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On the work of Givental’ relative to mirror symmetry. (English) Zbl 0944.14016

Appunti dei Corsi Tenuti da Docenti della Scuola. Pisa: Scuola Normale Superiore, ii, 90 p. (1998).
This work contains the notes of two seminars held in 1997 at the Università di Roma “La Sapienza” and at the Scuola Normale Superiore in Pisa. The aim of the seminars was to give a detailed analysis of the Givental’ paper [A. B. Givental’, Int. Math. Res. Not. 1996, No. 13, 613-663 (1996; Zbl 0881.55006)] about the mirror symmetry for projective complete intersections. Givental’s goal was to formulate in a proper way and prove the link between solutions of the Picard-Fuchs equation and numbers of rational curves on a complex projective algebraic variety \(X\). The numbers counting rational curves on \(X\), i.e. the Gromov-Witten invariants of \(X\), are all contained in a function called potential. Then, a family \(\{\nabla_h(s)\}\) of connections on the tangent bundle \(TH^*(X)\) to the cohomology of \(X\) is constructed. The structure constants of the connections are the third derivatives of the potential. A basis \(\{s_i\}\) of solutions of \(\nabla_h(s)=0\), suitably “manipulated”, gives a basis of solutions for the Picard-Fuchs equation.
The method implies a one parameter family of Calabi-Yau threefolds which are mirrors of the quintic \(X\). This means that from solutions of Picard-Fuchs equation of the mirror family, one can recover the numbers of rational curves of \(X\).
Reviewer: G.Zet (Iaşi)

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32Q25 Calabi-Yau theory (complex-analytic aspects)
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
81T70 Quantization in field theory; cohomological methods
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)

Citations:

Zbl 0881.55006
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