Divisors on elliptic Calabi-Yau 4-folds and the superpotential in F-theory, I

JOURNAL OF GEOMETRY AND PHYSICS

卷:28

Article

Grassi, A

关键词: string theory;    F-theory;    superpotential;    birational contractions;   

DOI  :  10.1016/S0393-0440(98)00004-7

来源: Elsevier

【 摘 要 】

Each smooth elliptic Calabi-Yau 4-fold determines both a three-dimensional physical theory (a compactification of M-theory) and a four-dimensional physical theory (using the F-theory construction). A key issue in both theories is the calculation of the superpotential of the theory, which by a result of Witten is determined by the divisors D on the 4-fold satisfying chi(O-D) = 1. Fire propose a systematic approach to identify these divisors, and derive some criteria to determine whether a given divisor indeed contributes. We then apply our techniques in explicit examples, in particular, when the base B of the elliptic fibration is a toric variety or a Fano 3-fold. When B is Fano, we show how divisors contributing to the superpotential are always exceptional (in some sense) for the Calabi-Yau 4-fold X. This naturally leads to certain transitions of X, i.e., birational transformations to a singular model (where the image of D no longer contributes) as well as certain smoothings of the singular model. The singularities which occur are canonical, the same type of singularities of a (singular) Weierstrass model. We work out the transitions. If a smoothing exists, then the Hedge numbers change. We speculate that divisors contributing to the superpotential are always exceptional (in some sense) for X, also in M-theory. In fact we show that this is a consequence of the (log)-minimal model algorithm in dimension 4, which is still conjectural in its generality, but it has been worked out in various cases, among which are toric varieties. (C) 1998 Elsevier Science B.V.

【 授权许可】

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