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Picard-Fuchs equations, special geometry and target space duality. (English) Zbl 0927.14020

Greene, B. (ed.) et al., Mirror symmetry II. Cambridge, MA: International Press, AMS/IP Stud. Adv. Math. 1, 281-353 (1997).
The holomorphic differential equations of special geometry are studied. Special geometry refers to a Kähler-Hodge manifold for which there exists a set of \(n+1\) sections \(L^A(z,\overline b)\) and \(\overline L^A(z, \overline b)\) of weight \((1,-1)\) and \((-1,1)\) respectively, satisfying some constraints expressed by Picard-Fuchs equations. The system of holomorphic differential identities implied by special Kählerian geometry of four-dimensional \(N=2\) supergravity is reviewed and the Picard-Fuchs equations for an arbitrary number of moduli are derived. The special case of superstring compactifications on Calabi-Yau manifolds is considered as an application. It is shown that the monodromy group of Picard-Fuchs equations is closely related to the target space duality symmetry group. Examples with one and two moduli are also considered.
For the entire collection see [Zbl 0905.00079].
Reviewer: G.Zet (Iaşi)

MSC:

14J32 Calabi-Yau manifolds (algebro-geometric aspects)
83E50 Supergravity
83E30 String and superstring theories in gravitational theory
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