Vafa, Nucl. Phys., B 644, No. 1–2, 3–20 (2002; Zbl 0999.81068), ibid. 644, No. 1–2, 21–39 (2002; Zbl 0999.81069)] for a special class of noncompact toric Calabi-Yau manifolds employing a remarkable relation between holomorphic Chern-Simons theory and random matrix models.

In contrast with the \(\mathbf A\)-model, the topological \(\mathbf B\)-model on compact Calabi-Yau spaces has not been given so far a rigorous mathematical description. However it is well known that the genus zero topological closed string amplitudes can be expressed in Hodge theoretic terms using the formalism of special geometry. On the other hand disc level topological open string amplitudes associated to D-branes wrapping curves in Calabi-Yau threefolds can also be given a geometric interpretation in terms of Abel-Jacobi maps. Higher genus amplitudes do not have a pure geometric interpretation. In principle one would have to quantize Kodaira-Spencer theory coupled to holomorphic Chern-Simons theory on a compact Calabi-Yau space, which is a very hard task at best.

In this paper we will formulate and prove a first order \(\mathbf B\)-model duality statement for general conifold transitions between compact Calabi-Yau spaces. By first order duality we mean a correspondence between topological disc amplitudes on the open string side and first order terms in a suitable expansion of the holomorphic prepotential on the closed string side. The expansion is taken around an appropriate stratum parameterizing nodal Calabi-Yau spaces that admit a projective crepant resolution.

Using special geometry, in section two we show that the first order terms in this expansion admit a intrinsic geometric interpretation in terms of degenerations of Hodge structures. In section three we will show that the first order duality statement follows from a Hodge theoretic result relating two different mixed Hodge structures. The main element in the proof is the Clemens-Schmid exact sequence.

A connection between mixed Hodge structures and \(\mathbf B\)-model topological disc amplitudes on toric Calabi-Yau manifolds has been previously developed in [W. Lerche, P. Mayr and N. Warner, Holomorphic \(N=1\) special geometry of open-closed type II strings, 2002, http://arxiv.org/abs/hep-th/0207259, \(N=1\) special geometry, mixed Hodge variations and toric geometry, 2002, http://arxiv.org/abs/hep-th/0208039, and B. Forbes, Open string mirror maps from Picard-Fuchs equations on relative cohomology, 2003, http://arxiv.org/abs/hep-th/0307167]. This machinery has been applied to first order large \(N\) duality for toric Calabi-Yau manifolds in [B. Forbes, Mod. Phys. Lett. A 20, No. 35, 2685–2697 (2005; Zbl 1075.81051), see also http://arxiv.org/abs/hep-th/0408167]. Our approach is different and can be used to extend the B-model large \(N\) duality beyond disc level. Some progress along these lines for an interesting class of noncompact transitions is reported in the companion paper [D.-E. Diaconescu et al., Nucl. Phys., B 752, No. 3, 329–390 (2006; Zbl 1215.14048)].

##### MSC:

81T45 | Topological field theories in quantum mechanics |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

32G20 | Period matrices, variation of Hodge structure; degenerations |

32Q25 | Calabi-Yau theory (complex-analytic aspects) |