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A bstractIt is now well known that the moduli space of a vector bundle for heterotic string compactifications to four dimensions is parameterized by a set of sections of a weighted projective space bundle of a particular kind, known as Looijenga’s weighted projective space bundle. We show that the requisite weighted projective spaces and the Weierstrass equations describing the spectral covers for gauge groups EN (N = 4, · · · , 8) and SU(n + 1) (n = 1, 2, 3) can be obtained systematically by a series of blowing-up procedures according to Tate’s algorithm, thereby the sections of correct line bundles claimed to arise by Looijenga’s theorem can be automatically obtained. They are nothing but the four-dimensional analogue of the set of independent polynomials in the six-dimensional F-theory parameterizing the complex structure, which is further confirmed in the constructions of D4, A5, D6, E3 and SU(2) × SU(2) bundles. We also explain why we can obtain them in this way by using the structure theorem of the Mordell-Weil lattice, which is also useful for understanding the relation between the singularity and the occurrence of chiral matter in F-theory.

Looijenga’s weighted projective space, Tate’s algorithm and Mordell-Weil lattice in F-theory and heterotic string theory